CHAPTER 11 PROBLEM 11.CQ1

CHAPTER 11

PROBLEM 11.CQ1 A bus travels the 100 miles between A and B at 50 mi/h and then another 100 miles between B and C at 70 mi/h. The average speed of the bus for the entire 200-mile trip is: (a) more than 60 mi/h (b) equal to 60 mi/h (c) less than 60 mi/h

SOLUTION The time required for the bus to travel from A to B is 2 h and from B to C is 100/70 = 1.43 h, so the total time is 3.43 h and the average speed is 200/3.43 = 58 mph. Answer: (c) 

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PROBLEM 11CQ2 Two cars A and B race each other down a straight road. The position of each car as a function of time is shown. Which of the following statements are true (more than one answer can be correct)? (a) At time t2 both cars have traveled the same distance (b) At time t1 both cars have the same speed (c) Both cars have the same speed at some time t < t1 (d) Both cars have the same acceleration at some time t < t1 (e) Both cars have the same acceleration at some time t1 < t < t2

SOLUTION The speed is the slope of the curve, so answer c) is true. The acceleration is the second derivative of the position. Since A’s position increases linearly the second derivative will always be zero. The second derivative of curve B is zero at the pont of inflection which occurs between t1 and t2. Answers: (c) and (e) 


PROBLEM 11.1 The motion of a particle is defined by the relation x = t 4 − 10t 2 + 8t + 12 , where x and t are expressed in inches and seconds, respectively. Determine the position, the velocity, and the acceleration of the particle when t = 1 s.

SOLUTION x = t 4 − 10t 2 + 8t + 12

At t = 1 s,

v=

dx = 4t 3 − 20t + 8 dt

a=

dv = 12t 2 − 20 dt

x = 1 − 10 + 8 + 12 = 11

x = 11.00 in. 

v = 4 − 20 + 8 = −8

v = −8.00 in./s 

a = 12 − 20 = −8 

a = −8.00 in./s 2 


PROBLEM 11.2 The motion of a particle is defined by the relation x = 2t 3 − 9t 2 + 12t + 10, where x and t are expressed in feet and seconds, respectively. Determine the time, the position, and the acceleration of the particle when v = 0.

SOLUTION x = 2t 3 − 9t 2 + 12t + 10

Differentiating,

v=

a=

dx = 6t 2 − 18t + 12 = 6(t 2 − 3t + 2) dt = 6(t − 2)(t − 1) dv = 12t − 18 dt

So v = 0 at t = 1 s and t = 2 s. At t = 1 s,

x1 = 2 − 9 + 12 + 10 = 15 a1 = 12 − 18 = −6

t = 1.000 s  x1 = 15.00 ft  a1 = −6.00 ft/s 2 

At t = 2 s, x2 = 2(2)3 − 9(2) 2 + 12(2) + 10 = 14

t = 2.00 s  x2 = 14.00 ft 

a2 = (12)(2) − 18 = 6

a2 = 6.00 ft/s 2 


PROBLEM 11.3 The vertical motion of mass A is defined by the relation x = 10 sin 2t + 15cos 2t + 100, where x and t are expressed in mm and seconds, respectively. Determine (a) the position, velocity and acceleration of A when t = 1 s, (b) the maximum velocity and acceleration of A.

SOLUTION x = 10sin 2t + 15cos 2t + 100 v=

dx = 20 cos 2t − 30sin 2t dt

a=

dv = −40sin 2t − 60 cos 2t dt

For trigonometric functions set calculator to radians: (a) At t = 1 s.

x1 = 10sin 2 + 15cos 2 + 100 = 102.9 v1 = 20cos 2 − 30sin 2 = −35.6 a1 = −40sin 2 − 60 cos 2 = −11.40

x1 = 102.9 mm  v1 = −35.6 mm/s  a1 = −11.40 mm/s 2 

(b) Maximum velocity occurs when a = 0. −40sin 2t − 60cos 2t = 0 tan 2t = −

60 = −1.5 40

2t = tan −1 (−1.5) = −0.9828 and −0.9828 + π

Reject the negative value. 2t = 2.1588 t = 1.0794 s t = 1.0794 s for vmax

so

vmax = 20cos(2.1588) − 30sin(2.1588) vmax = −36.1 mm/s 

= −36.056

Note that we could have also used vmax = 202 + 302 = 36.056

by combining the sine and cosine terms. For amax we can take the derivative and set equal to zero or just combine the sine and cosine terms. amax = 402 + 602 = 72.1 mm/s 2

amax = 72.1 mm/s 2 


PROBLEM 11.4 A loaded railroad car is rolling at a constant velocity when it couples with a spring and dashpot bumper system. After the coupling, the motion of the car is defined by the relation x = 60e−4.8t sin16t where x and t are expressed in mm and seconds, respectively. Determine the position, the velocity and the acceleration of the railroad car when (a) t = 0, (b) t = 0.3 s.

SOLUTION x = 60e−4.8t sin16t dx = 60(−4.8)e −4.8t sin16t + 60(16)e−4.8t cos16t dt v = −288e−4.8t sin16t + 960e −4.8t cos16t v=

a=

dv = 1382.4e−4.8t sin16t − 4608e−4.8t cos16t dt − 4608e−4.8t cos16t − 15360e−4.8t sin16t

a = −13977.6e −4.8t sin16t − 9216e−4.8 cos16t

(a) At t = 0,

x0 = 0

x0 = 0 mm 

v0 = 960 mm/s

v0 = 960 mm/s

a0 = −9216 mm/s 2

(b) At t = 0.3 s,

a0 = 9220 mm/s 2

 

e−4.8t = e −1.44 = 0.23692 sin16t = sin 4.8 = −0.99616 cos16t = cos 4.8 = 0.08750 x0.3 = (60)(0.23692)(−0.99616) = −14.16

x0.3 = 14.16 mm



v0.3 = 87.9 mm/s



v0.3 = −(288)(0.23692)(−0.99616) + (960)(0.23692)(0.08750) = 87.9 a0.3 = −(13977.6)(0.23692)( −0.99616) − (9216)(0.23692)(0.08750) = 3108

a0.3 = 3110 mm/s 2 

or 3.11 m/s 2




PROBLEM 11.5 The motion of a particle is defined by the relation x = 6t 4 − 2t 3 − 12t 2 + 3t + 3, where x and t are expressed in meters and seconds, respectively. Determine the time, the position, and the velocity when a = 0.

SOLUTION We have

x = 6t 4 − 2t 3 − 12t 2 + 3t + 3

Then

v=

dx = 24t 3 − 6t 2 − 24t + 3 dt

and

a=

dv = 72t 2 − 12t − 24 dt

When a = 0:

72t 2 − 12t − 24 = 12(6t 2 − t − 2) = 0 (3t − 2)(2t + 1) = 0

or t=

or At t =

2 s: 3

2 1 s and t = − s (Reject) 3 2 4

3

2

2 2 2 2 x2/3 = 6   − 2   − 12   + 3   + 3 3 3 3 3 3

t = 0.667 s  or

x2/3 = 0.259 m 

2

2 2 2 v2/3 = 24   − 6   − 24   + 3 3 3 3

or v2/3 = −8.56 m/s 


PROBLEM 11.6 The motion of a particle is defined by the relation x = t 3 − 9t 2 + 24t − 8, where x and t are expressed in inches and seconds, respectively. Determine (a) when the velocity is zero, (b) the position and the total distance traveled when the acceleration is zero.

SOLUTION We have

x = t 3 − 9t 2 + 24t − 8

Then

v=

dx = 3t 2 − 18t + 24 dt

and

a=

dv = 6 t − 18 dt

(a)

When v = 0:

3 t 2 − 18t + 24 = 3(t 2 − 6t + 8) = 0 (t − 2)(t − 4) = 0 t = 2.00 s and t = 4.00 s 

(b)

When a = 0:

6t − 18 = 0 or t = 3 s x3 = (3)3 − 9(3)2 + 24(3) − 8

At t = 3 s: First observe that 0 ≤ t < 2 s:

v>0

2 s < t ≤ 3 s:

v<0

or

x3 = 10.00 in. 

Now At t = 0:

x0 = −8 in.

At t = 2 s:

x2 = (2)3 − 9(2) 2 + 24(2) − 8 = 12 in.

Then

x2 − x0 = 12 − (−8) = 20 in. | x3 − x2 | = |10 − 12| = 2 in.

Total distance traveled = (20 + 2) in.

Total distance = 22.0 in. 


PROBLEM 11.7 The motion of a particle is defined by the relation x = 2t 3 − 15t 2 + 24t + 4, where x is expressed in meters and t in seconds. Determine (a) when the velocity is zero, (b) the position and the total distance traveled when the acceleration is zero.

SOLUTION x = 2t 3 − 15t 2 + 24t + 4 dx = 6t 2 − 30t + 24 v= dt dv a= = 12t − 30 dt

(a)

v = 0 when

6t 2 − 30t + 24 = 0 6(t − 1)(t − 4) = 0

(b)

a = 0 when

t = 1.000 s and t = 4.00 s 

12t − 30 = 0

t = 2.5 s

x2.5 = 2(2.5)3 − 15(2.5) 2 + 24(2.5) + 4

For t = 2.5 s:

x2.5 = +1.500 m 

To find total distance traveled, we note that x1 = 2(1)3 − 15(1)2 + 24(1) + 4

v = 0 when t = 1 s:

x1 = +15 m

For t = 0,

x0 = +4 m

Distance traveled From t = 0 to t = 1 s: From t = 1 s to t = 2.5 s:

x1 − x0 = 15 − 4 = 11 m x2.5 − x1 = 1.5 − 15 = 13.5 m

Total distance traveled = 11 m + 13.5 m

Total distance = 24.5 m 


PROBLEM 11.8 The motion of a particle is defined by the relation x = t 3 − 6t 2 − 36t − 40, where x and t are expressed in feet and seconds, respectively. Determine (a) when the velocity is zero, (b) the velocity, the acceleration, and the total distance traveled when x = 0.

SOLUTION We have

x = t 3 − 6t 2 − 36t − 40

Then

v=

dx = 3t 2 − 12t − 36 dt

and

a=

dv = 6t − 12 dt

(a)

When v = 0:

3t 2 − 12t − 36 = 3(t 2 − 4t − 12) = 0 (t + 2)(t − 6) = 0

or

t = −2 s (Reject) and t = 6 s

or (b)

When x = 0:

t = 6.00 s 

t 3 − 6t 2 − 36t − 40 = 0

Factoring

(t − 10)(t + 2)(t + 2) = 0 or t = 10 s

Now observe that

0 ≤ t < 6 s:

v<0

6 s < t ≤ 10 s:

v>0

and at t = 0: t = 6 s:

x0 = −40 ft x6 = (6)3 − 6(6)2 − 36(6) − 40 = −256 ft

t = 10 s:

Then

v10 = 3(10) 2 − 12(10) − 36

or v10 = 144.0 ft/s 

a10 = 6(10) − 12

or

a10 = 48.0 ft/s 2 

| x6 − x0 | = | − 256 − (−40)| = 216 ft x10 − x6 = 0 − (−256) = 256 ft

Total distance traveled = (216 + 256) ft

Total distance = 472 ft 


PROBLEM 11.9 The brakes of a car are applied, causing it to slow down at a rate of 10 m/s2. Knowing that the car stops in 100 m, determine (a) how fast the car was traveling immediately before the brakes were applied, (b) the time required for the car to stop.

SOLUTION a = −10 ft/s 2

(a)

Velocity at x = 0. v



0 v0

0−

(b)

dv = a = −10 dx vdv = −



xf

0

(−10) dx

v02 = −10 x f = −(10)(300) 2 v02 = 6000

v0 = 77.5 ft/s 2 

Time to stop. dv = a = −10 dx



0 v0

dv = −



tf

−10dt

0

0 − v0 = −10t f tf =

v0 77.5 = 10 10

t f = 7.75 s 


PROBLEM 11.10 The acceleration of a particle is directly proportional to the time t. At t = 0, the velocity of the particle is v = 16 in./s. Knowing that v = 15 in./s and that x = 20 in. when t = 1 s, determine the velocity, the position, and the total distance traveled when t = 7 s.

SOLUTION a = kt

We have

dv = a = kt dt

Now v

At t = 0, v = 16 in./s:



or

v − 16 =

dv =

16

0

kt dt

1 2 kt 2 1 2 kt (in./s) 2

15 in./s = 16 in./s + k = −2 in./s3

or

1 k (1 s) 2 2 and v = 16 − t 2

dx = v = 16 − t 2 dt

Also At t = 1 s, x = 20 in.:



t

v = 16 +

or At t = 1 s, v = 15 in./s:

k = constant



x

20

dx =



t

1

(16 − t 2 ) dt t

or or

1   x − 20 = 16t − t 3  3 1  1 13 x = − t 3 + 16t + (in.) 3 3

Then At t = 7 s:

When v = 0:

v7 = 16 − (7) 2

or v7 = −33.0 in./s 

1 13 x7 = − (7)3 + 16(7) + 3 3

or

x7 = 2.00 in. 

16 − t 2 = 0 or t = 4 s


PROBLEM 11.10 (Continued)

At t = 0:

x0 =

13 3

1 13 x4 = − (4)3 + 16(4) + = 47 in. 3 3

t = 4 s:

Now observe that

Then

0 ≤ t < 4 s:

v>0

4 s < t ≤ 7 s:

v<0

13 = 42.67 in. 3 | x7 − x4 | = |2 − 47| = 45 in. x4 − x0 = 47 −

Total distance traveled = (42.67 + 45) in.

Total distance = 87.7 in. 


PROBLEM 11.11 The acceleration of a particle is directly proportional to the square of the time t. When t = 0, the particle is at x = 24 m. Knowing that at t = 6 s, x = 96 m and v = 18 m/s, express x and v in terms of t.

SOLUTION a = kt 2

We have

k = constant

dv = a = kt 2 dt

Now v



or

1 v − 18 = k (t 3 − 216) 3

18

dv =



t

At t = 6 s, v = 18 m/s:

6

kt 2 dt

1 v = 18 + k (t 3 − 216)(m/s) 3

or

dx 1 = v = 18 + k (t 3 − 216) dt 3

Also

x

1  18 + k (t 3 − 216)  dt 3 



or

1 1  x − 24 = 18t + k  t 4 − 216t  3 4 

24

dx =



t

At t = 0, x = 24 m:

0 

Now At t = 6 s, x = 96 m: or Then or and or

1 1  96 − 24 = 18(6) + k  (6) 4 − 216(6)  3 4  k=

1 m/s 4 9

1  1  1  x − 24 = 18t +   t 4 − 216t  3  9  4  x(t ) =

1 4 t + 10t + 24 108



11 v = 18 +   (t 3 − 216) 3 9 v(t ) =

1 3 t + 10 27




PROBLEM 11.12 The acceleration of a particle is defined by the relation a = kt 2 . (a) Knowing that v = −8 m/s when t = 0 and that v = +8 m/s when t = 2 s, determine the constant k. (b) Write the equations of motion, knowing also that x = 0 when t = 2 s.

SOLUTION a = kt 2 dv = a = kt 2 dt

(1)

t = 0, v = −8 m/s and t = 2 s, v = +8 ft/s



(a)

8 −8

dv =



2 0

kt 2 dt

1 8 − ( −8) = k (2)3 3

(b)

k = 6.00 m/s 4 

Substituting k = 6 m/s 4 into (1) dv = a = 6t 2 dt

t = 0, v = −8 m/s:



v −8

dv =



t

0

a = 6t 2 

6t 2 dt

1 v − (−8) = 6(t )3 3

v = 2t 3 − 8 

dx = v = 2t 3 − 8 dt t = 2 s, x = 0:



x

0

dx =



t

2

(2t 3 − 8) dt ; x =

1 4 t − 8t 2

1  1  x =  t 4 − 8t  −  (2) 4 − 8(2)  2  2  1 x = t 4 − 8t − 8 + 16 2

t

2

x=

1 4 t − 8t + 8  2


PROBLEM 11.13 The acceleration of Point A is defined by the relation a = −1.8sin kt , where a and t are expressed in m/s 2 and seconds, respectively, and k = 3 rad/s. Knowing that x = 0 and v = 0.6 m/s when t = 0, determine the velocity and position of Point A when t = 0.5 s.

SOLUTION Given:

a = −1.8sin kt m/s 2 ,

v0 = 0.6 m/s, x0 = 0,

t

t

v − v0 =  0 a dt = −1.8  0 sin kt dt = v − 0.6 =

t

1.8 cos kt k

0

1.8 (cos kt − 1) = 0.6cos kt − 0.6 3

v = 0.6cos kt m/s

Velocity: t

t

x − x0 =  0 v dt = 0.6  0 cos kt dt = x−0=

0.6 sin kt k

t 0

0.6 (sin kt − 0) = 0.2sin kt 3

x = 0.2sin kt m

Position: When t = 0.5 s,

k = 3 rad/s

kt = (3)(0.5) = 1.5 rad v = 0.6cos1.5 = 0.0424 m/s

v = 42.4 mm/s 

x = 0.2sin1.5 = 0.1995 m

x = 199.5 mm 


PROBLEM 11.14 The acceleration of Point A is defined by the relation a = −1.08sin kt − 1.44cos kt , where a and t are expressed in m/s2 and seconds, respectively, and k = 3 rad/s. Knowing that x = 0.16 m and v = 0.36 m/s when t = 0, determine the velocity and position of Point A when t = 0.5 s.

SOLUTION Given:

a = −1.08sin kt − 1.44 cos kt m/s 2 , x0 = 0.16 m,

k = 3 rad/s

v0 = 0.36 m/s

t

t

t

v − v0 =  0 a dt = −1.08  0 sin kt dt − 1.44  0 cos kt dt v − 0.36 = =

1.08 cos kt k

t 0

−

1.44 sin kt k

t 0

1.08 1.44 (cos kt − 1) − (sin kt − 0) 3 3

= 0.36 cos kt − 0.36 − 0.48sin kt Velocity:

v = 0.36 cos kt − 0.48sin kt m/s t

t

t

x − x0 =  0 v dt = 0.36  0 cos kt dt − 0.48  0 sin kt dt x − 0.16 =

0.36 sin kt k

t 0

+

0.48 cos kt k

t 0

0.36 0.48 (sin kt − 0) + (cos kt − 1) 3 3 = 0.12sin kt + 0.16cos kt − 0.16

=

Position: When t = 0.5 s,

x = 0.12sin kt + 0.16cos kt m kt = (3)(0.5) = 1.5 rad

v = 0.36 cos1.5 − 0.48sin1.5 = −0.453 m/s

v = −453 mm/s 

x = 0.12sin1.5 + 0.16 cos1.5 = 0.1310 m

x = 131.0 mm 


PROBLEM 11.15 A piece of electronic equipment that is surrounded by packing material is dropped so that it hits the ground with a speed of 4 m/s. After contact the equipment experiences an acceleration of a = − kx, where k is a constant and x is the compression of the packing material. If the packing material experiences a maximum compression of 20 mm, determine the maximum acceleration of the equipment.

SOLUTION a=

vdv = −k x dx

Separate and integrate.



vf v0

vdv = −



xf

0

k x dx

1 2 1 2 1 v f − v0 = − kx 2 2 2 2

xf

0

1 = − k x 2f 2

Use v0 = 4 m/s, x f = 0.02 m, and v f = 0. Solve for k. 1 1 0 − (4) 2 = − k (0.02) 2 2 2

k = 40,000 s −2

Maximum acceleration. amax = −kxmax : ( −40,000)(0.02) = −800 m/s 2 a = 800 m/s 2




PROBLEM 11.16 A projectile enters a resisting medium at x = 0 with an initial velocity v0 = 900 ft/s and travels 4 in. before coming to rest. Assuming that the velocity of the projectile is defined by the relation v = v0 − kx, where v is expressed in ft/s and x is in feet, determine (a) the initial acceleration of the projectile, (b) the time required for the projectile to penetrate 3.9 in. into the resisting medium.

SOLUTION First note When x =

 4  0 = (900 ft/s) − k  ft   12 

4 ft, v = 0: 12

k = 2700

or (a)

1 s

We have

v = v0 − kx

Then

a=

or

a = −k (v0 − kx)

At t = 0:

1 a = 2700 (900 ft/s − 0) s

dv d = (v0 − kx) = −kv dt dt

a0 = −2.43 × 106 ft/s 2 

or (b)

dx = v = v0 − kx dt

We have At t = 0, x = 0: or



x

0

dx = v0 − kx

t

 dt 0

1 − [ln(v0 − kx)]0x = t k

or

t=

1  v0  1  1   ln   = ln  k  v0 − kx  k  1 − vk x  0  

When x = 3.9 in.:

t=

 1 ln  1 2700 s 1 −

1 2700 1/s 900 ft/s

(

3.9 12

 ft )  t = 1.366 × 10−3 s 

or


PROBLEM 11.17 The acceleration of a particle is defined by the relation a = −k/x. It has been experimentally determined that v = 15 ft/s when x = 0.6 ft and that v = 9 ft/s when x = 1.2 ft. Determine (a) the velocity of the particle when x = 1.5 ft, (b) the position of the particle at which its velocity is zero.

SOLUTION a=

vdv − k = dx x

Separate and integrate using x = 0.6 ft, v = 15 ft/s.



v

15

vdv = − k



x

0.6

v

dx x

1 2 v = − k ln x 2 15

x

0.6

1 2 1  x  v − (15) 2 = − k ln   2 2  0.6 

(1)

When v = 9 ft/s, x = 1.2 ft 1 2 1  1.2  (9) − (15) 2 = −k ln   2 2  0.6 

Solve for k. k = 103.874 ft 2 /s 2

(a)

Velocity when x = 65 ft. Substitute

k = 103.874 ft 2 /s 2

and

x = 1.5 ft into (1).

1 2 1  1.5  v − (15) 2 = −103.874 ln   2 2  0.6  v = 5.89 ft/s 

(b)

Position when for v = 0, 1  x  0 − (15) 2 = −103.874 ln   2  0.6   x  ln   = 1.083  0.6 





 x = 1.772 ft 


PROBLEM 11.18 A brass (nonmagnetic) block A and a steel magnet B are in equilibrium in a brass tube under the magnetic repelling force of another steel magnet C located at a distance x = 0.004 m from B. The force is inversely proportional to the square of the distance between B and C. If block A is suddenly removed, the acceleration of block B is a = −9.81 + k /x 2 , where a and x are expressed in m/s2 and m, respectively, and k = 4 × 10−4 m3 /s 2 . Determine the maximum velocity and acceleration of B.

SOLUTION The maximum veolocity occurs when a = 0. xm2 =

0 = −9.81 +

k 4 × 10−4 = = 40.775 × 10−6 m 2 9.81 9.81

k xm2

xm = 0.0063855 m

The acceleration is given as a function of x. v

dv k = a = −9.81 + 2 dx x

Separate variables and integrate: vdv = −9.81dx +



v 0



vdv = −9.81

x x0

k dx x2

dx + k



x x0

dx x2

1 1  1 2 v = −9.81( x − x0 ) − k  −  2  x x0   1 1 2 1  vm = −9.81( xm − x0 ) − k  −  2  xm x0  1 1   = −9.81(0.0063855 − 0.004) − (4 × 10−4 )  −  0.0063855 0.004   = −0.023402 + 0.037358 = 0.013956 m 2 /s2

Maximum velocity:

vm = 0.1671 m/s

vm = 167.1 mm/s 

The maximum acceleration occurs when x is smallest, that is, x = 0.004 m. am = −9.81 +

4 × 10−4 (0.004) 2

am = 15.19 m/s 2




PROBLEM 11.19 Based on experimental observations, the acceleration of a particle is defined by the relation a = −(0.1 + sin x/b), where a and x are expressed in m/s2 and meters, respectively. Knowing that b = 0.8 m and that v = 1 m/s when x = 0, determine (a) the velocity of the particle when x = −1 m, (b) the position where the velocity is maximum, (c) the maximum velocity.

SOLUTION We have When x = 0, v = 1 m/s:

v



v 1

dv x   = a = −  0.1 + sin dx 0.8  

vdv =



x   −  0.1 + sin dx 0  0.8  x

x

1 2 x   (v − 1) = − 0.1x − 0.8 cos 2 0.8  0 

or

x 1 2 v = −0.1x + 0.8 cos − 0.3 2 0.8

or (a)

When x = −1 m:

1 2 −1 v = −0.1(−1) + 0.8 cos − 0.3 2 0.8 v = ± 0.323 m/s 

or (b)

When v = vmax, a = 0: or

(c)

x   −  0.1 + sin =0 0.8   x = −0.080 134 m

x = −0.0801 m 

When x = −0.080 134 m: 1 2 −0.080 134 vmax = −0.1(−0.080 134) + 0.8 cos − 0.3 2 0.8 = 0.504 m 2 /s 2 vmax = 1.004 m/s 

or


PROBLEM 11.20 A spring AB is attached to a support at A and to a collar. The unstretched length of the spring is l. Knowing that the collar is released from rest at x = x0 and has an acceleration defined by the relation a = −100( x − lx / l 2 + x 2 ) , determine the velocity of the collar as it passes through Point C.

SOLUTION a=v

Since a is function of x,

 dv lx = −100  x −  dx l 2 + x2 

   

Separate variables and integrate:



vf v0

vdv = −100



 lx x− 2 x0  l + x2  0

 x2 1 2 1 2 − l l 2 + x2 v f − v0 = −100  2 2 2 

  

  dx   0

x0

 x2 1 2 v f − 0 = −100  − 0 − l 2 + l l 2 + x02  2 2 

  

1 2 100 2 (−l + x02 − l 2 − 2l l 2 + x02 ) vf = 2 2 100 ( l 2 + x02 − l )2 = 2 v f = 10( l 2 + x02 − l ) 


PROBLEM 11.21 The acceleration of a particle is defined by the relation a = −0.8v where a is expressed in m/s2 and v in m/s. Knowing that at t = 0 the velocity is 1 m/s, determine (a) the distance the particle will travel before coming to rest, (b) the time required for the particle’s velocity to be reduced by 50 percent of its initial value.

SOLUTION (a)

Determine relationship between x and v. a=

vdv = −0.8v dx

dv = −0.8dx

Separate and integrate with v = 1 m/s when x = 0.



v

1

dv = −0.8



x 0

dx

v − 1 = −0.8 x

Distance traveled. For v = 0, (b)

x=

−1  −0.8

x = 1.25 m 

Determine realtionship between v and t. a=



v

1

dv = 0.8v dt

dv =− v



x 0

0.8dt

v ln   = −0.8t 1

1 t = 1.25ln   v

For v = 0.5(1 m/s) = 0.5 m/s,  1  t = 1.25ln    0.5 

t = 0.866 s 


PROBLEM 11.22 Starting from x = 0 with no initial velocity, a particle is given an acceleration a = 0.1 v 2 + 16, where a and v are expressed in ft/s2 and ft/s, respectively. Determine (a) the position of the particle when v = 3ft/s, (b) the speed and acceleration of the particle when x = 4 ft.

SOLUTION a=

vdv = 0.1(v 2 + 16)1/ 2 dx

(1)

Separate and integrate.



v 0

vdv v 2 + 16

=



x 0

0.1 dx

v

(v 2 + 16)1/2 = 0.1 x 0

2

1/2

(v + 16)

(a)

(b)

− 4 = 0.1 x x = 10[(v 2 + 16)1/2 − 4]

(2)

x = 10[(32 + 16)1/2 − 4]

x = 10.00 ft 

v = 3 ft/s.

x = 4 ft.

From (2),

(v 2 + 16)1/2 = 4 + 0.1x = 4 + (0.1)(4) = 4.4 v 2 + 16 = 19.36 v 2 = 3.36 ft 2 /s 2

From (1),

a = 0.1(1.8332 + 16)1/2

v = 1.833 ft/s  a = 0.440 ft/s 2 


PROBLEM 11.23 A ball is dropped from a boat so that it strikes the surface of a lake with a speed of 16.5 ft/s. While in the water the ball experiences an acceleration of a = 10 − 0.8v, where a and v are expressed in ft/s2 and ft/s, respectively. Knowing the ball takes 3 s to reach the bottom of the lake, determine (a) the depth of the lake, (b) the speed of the ball when it hits the bottom of the lake.

SOLUTION a=

dv = 10 − 0.8v dt

Separate and integrate:



dv = v0 10 − 0.8v v



t 0

dt

v

−

1 ln(10 − 0.8v) = t 0.8 v0

 10 − 0.8v  ln   = −0.8t  10 − 0.8v0  10 − 0.8v = (10 − 0.8v0 )e −0.8t

or

0.8v = 10 − (10 − 0.8v0 )e−0.8t v = 12.5 − (12.5 − v0 )e −0.8t

With v0 = 16.5ft/s

v = 12.5 + 4e−0.8t



Integrate to determine x as a function of t. dx = 12.5 + 4e−0.8t v= dt



x 0

dx =



t 0

(12.5 + 4e−0.8t )dt

x = 12.5t − 5e −0.8t

(a)

t 0

= 12.5t − 5e−0.8t + 5

At t = 35 s, x = 12.5(3) − 5e −2.4 + 5 = 42.046 ft

(b)

x = 42.0 ft 

v = 12.5 + 4e −2.4 = 12.863 ft/s

v = 12.86 ft/s 


PROBLEM 11.24 The acceleration of a particle is defined by the relation a = − k v , where k is a constant. Knowing that x = 0 and v = 81 m/s at t = 0 and that v = 36 m/s when x = 18 m, determine (a) the velocity of the particle when x = 20 m, (b) the time required for the particle to come to rest.

SOLUTION (a)

We have

dv = a = −k v dx

v

v dv = − k dx

so that v



or

2 3/2 v [v ]81 = −kx 3

or

2 3/2 [v − 729] = −kx 3

When x = 18 m, v = 36 m/s:

v dv =

81



x

When x = 0, v = 81 m/s:

0

− k dx

2 (363/2 − 729) = − k (18) 3 k = 19 m/s 2

or Finally When x = 20 m:

2 3/2 (v − 729) = −19(20) 3 v3/2 = 159

or (b)

We have

dv = a = −19 v dt

At t = 0, v = 81 m/s:



v

dv

81

v

=



t 0

−19dt

or

v 2[ v ]81 = −19t

or

2( v − 9) = −19t

When v = 0:

v = 29.3 m/s 

2(−9) = −19t t = 0.947 s 

or


PROBLEM 11.25 A particle is projected to the right from the position x = 0 with an initial velocity of 9 m/s. If the acceleration of the particle is defined by the relation a = −0.6v3/ 2 , where a and v are expressed in m/s2 and m/s, respectively, determine (a) the distance the particle will have traveled when its velocity is 4 m/s, (b) the time when v = 1 m/s, (c) the time required for the particle to travel 6 m.

SOLUTION (a)

We have When x = 0, v = 9 m/s:

v



v 9

dv = a = −0.6v3/2 dx

v − (1/2) dv =



x 0

−0.6dx

or

2[v1/2 ]9v = −0.6 x

or

x=

1 (3 − v1/ 2 ) 0.3

When v = 4 m/s:

x=

1 (3 − 41/ 2 ) 0.3 x = 3.33 m 

or (b)

dv = a = −0.6v3/2 dt

We have v



or

−2[v − (1/2) ]9v = −0.6t

When v = 1 m/s:

9

v − (3/2) dv =

1 v 1 1



t

When t = 0, v = 9 m/s:

or

0

−0.6dt

−

1 = 0.3t 3

−

1 = 0.3t 3 t = 2.22 s 

or (c)

We have

(1)

1 v

−

1 = 0.3t 3 2

or Now

9  3  v=  = 1 0.9 t + (1 + 0.9t ) 2   dx 9 =v= dt (1 + 0.9t ) 2





At t = 0, x = 0:

x 0



dx =

9 dt 0 (1 + 0.9t ) 2 t

t

1   1 x = 9 −   0.9 1 + 0.9t  0

or

1   = 10 1 −   1 + 0.9t  9t = 1 + 0.9t

When x = 6 m:

6=

9t 1 + 0.9t t = 1.667 s 

or An alternative solution is to begin with Eq. (1). x=

1 (3 − v1/ 2 ) 0.3

dx = v = (3 − 0.3x) 2 dt

Then Now At t = 0, x = 0:



x 0

dx = (3 − 0.3x) 2



t 0

dt x

or

1  1  x = t=   0.3  3 − 0.3x  0 9 − 0.9 x

which leads to the same equation as above.


PROBLEM 11.26 The acceleration of a particle is defined by the relation a = 0.4(1 − kv), where k is a constant. Knowing that at t = 0 the particle starts from rest at x = 4 m and that when t = 15 s, v = 4 m/s, determine (a) the constant k, (b) the position of the particle when v = 6 m/s, (c) the maximum velocity of the particle.

SOLUTION (a)

dv = a = 0.4(1 − kv) dt

We have



At t = 0, v = 0:

dv = 0 1− kv v



t 0

0.4dt

1 − [ln(1 − k v)]v0 = 0.4t k

or or

ln(1 − k v) = −0.4 kt

At t = 15 s, v = 4 m/s:

ln(1 − 4k ) = −0.4k (15)

(1)

= −6k k = 0.145703 s/m

Solving yields

k = 0.1457 s/m 

or (b)

We have

v



When x = 4 m, v = 0:

vdv = 0 1− kv v

x



4

0.4 dx

v 1 1/k =− + 1− kv k 1− kv

Now Then

dv = a = 0.4(1 − kv) dx



v

 1 1 − +   dv = 0  k k (1 − k v) 



x 4

0.4 dx

v

or

 v 1  x  − k − k 2 ln(1 − k v)  = 0.4[ x]4  0

or

v 1  −  + 2 ln(1 − kv)  = 0.4( x − 4) k k  

When v = 6 m/s:

  6 1 ln(1 − 0.145 703 × 6)  = 0.4( x − 4) − + 2  0.145 703 (0.145 703) 

or

0.4( x − 4) = 56.4778 x = 145.2 m 

or



(c)

The maximum velocity occurs when a = 0. a = 0: 0.4(1 − k vmax ) = 0 vmax =

or

1 0.145 703 vmax = 6.86 m/s 

or An alternative solution is to begin with Eq. (1). ln(1 − k v) = −0.4 kt v=

Then Thus, vmax is attained as t

1 (1 − k −0.4 kt ) k

∞

vmax =

1 k

as above.


PROBLEM 11.27 Experimental data indicate that in a region downstream of a given louvered supply vent the velocity of the emitted air is defined by v = 0.18v0 /x, where v and x are expressed in m/s and meters, respectively, and v0 is the initial discharge velocity of the air. For v0 = 3.6 m/s, determine (a) the acceleration of the air at x = 2 m, (b) the time required for the air to flow from x = 1 to x = 3 m.

SOLUTION (a)

dv dx 0.18v0 d  0.18v0  = x dx  x 

a=v

We have

=− a=−

When x = 2 m:

0.0324v02 x3 0.0324(3.6) 2 (2)3 a = −0.0525 m/s 2 

or (b)

0.18v0 dx =v= dt x

We have From x = 1 m to x = 3 m:



3 1

xdx =



t3 t1

0.18v0 dt

3

or

1 2   2 x  = 0.18v0 (t3 − t1 )  1

or

(t3 − t1 ) =

1 2

(9 − 1)

0.18(3.6) t3 − t1 = 6.17 s 

or


PROBLEM 11.28 Based on observations, the speed of a jogger can be approximated by the relation v = 7.5(1 − 0.04 x)0.3 , where v and x are expressed in mi/h and miles, respectively. Knowing that x = 0 at t = 0, determine (a) the distance the jogger has run when t = 1 h, (b) the jogger’s acceleration in ft/s2 at t = 0, (c) the time required for the jogger to run 6 mi.

SOLUTION (a)

dx = v = 7.5(1 − 0.04 x)0.3 dt

We have



At t = 0, x = 0: or

x

0

dx = (1 − 0.04 x)0.3

t

 7.5dt 0

1  1  [(1 − 0.04 x)0.7 ]0x = 7.5t −  0.7  0.04  1 − (1 − 0.04 x)0.7 = 0.21t

or or

x=

1 [1 − (1 − 0.21t )1/0.7 ] 0.04

At t = 1 h:

x=

1 {1 − [1 − 0.21(1)]1/0.7 } 0.04

(1)

x = 7.15 mi 

or (b)

We have

a=v

dv dx

d [7.5(1 − 0.04 x)0.3 ] dx = 7.52 (1 − 0.04 x)0.3 [(0.3)(−0.04)(1 − 0.04 x) −0.7 ] = 7.5(1 − 0.04 x)0.3

= −0.675(1 − 0.04 x) −0.4

At t = 0, x = 0:

a0 = −0.675 mi/h 2 ×

5280 ft  1 h  ×  1 mi  3600 s 

a0 = −275 × 10−6 ft/s 2 

or (c)

2

From Eq. (1)

t=

When x = 6 mi:

t=

1 [1 − (1 − 0.04 x)0.7 ] 0.21

1 {1 − [1 − 0.04(6)]0.7 } 0.21 = 0.83229 h t = 49.9 min 

or


PROBLEM 11.29 The acceleration due to gravity at an altitude y above the surface of the earth can be expressed as a=

−32.2 [1 + ( y/20.9 × 106 )]2

where a and y are expressed in ft/s2 and feet, respectively. Using this expression, compute the height reached by a projectile fired vertically upward from the surface of the earth if its initial velocity is (a) 1800 ft/s, (b) 3000 ft/s, (c) 36,700 ft/s.

SOLUTION We have

v

dv =a=− dy

32.2

(1 +

y 20.9 × 106

When

y = 0,

provided that v does reduce to zero,

y = ymax , v = 0



Then

v0

v dv =



v = v0

ymax

0

(1 +

−32.2 y 20.9 × 106

)

2

dy

 1 v02 = 1345.96 × 106 1 −  1 + ymax 20.9 × 106 

or

ymax =

or

v0 = 1800 ft/s:

ymax =

v0 = 3000 ft/s:

y

 max   0    

v02 64.4 −

v02

20.9 × 106

(1800)2 64.4 −

(1800)2 20.9 × 106

ymax = 50.4 × 103 ft 

or (b)

2

 1 1 − v02 = −32.2  −20.9 × 106 y  2 1+ × 106 20.9 

or

(a)

0

)

ymax =

(3000) 2 64.4 −

(3000)2 20.9 × 106

ymax = 140.7 × 103 ft 

or



(c)

v0 = 36,700 ft/s:

ymax =

(36, 700) 2 64.4 −

(36,700)2 20.9 × 106

= −3.03 × 1010 ft

This solution is invalid since the velocity does not reduce to zero. The velocity 36,700 ft/s is above the escape velocity vR from the earth. For vR and above. ymax

∞ 


PROBLEM 11.30 The acceleration due to gravity of a particle falling toward the earth is a = − gR 2 /r 2, where r is the distance from the center of the earth to the particle, R is the radius of the earth, and g is the acceleration due to gravity at the surface of the earth. If R = 3960 mi, calculate the escape velocity, that is, the minimum velocity with which a particle must be projected vertically upward from the surface of the earth if it is not to return to the earth. (Hint: v = 0 for r = ∞.)

SOLUTION We have

v

dv gR 2 =a=− 2 dr r r = R, v = ve

When

r = ∞, v = 0 0

gR 2 dr r2



or

1 1  − ve2 = gR 2   2  r R

or

ve = 2 gR

ve

vdv =



∞

then

R

−

∞

1/2

5280 ft   =  2 × 32.2 ft/s 2 × 3960 mi ×  1 mi  

ve = 36.7 × 103 ft/s 

or


PROBLEM 11.31 The velocity of a particle is v = v0 [1 − sin (π t/T )]. Knowing that the particle starts from the origin with an initial velocity v0 , determine (a) its position and its acceleration at t = 3T , (b) its average velocity during the interval t = 0 to t = T .

SOLUTION (a)

 dx  π t  = v = v0 1 − sin    dt  T  

We have

At t = 0, x = 0:



x

0

dx =



  π t  v0 1 − sin    dt 0  T   t

t

 T  T  π t   πt  T  x = v0 t + cos    = v0 t + cos   −   T 0 T  π  π  π

At t = 3T :

 T 2T   π × 3T  T   −  = v0  3T − x3T = v0 3T + cos   π π   T  π  

a=

At t = 3T : (b)

(1)

x3T = 2.36 v0T 

π πt dv d    π t    = v0 1 − sin     = −v0 cos dt dt   T T  T   

a3T = −v0

π T

cos

π × 3T

a3T =

T

π v0 T



Using Eq. (1) At t = 0: At t = T : Now

T T  x0 = v0 0 + cos(0) −  = 0 π π    T  πT xT = v0 T + cos  π  T 

vave =

2T  T   − π  = v0  T − π   

xT − x0 0.363v0T − 0 = Δt T −0

  = 0.363v0T 

vave = 0.363v0 


PROBLEM 11.32 The velocity of a slider is defined by the relation v = v′ sin (ωn t + φ ). Denoting the velocity and the position of the slider at t = 0 by v0 and x0 , respectively, and knowing that the maximum displacement of the slider is 2 x0 , show that (a) v′ = (v02 + x02ωn2 )/2 x0ωn , (b) the maximum value of the velocity occurs when x = x0 [3 − (v0 /x0ωn ) 2 ]/2.

SOLUTION (a)

v0 = v′ sin (0 + φ ) = v′ sin φ

At t = 0, v = v0 : Then

2

cos φ = v′ − v02 v′ dx = v = v′ sin (ωn t + φ ) dt

Now x



or

 1  x − x0 = v′  − cos (ωn t + φ )   ωn 0

x0

dx =



t

At t = 0, x = x0 :

0

v′ sin (ωn t + φ )dt t

or

x = x0 +

v′

ωn

[cos φ − cos (ωn t + φ )]

Now observe that xmax occurs when cos (ωn t + φ ) = −1. Then xmax = 2 x0 = x0 +

ωn

[cos φ − ( −1)]

2 2  v′  v′ − v0  x0 = + 1  ωn  v1  

Substituting for cos φ

or

v′

x0ωn − v′ = v′2 − v02

Squaring both sides of this equation x02ωn2 − 2 x0ωn + v′2 = v′2 − v02

or

v′ =

v02 + x02ωn2 2 x0ωn

Q. E. D.



(b)

First observe that vmax occurs when ωn t + φ = π2 . The corresponding value of x is v′   π  cos φ − cos    ωn   2  v′ = x0 + cos φ

xvmax = x0 +

ωn

Substituting first for cos φ and then for v′ xvmax = x0 +

2

v′ − v02 v′

v′

ωn

1/2

2   1  v02 + x02ωn2  2 = x0 +   − v0   ωn  2 x0ωn    1 v04 + 2v02 x02ωn2 + x04ωn4 − 4 x02ωn2 v02 = x0 + 2 2 x0ωn

(

= x0 + = x0 + x = 0 2

1

(

 x 2ω 2 − v 2 0 n 0 2 x0ωn2 

)

1/2

1/ 2 2

) 

x02ωn2 − v02 2 x0ωn2

  v 2  3 −  0     x0ωn    

Q. E. D.


PROBLEM 11.33 A stone is thrown vertically upward from a point on a bridge located 40 m above the water. Knowing that it strikes the water 4 s after release, determine (a) the speed with which the stone was thrown upward, (b) the speed with which the stone strikes the water.

SOLUTION Uniformly accelerated motion. Origin at water.

1 y = y0 + v0t + a t 2 2 v = v0 + a t where y0 = 40 m and a = −9.81 m/s2 . (a)

Initial speed.

y = 0 when t = 4 s. 1 0 = 40 + v0 (4) − (9.81)(4)2 2 v0 = 9.62 m/s (b)

v0 = 9.62 m/s 

Speed when striking the water. (v at t = 4 s)

v = 9.62 − (9.81)(4) = −29.62 m/s

v = 29.6 m/s 


PROBLEM 11.34 A motorist is traveling at 54 km/h when she observes that a traffic light 240 m ahead of her turns red. The traffic light is timed to stay red for 24 s. If the motorist wishes to pass the light without stopping just as it turns green again, determine (a) the required uniform deceleration of the car, (b) the speed of the car as it passes the light.

SOLUTION Uniformly accelerated motion: x0 = 0 v0 = 54 km/h = 15 m/s

(a)

x = x0 + v0t +

1 2 at 2

when t = 24s, x = 240 m: 240 m = 0 + (15 m/s)(24 s) +

1 a (24 s)2 2

a = −0.4167 m/s 2

(b)

a = −0.417 m/s 2 

v = v0 + a t

when t = 24s: v = (15 m/s) + (−0.4167 m/s)(24 s) v = 5.00 m/s v = 18.00 km/h

v = 18.00 km/h 


PROBLEM 11.35 A motorist enters a freeway at 30 mi/h and accelerates uniformly to 60 mi/h. From the odometer in the car, the motorist knows that she traveled 550 ft while accelerating. Determine (a) the acceleration of the car, (b) the time required to reach 60 mi/h.

SOLUTION (a)

Acceleration of the car. v12 = v02 + 2a( x1 − x0 ) a=

Data:

v12 − v02 2( x1 − x0 )

v0 = 30 mi/h = 44 ft/s v1 = 60 mi/h = 88 ft/s x0 = 0 x1 = 550 ft a=

(b)

(88) 2 − (44) 2 (2)(55 − 0)

a = 5.28 ft/s 2 

Time to reach 60 mi/h. v1 = v0 + a (t1 − t0 ) v1 − v0 a 88 − 44 = 5.28 = 8.333 s

t1 − t0 =

t1 − t0 = 8.33 s 


PROBLEM 11.36 A group of students launches a model rocket in the vertical direction. Based on tracking data, they determine that the altitude of the rocket was 89.6 ft at the end of the powered portion of the flight and that the rocket landed 16 s later. Knowing that the descent parachute failed to deploy so that the rocket fell freely to the ground after reaching its maximum altitude and assuming that g = 32.2 ft/s 2 , determine (a) the speed v1 of the rocket at the end of powered flight, (b) the maximum altitude reached by the rocket.

SOLUTION (a)

1 2 at 2

We have

y = y1 + v1t +

At tland ,

y=0

Then

0 = 89.6 ft + v1 (16 s) +

1 (−32.2 ft/s 2 )(16 s) 2 2 v1 = 252 ft/s 

or (b)

v 2 = v12 + 2a( y − y1 )

We have At

y = ymax , v = 0

Then

0 = (252 ft/s) 2 + 2( −32.2 ft/s 2 )( ymax − 89.6) ft ymax = 1076 ft 

or


PROBLEM 11.37 A small package is released from rest at A and moves along the skate wheel conveyor ABCD. The package has a uniform acceleration of 4.8 m/s 2 as it moves down sections AB and CD, and its velocity is constant between B and C. If the velocity of the package at D is 7.2 m/s, determine (a) the distance d between C and D, (b) the time required for the package to reach D.

SOLUTION (a)

For A

B and C

D we have v 2 = v02 + 2a( x − x0 ) 2 vBC = 0 + 2(4.8 m/s 2 )(3 − 0) m

Then, at B

= 28.8 m 2 /s 2

(vBC = 5.3666 m/s)

2 vD2 = vBC + 2aCD ( xD − xC )

and at D

d = xD − xC

(7.2 m/s)2 = (28.8 m 2 /s 2 ) + 2(4.8 m/s 2 )d

or

d = 2.40 m 

or (b)

For A

B and C

D we have v = v0 + at

Then A

B

5.3666 m/s = 0 + (4.8 m/s 2 )t AB t AB = 1.11804 s

or and C

7.2 m/s = 5.3666 m/s + (4.8 m/s 2 )tCD

D

tCD = 0.38196 s

or Now, for B

C, we have

xC = xB + vBC t BC

or

3 m = (5.3666 m/s)t BC

or

t BC = 0.55901 s

Finally,

t D = t AB + t BC + tCD = (1.11804 + 0.55901 + 0.38196) s t D = 2.06 s 

or


PROBLEM 11.38 A sprinter in a 100-m race accelerates uniformly for the first 35 m and then runs with constant velocity. If the sprinter’s time for the first 35 m is 5.4 s, determine (a) his acceleration, (b) his final velocity, (c) his time for the race.

SOLUTION Given:

0 ≤ x ≤ 35 m, a = constant 35 m < x ≤ 100 m, v = constant

At t = 0, v = 0 when

x = 35 m, t = 5.4 s

Find: (a)

a

(b)

v when x = 100 m

(c)

t when x = 100 m

(a)

We have At t = 5.4 s: or

x = 0 + 0t + 35 m =

1 2 at 2

for

0 ≤ x ≤ 35 m

1 a(5.4 s) 2 2

a = 2.4005 m/s 2 a = 2.40 m/s 2 

(b)

First note that v = vmax for 35 m ≤ x ≤ 100 m. Now

(c)

v 2 = 0 + 2a( x − 0)

for

0 ≤ x ≤ 35 m

When x = 35 m:

2 vmax = 2(2.4005 m/s 2 )(35 m)

or

vmax = 12.9628 m/s

We have When x = 100 m:

vmax = 12.96 m/s 

x = x1 + v0 (t − t1 )

for

35 m < x ≤ 100 m

100 m = 35 m + (12.9628 m/s)(t2 − 5.4) s t2 = 10.41 s 

or


PROBLEM 11.39 As relay runner A enters the 20-m-long exchange zone with a speed of 12.9 m/s, he begins to slow down. He hands the baton to runner B 1.82 s later as they leave the exchange zone with the same velocity. Determine (a) the uniform acceleration of each of the runners, (b) when runner B should begin to run.

SOLUTION (a)

For runner A: At t = 1.82 s:

x A = 0 + (v A )0 t +

1 a At 2 2

20 m = (12.9 m/s)(1.82 s) +

1 a A (1.82 s) 2 2 a A = −2.10 m/s 2 

or Also At t = 1.82 s:

v A = (v A ) 0 + a A t (v A )1.82 = (12.9 m/s) + ( −2.10 m/s 2 )(1.82 s) = 9.078 m/s

For runner B:

vB2 = 0 + 2aB [ xB − 0]

When

xB = 20 m, vB = v A : (9.078 m/s) 2 = 2aB (20 m)

or

aB = 2.0603 m/s 2 aB = 2.06 m/s 2 

(b)

For runner B:

vB = 0 + aB (t − t B )

where t B is the time at which he begins to run. At t = 1.82 s: or

9.078 m/s = (2.0603 m/s 2 )(1.82 − t B ) s t B = −2.59 s

Runner B should start to run 2.59 s before A reaches the exchange zone. 


PROBLEM 11.40 In a boat race, boat A is leading boat B by 50 m and both boats are traveling at a constant speed of 180 km/h. At t = 0, the boats accelerate at constant rates. Knowing that when B passes A, t = 8 s and v A = 225 km/h, determine (a) the acceleration of A, (b) the acceleration of B.

SOLUTION (a)

We have

v A = (v A ) 0 + a A t (v A )0 = 180 km/h = 50 m/s

At t = 8 s: Then

v A = 225 km/h = 62.5 m/s 62.5 m/s = 50 m/s + a A (8 s) a A = 1.563 m/s 2 

or (b)

We have x A = ( x A ) 0 + (v A )0 t +

1 1 a At 2 = 50 m + (50 m/s)(8 s) + (1.5625 m/s 2 )(8 s) 2 = 500 m 2 2

and

xB = 0 + (vB )0 t +

At t = 8 s:

x A = xB

1 aB t 2 2

500 m = (50 m/s)(8 s) +

(vB )0 = 50 m/s

1 aB (8 s) 2 2 aB = 3.13 m/s 2 

or


PROBLEM 11.41 A police officer in a patrol car parked in a 45 mi/h speed zone observes a passing automobile traveling at a slow, constant speed. Believing that the driver of the automobile might be intoxicated, the officer starts his car, accelerates uniformly to 60 mi/h in 8 s, and, maintaining a constant velocity of 60 mi/h, overtakes the motorist 42 s after the automobile passed him. Knowing that 18 s elapsed before the officer began pursuing the motorist, determine (a) the distance the officer traveled before overtaking the motorist, (b) the motorist’s speed.

SOLUTION

(vP )18 = 0

(a)

(vP )26 = 60 mi/h = 88 ft/s

(vP ) 42 = 90 mi/h = 88 ft/s

Patrol car: For 18 s < t ≤ 26 s: At t = 26 s:

vP = 0 + aP (t − 18) 88 ft/s = aP (26 − 18) s

or

aP = 11 ft/s 2

Also,

xP = 0 + 0(t − 18) −

At t = 26 s: For 26 s < t ≤ 42 s: At t = 42 s:

(xP ) 26 =

1 aP (t − 18) 2 2

1 (11 ft/s 2 )(26 − 18) 2 = 352 ft 2

xP = ( xP )26 + (vP ) 26 (t − 26) (xP ) 42 = 352 m + (88 ft/s)(42 − 26) s = 1760 ft ( xP ) 42 = 1760 ft 

(b)

For the motorist’s car: At t = 42 s, xM = xP : or

xM = 0 + vM t 1760 ft = vM (42 s) vM = 41.9048 ft/s vM = 28.6 mi/h 

or


PROBLEM 11.42 Automobiles A and B are traveling in adjacent highway lanes and at t = 0 have the positions and speeds shown. Knowing that automobile A has a constant acceleration of 1.8 ft/s 2 and that B has a constant deceleration of 1.2 ft/s 2 , determine (a) when and where A will overtake B, (b) the speed of each automobile at that time.

SOLUTION a A = +1.8 ft/s 2

aB = −1.2 ft/s 2

(v A )0 = 24 mi/h = 35.2 ft/s

A overtakes B

(vB )0 = 36 mi/h = 52.8 ft/s

Motion of auto A: v A = (v A )0 + a At = 35.2 + 1.8t x A = ( x A ) 0 + (v A ) 0 t +

1 1 a At 2 = 0 + 35.2t + (1.8)t 2 2 2

(1) (2)

Motion of auto B: vB = (vB )0 + aB t = 52.8 − 1.2t x B = ( xB ) 0 + ( vB ) 0 t +

(a)

1 1 aB t 2 = 75 + 52.8t + (−1.2)t 2 2 2

(3) (4)

A overtakes B at t = t1 . x A = xB : 35.2t + 0.9t12 = 75 + 52.8t1 − 0.6t12 1.5t12 − 17.6t1 − 75 = 0 t1 = −3.22 s and t1 = 15.0546

Eq. (2):

x A = 35.2(15.05) + 0.9(15.05) 2

t1 = 15.05 s  x A = 734 ft 



(b)

Velocities when t1 = 15.05 s Eq. (1):

v A = 35.2 + 1.8(15.05) v A = 62.29 ft/s

Eq. (3):

v A = 42.5 mi/h



vB = 23.7 mi/h



vB = 52.8 − 1.2(15.05) vB = 34.74 ft/s


PROBLEM 11.43 Two automobiles A and B are approaching each other in adjacent highway lanes. At t = 0, A and B are 3200 ft apart, their speeds are v A = 65 mi/h and vB = 40 mi/h, and they are at Points P and Q, respectively. Knowing that A passes Point Q 40 s after B was there and that B passes Point P 42 s after A was there, determine (a) the uniform accelerations of A and B, (b) when the vehicles pass each other, (c) the speed of B at that time.

SOLUTION (a)

x A = 0 + (v A ) 0 t +

We have (x is positive

3200 m = (95.333 m/s)(40 s) +

Also, xB = 0 + (vB )0 t +

1 aB t 2 2

Then (95.333 ft/s)t AB +

(c)

1 aB (42 s) 2 2

aB = 0.83447 ft/s 2

When the cars pass each other

Solving

a A = −0.767 ft/s 2 

(vB )0 = 40 mi/h = 58.667 ft/s

3200 ft = (58.667 ft/s)(42 s) +

or

or

1 a A (40 s) 2 2

; origin at Q.)

At t = 42 s:

(b)

(v A )0 = 65 mi/h = 95.33 ft/s

; origin at P.)

At t = 40 s:

(xB is positive

1 a At 2 2

aB = 0.834 ft/s 2 

x A + xB = 3200 ft

1 1 2 2 + (58.667 ft/s)t AB + (0.83447 ft/s 2 )t AB = 3200 ft (−0.76667 ft/s)t AB 2 2 2 0.03390t AB + 154t AB − 3200 = 0

t = 20.685 s and t = −4563 s

We have

vB = ( vB ) 0 + a B t

At t = t AB :

vB = 58.667 ft/s + (0.83447 ft/s 2 )(20.685 s) = 75.927 ft/s

t >0

t AB = 20.7 s 

vB = 51.8 mi/h 


PROBLEM 11.44 An elevator is moving upward at a constant speed of 4 m/s. A man standing 10 m above the top of the elevator throws a ball upward with a speed of 3 m/s. Determine (a) when the ball will hit the elevator, (b) where the ball will hit the elevator with respect to the location of the man.

SOLUTION Place the origin of the position coordinate at the level of the standing man, the positive direction being up. The ball undergoes uniformly accelerated motion. yB = ( yB )0 + (vB )0 t −

1 2 gt 2

with ( yB )0 = 0, (vB )0 = 3 m/s, and g = 9.81 m/s 2 . yB = 3t − 4.905t 2

The elevator undergoes uniform motion. y E = ( y E ) 0 + vE t

with ( yE )0 = −10 m and vE = 4 m/s. (a)

Set yB = yE

Time of impact.

3t − 4.905t 2 = −10 + 4t 4.905t 2 + t − 10 = 0 t = 1.3295 and −1.5334

(b)

t = 1.330 s 

Location of impact. yB = (3)(1.3295) − (4.905)(1.3295)2 = −4.68 m yE = −10 + (4)(1.3295) = −4.68 m

(checks)

4.68 m below the man 


PROBLEM 11.45 Two rockets are launched at a fireworks display. Rocket A is launched with an initial velocity v0 = 100 m/s and rocket B is launched t1 seconds later with the same initial velocity. The two rockets are timed to explode simultaneously at a height of 300 m as A is falling and B is rising. Assuming a constant acceleration g = 9.81 m/s2 , determine (a) the time t1, (b) the velocity of B relative to A at the time of the explosion.

SOLUTION Place origin at ground level. The motion of rockets A and B is Rocket A:

v A = (v A )0 − gt = 100 − 9.81t y A = ( y A ) 0 + (v A ) 0 t −

Rocket B:

1 2 gt = 100t − 4.905t 2 2

(1) (2)

vB = (vB )0 − g (t − t1 ) = 100 − 9.81(t − t1 )

(3)

1 g (t − t1 ) 2 2 = 100(t − t1 ) − 4.905(t − t1 )2

(4)

yB = ( yB )0 + (vB )0 (t − t1 ) −

Time of explosion of rockets A and B. y A = yB = 300 ft From (2),

300 = 100t − 4.905t 2 4.905t 2 − 100t + 300 = 0 t = 16.732 s and 3.655 s

From (4),

300 = 100(t − t1 ) − 4.905(t − t12 ) t − t1 = 16.732 s and 3.655 s

Since rocket A is falling, Since rocket B is rising, (a)

Time t1:

(b)

Relative velocity at explosion.

t = 16.732 s t − t1 = 3.655 s t1 = t − (t − t1 )

From (1),

v A = 100 − (9.81)(16.732) = −64.15 m/s

From (3),

vB = 100 − (9.81)(16.732 − 13.08) = 64.15 m/s

Relative velocity:

vB/A = vB − v A

t1 = 13.08 s 

vB/A = 128.3 m/s 


PROBLEM 11.46 Car A is parked along the northbound lane of a highway, and car B is traveling in the southbound lane at a constant speed of 60 mi/h. At t = 0, A starts and accelerates at a constant rate a A , while at t = 5 s, B begins to slow down with a constant deceleration of magnitude a A /6. Knowing that when the cars pass each other x = 294 ft and v A = vB , determine (a) the acceleration a A , (b) when the vehicles pass each other, (c) the distance d between the vehicles at t = 0.

SOLUTION

For t ≥ 0:

v A = 0 + a At xA = 0 + 0 +

1 a At 2 2

0 ≤ t < 5 s:

xB = 0 + (vB )0 t (vB )0 = 60 mi/h = 88 ft/s

At t = 5 s:

xB = (88 ft/s)(5 s) = 440 ft

For t ≥ 5 s:

vB = (vB )0 + aB (t − 5)

1 aB = − a A 6

xB = ( xB ) S + (vB )0 (t − 5) +

1 aB (t − 5) 2 2

Assume t > 5 s when the cars pass each other. At that time (t AB ), v A = vB :

a At AB = (88 ft/s) −

x A = 294 ft:

294 ft =

Then or

a A( 76 t AB − 65 ) 1 2

2 a At AB

=

aA (t AB − 5) 6

1 2 a At AB 2 88 294

2 44t AB − 343t AB + 245 = 0



Solving (a)

With t AB > 5 s,

t AB = 0.795 s and t AB = 7.00 s 294 ft =

1 a A (7.00 s) 2 2 a A = 12.00 ft/s 2 

or (b)

t AB = 7.00 s 

From above

Note: An acceptable solution cannot be found if it is assumed that t AB ≤ 5 s. (c)

We have

d = x + ( xB )t AB = 294 ft + 440 ft + (88 ft/s)(2.00 s) 1 1  +  − × 12.00 ft/s 2  (2.00 s)2 2 6  d = 906 ft 

or


PROBLEM 11.47 The elevator shown in the figure moves downward with a constant velocity of 4 m/s. Determine (a) the velocity of the cable C, (b) the velocity of the counterweight W, (c) the relative velocity of the cable C with respect to the elevator, (d ) the relative velocity of the counterweight W with respect to the elevator.

SOLUTION Choose the positive direction downward. (a)

Velocity of cable C. yC + 2 yE = constant vC + 2vE = 0 vE = 4 m/s

But, or (b)

vC = −2vE = −8 m/s

vC = 8.00 m/s 

Velocity of counterweight W. yW + yE = constant vW + vE = 0 vW = −vE = −4 m/s

(c)

vW = 4.00 m/s 

Relative velocity of C with respect to E. vC/E = vC − vE = (−8 m/s) − ( +4 m/s) = −12 m/s vC/E = 12.00 m/s 

(d )

Relative velocity of W with respect to E. vW /E = vW − vE = (−4 m/s) − (4 m/s) = −8 m/s vW /E = 8.00 m/s 


PROBLEM 11.48 The elevator shown starts from rest and moves upward with a constant acceleration. If the counterweight W moves through 30 ft in 5 s, determine (a) the acceleration of the elevator and the cable C, (b) the velocity of the elevator after 5 s.

SOLUTION We choose positive direction downward for motion of counterweight. yW =

At t = 5 s,

1 aW t 2 2

yW = 30 ft 30 ft =

1 aW (5 s) 2 2

aW = 2.4 ft/s 2

(a)

Accelerations of E and C. Since Thus: Also, Thus:

(b)

aW = 2.4 ft/s 2

yW + yE = constant vW + vE = 0, and aW + aE = 0 aE = − aW = −(2.4 ft/s 2 ),

a E = 2.40 ft/s 2 

yC + 2 yE = constant, vC + 2vE = 0, and aC + 2aE = 0 aC = −2aE = −2(−2.4 ft/s 2 ) = +4.8 ft/s 2 ,

aC = 4.80 ft/s 2 

Velocity of elevator after 5 s. vE = (vE )0 + aE t = 0 + (−2.4 ft/s 2 )(5 s) = −12 ft/s

( v E )5 = 12.00 ft/s 


PROBLEM 11.49 Slider block A moves to the left with a constant velocity of 6 m/s. Determine (a) the velocity of block B, (b) the velocity of portion D of the cable, (c) the relative velocity of portion C of the cable with respect to portion D.

SOLUTION From the diagram, we have x A + 3 yB = constant

Then

v A + 3vB = 0

(1)

and

a A + 3aB = 0

(2)

(a)

Substituting into Eq. (1)

6 m/s + 3vB = 0 v B = 2.00 m/s 

or (b)

From the diagram

yB + yD = constant

Then

vB + vD = 0



v D = 2.00 m/s 

(c)

From the diagram

x A + yC = constant

Then

v A + vC = 0

Now

vC = −6 m/s

vC/D = vC − vD = (−6 m/s) − (2 m/s) = −8 m/s vC/D = 8.00 m/s 


PROBLEM 11.50 Block B starts from rest and moves downward with a constant acceleration. Knowing that after slider block A has moved 9 in. its velocity is 6 ft/s, determine (a) the accelerations of A and B, (b) the velocity and the change in position of B after 2 s.

SOLUTION From the diagram, we have x A + 3 yB = constant

Then

v A + 3vB = 0

(1)

and

a A + 3aB = 0

(2)

(a)

Eq. (2): a A + 3aB = 0 and a B is constant and positive  a A is constant and negative Also, Eq. (1) and (vB )0 = 0  (v A )0 = 0 v A2 = 0 + 2a A [ x A − ( x A )0 ]

Then When |Δ x A | = 0.4 m:

(6 ft/s) 2 = 2a A (9/12 ft) a A = 24.0 ft/s 2

or



Then, substituting into Eq. (2): −24 ft/s 2 + 3aB = 0 aB =

or

24 ft/s 2 3

a B = 8.00 ft/s 2 



(b)

We have

vB = 0 + a B t

At t = 2 s:

 24  vB =  ft/s 2  (2 s)  3 

v B = 16.00 ft/s 

or yB = ( yB )0 + 0 +

Also At t = 2 s: or

y B − ( y B )0 =

1 aB t 2 2

1  24  ft/s 2  (2 s) 2  2  3 

y B − (y B )0 = 16.00 ft 




PROBLEM 11.51 Slider block B moves to the right with a constant velocity of 300 mm/s. Determine (a) the velocity of slider block A, (b) the velocity of portion C of the cable, (c) the velocity of portion D of the cable, (d ) the relative velocity of portion C of the cable with respect to slider block A.

SOLUTION

From the diagram

xB + ( xB − x A ) − 2 x A = constant

Then

2vB − 3v A = 0

(1)

and

2aB − 3a A = 0

(2)

Also, we have

vD + v A = 0

Then (a)

− xD − x A = constant

Substituting into Eq. (1)

2(300 mm/s) − 3v A = 0

or (b)

From the diagram Then Substituting

(3)

v A = 200 mm/s



vC = 600 mm/s



xB + ( xB − xC ) = constant 2vB − vC = 0 2(300 mm/s) − vC = 0

or



(c)

From the diagram Then Substituting

( xC − x A ) + ( xD − x A ) = constant vC − 2v A + vD = 0 600 mm/s − 2(200 mm/s) + vD = 0

v D = 200 mm/s

or (d)

We have



vC/A = vC − v A = 600 mm/s − 200 mm/s

vC/A = 400 mm/s

or




PROBLEM 11.52 At the instant shown, slider block B is moving with a constant acceleration, and its speed is 150 mm/s. Knowing that after slider block A has moved 240 mm to the right its velocity is 60 mm/s, determine (a) the accelerations of A and B, (b) the acceleration of portion D of the cable, (c) the velocity and change in position of slider block B after 4 s.

SOLUTION

xB + ( xB − x A ) − 2 x A = constant

From the diagram Then

2vB − 3v A = 0

(1)

and

2aB − 3a A = 0

(2)

(a)

First observe that if block A moves to the right, v A → and Eq. (1)  v B → . Then, using Eq. (1) at t = 0 2(150 mm/s) − 3(v A )0 = 0 (v A )0 = 100 mm/s

or

Also, Eq. (2) and aB = constant  a A = constant v A2 = (v A )02 + 2a A [ x A − ( x A )0 ]

Then When x A − ( x A )0 = 240 mm:

(60 mm/s) 2 = (100 mm/s) 2 + 2a A (240 mm)

or

aA = −

40 mm/s 2 3

a A = 13.33 mm/s 2

or





Then, substituting into Eq. (2)  40  2aB − 3  − mm/s 2  = 0  3  aB = −20 mm/s 2

or (b)

a B = 20.0 mm/s 2



From the diagram, − xD − x A = constant vD + v A = 0

Then Substituting

aD + a A = 0  40  aD +  − mm/s 2  = 0 3  

or (c)

We have

vB = ( vB ) 0 + a B t

At t = 4 s:

vB = 150 mm/s + ( −20.0 mm/s 2 )(4 s)

or Also At t = 4 s:

x B = ( xB ) 0 + ( vB ) 0 t +

a D = 13.33 mm/s 2



v B = 70.0 mm/s



1 aB t 2 2

xB − ( xB )0 = (150 mm/s)(4 s) +

1 (−20.0 mm/s 2 )(4 s) 2 2

x B − (x B )0 = 440 mm

or




PROBLEM 11.53 Collar A starts from rest and moves upward with a constant acceleration. Knowing that after 8 s the relative velocity of collar B with respect to collar A is 24 in./s, determine (a) the accelerations of A and B, (b) the velocity and the change in position of B after 6 s.

SOLUTION From the diagram 2 y A + yB + ( yB − y A ) = constant

Then

v A + 2 vB = 0

(1)

and

a A + 2aB = 0

(2)

(a)

Eq. (1) and (v A )0 = 0  (vB )0 = 0 Also, Eq. (2) and a A is constant and negative  a B is constant and positive. Then Now From Eq. (2) So that

v A = 0 + a At

vB = 0 + a B t

vB/A = vB − v A = (aB − a A )t 1 aB = − a A 2 3 vB/A = − a At 2



At t = 8 s:

3 24 in./s = − a A (8 s) 2

a A = 2.00 in./s 2 

or and then

1 aB = − (−2 in./s 2 ) 2

a B = 1.000 in./s 2 

or (b)

At t = 6 s:

vB = (1 in./s 2 )(6 s)

v B = 6.00 in./s 

or 

Now At t = 6 s:

yB = ( yB )0 + 0 + y B − ( y B )0 =

1 aB t 2 2

1 (1 in./s 2 )(6 s) 2 2

y B − (y B )0 = 18.00 in. 

or


PROBLEM 11.54 The motor M reels in the cable at a constant rate of 100 mm/s. Determine (a) the velocity of load L, (b) the velocity of pulley B with respect to load L.

SOLUTION

Let xB and xL be the positions, respectively, of pulley B and load L measured downward from a fixed elevation above both. Let xM be the position of a point on the cable about to enter the reel driven by the motor. Then, considering the lengths of the two cables, xM + 3xB = constant

vM + 3vB = 0

xL + ( xL − xB ) = constant

2v L + v B = 0

vM = 100 mm/s

with

vB = − vL =

vM = −33.333 m/s 3

vB = −16.667 mm/s 2

v L = 16.67 mm/s 

(a)

Velocity of load L.

(b)

Velocity of pulley B with respect to load L. vB/L = vB − vL = −33.333 − (−16.667) = −16.667 v B/L = 16.67 mm/s 


PROBLEM 11.55 Block C starts from rest at t = 0 and moves downward with a constant acceleration of 4 in./s2. Knowing that block B has a constant velocity of 3 in./s upward, determine (a) the time when the velocity of block A is zero, (b) the time when the velocity of block A is equal to the velocity of block D, (c) the change in position of block A after 5 s.

SOLUTION From the diagram: Cord 1:

2 y A + 2 y B + yC = constant

Then

2v A + 2vB + vC = 0

and

2a A + 2aB + aC = 0

Cord 2:

(1)

( y D − y A ) + ( y D − y B ) = constant

Then

2vD − v A − v B = 0

and

2aD − a A − aB = 0

(2)

Use units of inches and seconds. Motion of block C:

vC = vC 0 + aC t where aC = −4 in./s 2

= 0 + 4t

Motion of block B:

vB = −3 in./s;

Motion of block A:

From (1) and (2),

aB = 0

1 1 v A = −vB − vC = 3 − (4t ) = 3 − 2t in./s 2 2 1 1 a A = −aB − aC = 0 − (4) = −2 in./s 2 2 2



(a)

Time when vB is zero. 3 − 2t = 0

Motion of block D:

From (3), vD =

(b)

t = 1.500 s 

1 1 1 1 v A + vB = (3 − 2t ) − (3) = −1t 2 2 2 2

Time when vA is equal to v0. 3 − 2t = −t

(c)

t = 3.00 s 

Change in position of block A (t = 5 s). 1 a At 2 2 1 = (3)(5) + (−2)(5)2 = −10 in. 2

Δy A = ( v A ) 0 t +

Change in position = 10.00 in. 


PROBLEM 11.56 Block A starts from rest at t = 0 and moves downward with a constant acceleration of 6 in./s2. Knowing that block B moves up with a constant velocity of 3 in./s, determine (a) the time when the velocity of block C is zero, (b) the corresponding position of block C.

SOLUTION

The cable lengths are constant. L1 = 2 yC + 2 yD + constant L 2 = y A + yB + ( yB − yD ) + constant

Eliminate yD. L1 + 2 L 2 = 2 yC + 2 yD + 2 y A + 2 yB + 2( yB − yD ) + constant 2( yC + y A + 2 yB ) = constant

Differentiate to obtain relationships for velocities and accelerations, positive downward. vC + v A + 2vB = 0

(1)

aC + a A + 2aB = 0

(2)

Use units of inches and seconds. Motion of block A:

v A = a At + 6t Δy A =

Motion of block B:

1 1 a At 2 = (6)t 2 = 3t 2 2 2

v B = 3 in./s

vB = −3 in./s

ΔyB = vB t = −3t

Motion of block C:

From (1), vC = −v A − 2vB = −6t − 2(−3) = 6 − 6t ΔyC =

(a)

Time when vC is zero.

(b)

Corresponding position.



t

0

vC dt = 6t − 3t 2

6 − 6t = 0

t = 1.000 s 

ΔyC = (6)(1) − (3)(1)2 = 3 in.

ΔyC = 3.00 in. 


PROBLEM 11.57 Block B starts from rest, block A moves with a constant acceleration, and slider block C moves to the right with a constant acceleration of 75 mm/s 2 . Knowing that at t = 2 s the velocities of B and C are 480 mm/s downward and 280 mm/s to the right, respectively, determine (a) the accelerations of A and B, (b) the initial velocities of A and C, (c) the change in position of slider block C after 3 s.

SOLUTION

From the diagram 3 y A + 4 yB + xC = constant

Then

3v A + 4vB + vC = 0

(1)

and

3a A + 4aB + aC = 0

(2)

(vB ) = 0,

Given:

a A = constent (aC ) = 75 mm/s 2

At t = 2 s,

v B = 480 mm/s vC = 280 mm/s

(a)

Eq. (2) and a A = constant and aC = constant  aB = constant vB = 0 + a B t

Then At t = 2 s:

480 mm/s = aB (2 s) aB = 240 mm/s 2



or

a B = 240 mm/s 2 

or

a A = 345 mm/s 2 

Substituting into Eq. (2) 3a A + 4(240 mm/s 2 ) + (75 mm/s 2 ) = 0 a A = −345 mm/s

 



(b)

vC = (vC )0 + aC t

We have At t = 2 s:

280 mm/s = (vC )0 + (75 mm/s)(2 s) vC = 130 mm/s



or

( vC )0 = 130.0 mm/s



Then, substituting into Eq. (1) at t = 0 3(v A )0 + 4(0) + (130 mm/s) = 0 v A = −43.3 mm/s

(c)

We have At t = 3 s:

xC = ( xC )0 + (vC )0 t +

or 1 aC t 2 2

xC − ( xC )0 = (130 mm/s)(3 s) + = 728 mm

(v A )0 = 43.3 mm/s 

1 (75 mm/s 2 )(3 s) 2 2

or

xC − (xC )0 = 728 mm




PROBLEM 11.58 Block B moves downward with a constant velocity of 20 mm/s. At t = 0, block A is moving upward with a constant acceleration, and its velocity is 30 mm/s. Knowing that at t = 3 s slider block C has moved 57 mm to the right, determine (a) the velocity of slider block C at t = 0, (b) the accelerations of A and C, (c) the change in position of block A after 5 s.

SOLUTION

From the diagram 3 y A + 4 yB + xC = constant

Then

3v A + 4vB + vC = 0

(1)

and

3a A + 4aB + aC = 0

(2)

v B = 20 mm/s ;

Given:

( v A )0 = 30 mm/s

(a)

Substituting into Eq. (1) at t = 0 3(−30 mm/s) + 4(20 mm/s) + (vC )0 = 0 (vC )0 = 10 mm/s

(b)

We have At t = 3 s:

( vC )0 = 10.00 mm/s

or

xC = ( xC )0 + (vC )0 t +

1 aC t 2 2

57 mm = (10 mm/s)(3 s) +

1 aC (3 s)2 2

aC = 6 mm/s 2



Now



or

aC = 6.00 mm/s 2



v B = constant → aB = 0



Then, substituting into Eq. (2) 3a A + 4(0) + (6 mm/s 2 ) = 0 a A = −2 mm/s 2

(c)

We have At t = 5 s:

y A = ( y A )0 + (v A )0 t +

or

a A = 2.00 mm/s 2 

1 a At 2 2

y A − ( y A )0 = (−30 mm/s)(5 s) +

1 (−2 mm/s 2 )(5 s)2 2

= −175 mm y A − (y A )0 = 175.0 mm 

or


PROBLEM 11.59 The system shown starts from rest, and each component moves with a constant acceleration. If the relative acceleration of block C with respect to collar B is 60 mm/s 2 upward and the relative acceleration of block D with respect to block A is 110 mm/s 2 downward, determine (a) the velocity of block C after 3 s, (b) the change in position of block D after 5 s.

SOLUTION From the diagram 2 y A + 2 yB + yC = constant

Cable 1: Then

2v A + 2vB + vC = 0

(1)

and

2a A + 2aB + aC = 0

(2)

Cable 2:

( yD − y A ) + ( yD − yB ) = constant

Then and

− v A − v B + 2 vD = 0

(3)

− a A − aB + 2aD = 0

(4)

Given: At t = 0, v = 0; all accelerations constant; aC/B = 60 mm/s 2 , aD /A = 110 mm/s 2 (a)

We have

aC/B = aC − aB = −60 or aB = aC + 60

and

aD/A = aD − a A = 110 or a A = aD − 110

Substituting into Eqs. (2) and (4) Eq. (2):

2(aD − 110) + 2( aC + 60) + aC = 0 3aC + 2aD = 100

or Eq. (4):

−(aD − 110) − ( aC + 60) + 2aD = 0 − aC + aD = −50

or

(5)

(6)



Solving Eqs. (5) and (6) for aC and aD aC = 40 mm/s 2 aD = −10 mm/s 2

Now

vC = 0 + aC t

At t = 3 s:

vC = (40 mm/s 2 )(3 s) vC = 120.0 mm/s 

or (b)

We have At t = 5 s:

yD = ( yD )0 + (0)t + yD − ( yD )0 =

1 aD t 2 2

1 ( −10 mm/s 2 )(5 s) 2 2 y D − (y D )0 = 125.0 mm 

or


PROBLEM 11.60* The system shown starts from rest, and the length of the upper cord is adjusted so that A, B, and C are initially at the same level. Each component moves with a constant acceleration, and after 2 s the relative change in position of block C with respect to block A is 280 mm upward. Knowing that when the relative velocity of collar B with respect to block A is 80 mm/s downward, the displacements of A and B are 160 mm downward and 320 mm downward, respectively, determine (a) the accelerations of A and B if aB > 10 mm/s 2 , (b) the change in position of block D when the velocity of block C is 600 mm/s upward.

SOLUTION From the diagram 2 y A + 2 yB + yC = constant

Cable 1: Then

2v A + 2vB + vC = 0

(1)

and

2a A + 2aB + aC = 0

(2)

Cable 2: ( yD − y A ) + ( yD − yB ) = constant Then

− v A − vB − 2 vD = 0

(3)

and

− a A − aB + 2aD = 0

(4)

t =0

Given: At

v=0 ( y A )0 = ( yB )0 = ( yC )0

All accelerations constant. At t = 2 s yC /A = 280 mm vB /A = 80 mm/s

When

y A − ( y A )0 = 160 mm yB − ( yB )0 = 320 mm aB > 10 mm/s 2


PROBLEM 11.60* (Continued)

(a)

We have

y A = ( y A )0 + (0)t +

1 a At 2 2

and

yC = ( yC )0 + (0)t +

1 aC t 2 2

Then

yC/A = yC − y A =

1 (aC − a A )t 2 2

At t = 2 s, yC/A = −280 mm: −280 mm =

or

1 (aC − a A )(2 s) 2 2

aC = a A − 140

(5)

Substituting into Eq. (2) 2a A + 2aB + (a A − 140) = 0

or

1 a A = (140 − 2aB ) 3

Now

vB = 0 + a B t

(6)

v A = 0 + a At vB/A = vB − v A = (aB − a A )t

Also When



yB = ( yB )0 + (0)t + v B /A = 80 mm/s :

1 aB t 2 2 80 = (aB − a A )t

Δy A = 160 mm :

160 =

1 a At 2 2

ΔyB = 320 mm :

320 =

1 aB t 2 2

(7)

1 (aB − a A )t 2 2

Then

160 =

Using Eq. (7)

320 = (80)t or t = 4 s

Then

160 =

1 a A (4)2 2

or

a A = 20.0 mm/s 2 

and

320 =

1 aB (4) 2 2

or

a B = 40.0 mm/s 2 

Note that Eq. (6) is not used; thus, the problem is over-determined.



(b)

Substituting into Eq. (5) aC = 20 − 140 = −120 mm/s 2

and into Eq. (4) −(20 mm/s 2 ) − (40 mm/s 2 ) + 2aD = 0

or

aD = 30 mm/s 2

Now

vC = 0 + aC t

When vC = −600 mm/s: or Also At t = 5 s:

−600 mm/s = ( −120 mm/s 2 )t t =5s yD = ( yD )0 + (0)t + yD − ( yD )0 =

1 aD t 2 2

1 (30 mm/s 2 )(5 s)2 2 y D − (y D )0 = 375 mm 

or


PROBLEM 11.61 A particle moves in a straight line with the acceleration shown in the figure. Knowing that it starts from the origin with v0 = −14 ft/s, plot the v−t and x−t curves for 0 < t < 15 s and determine (a) the maximum value of the velocity of the particle, (b) the maximum value of its position coordinate.

SOLUTION

v0 = −14 ft/s

Change in v = area under a−t curve.

t = 0 to t = 2 s:

v2 − v0 = (3 ft/s 2 )(2 s) = +6 ft/s

v2 = −8 ft/s

t = 2 s to t = 5 s:

v5 − v2 = (8 ft/s2 )(3 s) = +24 ft/s

v5 = +16 ft/s

t = 5 s to t = 8 s:

v8 − v5 = (3 ft/s 2 )(3 s) = +9 ft/s

v8 = +25 ft/s

t = 8 s to t = 10 s:

v10 − v8 = (−5 ft/s2 )(2 s) = −10 ft/s

v10 = +15 ft/s

t = 10 s to t = 15 s:

v15 − v10 = (−5 ft/s2 )(5 s) = −25 ft/s

v15 = −10 ft/s



Plot v−t curve. Then by similar triangles Δ’s find t for v = 0.

x0 = 0

Change in x = area under v−t curve

t = 0 to t = 2 s:

1 x2 − x0 = (−14 − 8)(2) = −22 ft 2

x2 = −22 ft

t = 2 s to t = 3 s:

1 x3 − x2 = (−8)(1) = −4 ft 2

x8 = −26 ft

t = 3 s to t = 5 s:

1 x5 − x3 = (+16)(2) = +16 ft 2

x5 = −10 ft

t = 5 s to t = 8 s:

1 x8 − x5 = (+16 + 25)(3) = +61.5 ft 2

x8 = +51.6 ft

t = 8 s to t = 10 s:

1 x10 − x8 = (+25 + 15)(2) = + 40 ft 2

x10 = +91.6 ft

t = 10 s to t = 13 s:

1 x13 − x10 = (+15)(3) = +22.5 ft 2

x13 = +114 ft

t = 13 s to t = 15 s:

1 x15 − x13 = (−10)(2) = −10 ft 2

x15 = +94 ft

(a)

Maximum velocity: When t = 8 s,

(b)

Maximum x: When t = 13 s,

vm = 25.0 ft/s  xm = 114.0 ft 


PROBLEM 11.62 For the particle and motion of Problem 11.61, plot the v−t and x−t curves for 0 < t < 15 s and determine the velocity of the particle, its position, and the total distance traveled after 10 s. PROBLEM 11.61 A particle moves in a straight line with the acceleration shown in the figure. Knowing that it starts from the origin with v0 = −14 ft/s, plot the v−t and x−t curves for 0 < t < 15 s and determine (a) the maximum value of the velocity of the particle, (b) the maximum value of its position coordinate.

SOLUTION

v0 = −14 ft/s

Change in v = area under a−t curve.

t = 0 to t = 2 s:

v2 − v0 = (3 ft/s 2 )(2 s) = +6 ft/s

v2 = −8 ft/s

t = 2 s to t = 5 s:

v5 − v2 = (8 ft/s2 )(3 s) = +24 ft/s

v5 = +16 ft/s

t = 5 s to t = 8 s:

v8 − v5 = (3 ft/s 2 )(3 s) = +9 ft/s

v8 = +25 ft/s

t = 8 s to t = 10 s:

v10 − v8 = (−5 ft/s2 )(2 s) = −10 ft/s

v10 = +15 ft/s

t = 10 s to t = 15 s:

v15 − v10 = (−5 ft/s2 )(5 s) = −25 ft/s

v15 = −10 ft/s



Plot v−t curve. Then by similar triangles Δ’s find t for v = 0.

x0 = 0

Change in x = area under v−t curve

t = 0 to t = 2 s:

1 x2 − x0 = (−14 − 8)(2) = −22 ft 2

x2 = −22 ft

t = 2 s to t = 3 s:

1 x3 − x2 = (−8)(1) = −4 ft 2

x8 = −26 ft

t = 3 s to t = 5 s:

1 x5 − x3 = (+16)(2) = +16 ft 2

x5 = −10 ft

t = 5 s to t = 8 s:

1 x8 − x5 = (+16 + 25)(3) = +61.5 ft 2

x8 = +51.6 ft

t = 8 s to t = 10 s:

1 x10 − x8 = (+25 + 15)(2) = + 40 ft 2

x10 = +91.6 ft

t = 10 s to t = 13 s:

1 x13 − x10 = (+15)(3) = +22.5 ft 2

x13 = +114 ft

t = 13 s to t = 15 s:

1 x15 − x13 = (−10)(2) = −10 ft 2

x15 = +94 ft

when t = 10 s:

v10 = +15 ft/s  x10 = +91.5 ft/s 

Distance traveled: t = 0 to t = 105 t = 0 to t = 3 s:

Distance traveled = 26 ft

t = 3 s to t = 10 s

Distance traveled = 26 ft + 91.5 ft = 117.5 ft Total distance traveled = 26 + 117.5 = 143.5 ft 


PROBLEM 11.63 A particle moves in a straight line with the velocity shown in the figure. Knowing that x = −540 m at t = 0, (a) construct the a −t and x−t curves for 0 < t < 50 s, and determine (b) the total distance traveled by the particle when t = 50 s, (c) the two times at which x = 0.

SOLUTION (a)

at = slope of v −t curve at time t

From t = 0 to t = 10 s:

v = constant  a = 0 −20 − 60 = −5 m/s 2 26 − 10

t = 10 s to t = 26 s:

a=

t = 26 s to t = 41 s:

v = constant  a = 0

t = 41 s to t = 46 s:

a=

t = 46 s:

−5 − (−20) = 3 m/s 2 46 − 41


x2 = x1 + (area under v −t curve from t1 to t2 )

At t = 10 s:

x10 = −540 + 10(60) + 60 m

Next, find time at which v = 0. Using similar triangles tv = 0 − 10 60

At

t = 22 s: t = 26 s: t = 41 s: t = 46 s: t = 50 s:

=

26 − 10 80

or

tv = 0 = 22 s

1 x22 = 60 + (12)(60) = 420 m 2 1 x26 = 420 − (4)(20) = 380 m 2 x41 = 380 − 15(20) = 80 m  20 + 5  x46 = 80 − 5   = 17.5 m  2  x50 = 17.5 − 4(5) = −2.5 m



(b)

From t = 0 to t = 22 s: Distance traveled = 420 − (−540) = 960 m t = 22 s to t = 50 s: Distance traveled = |− 2.5 − 420| = 422.5 m Total distance traveled = (960 + 422.5) ft = 1382.5 m Total distance traveled = 1383 m 

(c)

Using similar triangles Between 0 and 10 s:

(t x = 0 )1 − 0 10 = 540 600 (t x = 0 )1 = 9.00 s 

Between 46 s and 50 s:

(t x = 0 )2 − 46 4 = 17.5 20 (t x =0 )2 = 49.5 s 


PROBLEM 11.64 A particle moves in a straight line with the velocity shown in the figure. Knowing that x = −540 m at t = 0, (a) construct the a −t and x −t curves for 0 < t < 50 s, and determine (b) the maximum value of the position coordinate of the particle, (c) the values of t for which the particle is at x = 100 m.

SOLUTION (a)

at = slope of v −t curve at time t v = constant  a = 0

From t = 0 to t = 10 s:

−20 − 60 = −5 m/s 2 26 − 10

t = 10 s to t = 26 s:

a=

t = 26 s to t = 41 s:


t = 41 s to t = 46 s:

a=

−5 − (−20) = 3 m/s 2 46 − 41


t = 46 s:

x2 = x1 + (area under v −t curve from t1 to t2 )

At t = 10 s:

x10 = −540 + 10(60) = 60 m

Next, find time at which v = 0. Using similar triangles tv = 0 − 10 60

At

t = 22 s: t = 26 s: t = 41 s: t = 46 s: t = 50 s:

=

26 − 10 80

or

tv = 0 = 22 s

1 x22 = 60 + (12)(60) = 420 m 2 1 x26 = 420 − (4)(20) = 380 m 2 x41 = 380 − 15(20) = 80 m  20 + 5  x46 = 80 − 5   = 17.5 m  2  x50 = 17.5 − 4(5) = −2.5 m



(b)

Reading from the x −t curve

(c)

Between 10 s and 22 s

xmax = 420 m 

100 m = 420 m − (area under v −t curve from t , to 22 s) m 100 = 420 −

1 (22 − t1 )(v1 ) 2

(22 − t1 )(v1 ) = 640

Using similar triangles v1 60 = 22 − t1 12

Then

v1 = 5(22 − t1 )

or

(22 − t1 )[5(22 − t1 )] = 640 t1 = 10.69 s and t1 = 33.3 s 10 s < t1 < 22 s 

We have

t1 = 10.69 s 

Between 26 s and 41 s: Using similar triangles 41 − t2 15 = 20 300 t2 = 40.0 s 


PROBLEM 11.65 During a finishing operation the bed of an industrial planer moves alternately 30 in. to the right and 30 in. to the left. The velocity of the bed is limited to a maximum value of 6 in./s to the right and 12 in./s to the left; the acceleration is successively equal to 6 in./s2 to the right, zero 6 in./s2 to the left, zero, etc. Determine the time required for the bed to complete a full cycle, and draw the v−t and x−t curves.

SOLUTION We choose positive to the right, thus the range of permissible velocities is −12 in./s < v < 6 in./s since acceleration is −6 in./s 2 , 0, or + 6 in./s 2 . The slope the v−t curve must also be −6 in./s2, 0, or +6 in./s2.

Planer moves = 30 in. to right: +30 in. = 3 + 6t1 + 3

t1 = 4.00 s

Planer moves = 30 in. to left: −30 in. = −12 − 12t2 + 12

t2 = 0.50 s

Total time = 1 s + 4 s + 1 s + 2 s + 0.5 s + 2 s = 10.5 s

ttotal = 10.50 s 


PROBLEM 11.66 A parachutist is in free fall at a rate of 200 km/h when he opens his parachute at an altitude of 600 m. Following a rapid and constant deceleration, he then descends at a constant rate of 50 km/h from 586 m to 30 m, where he maneuvers the parachute into the wind to further slow his descent. Knowing that the parachutist lands with a negligible downward velocity, determine (a) the time required for the parachutist to land after opening his parachute, (b) the initial deceleration.

SOLUTION Assume second deceleration is constant. Also, note that 200 km/h = 55.555 m/s, 50 km/h = 13.888 m/s



(a)

Now Δ x = area under v −t curve for given time interval Then  55.555 + 13.888  (586 − 600) m = −t1   m/s 2  

t1 = 0.4032 s

(30 − 586) m = −t2 (13.888 m/s) t2 = 40.0346 s 1 (0 − 30) m = − (t3 )(13.888 m/s) 2 t3 = 4.3203 s ttotal = (0.4032 + 40.0346 + 4.3203) s

(b)

We have

ainitial = =

ttotal = 44.8 s 

Δ vinitial t1 [−13.888 − (−55.555)] m/s 0.4032 s

= 103.3 m/s 2 ainitial = 103.3 m/s 2 


PROBLEM 11.67 A commuter train traveling at 40 mi/h is 3 mi from a station. The train then decelerates so that its speed is 20 mi/h when it is 0.5 mi from the station. Knowing that the train arrives at the station 7.5 min after beginning to decelerate and assuming constant decelerations, determine (a) the time required for the train to travel the first 2.5 mi, (b) the speed of the train as it arrives at the station, (c) the final constant deceleration of the train.

SOLUTION Given: At t = 0, v = 40 mi/h, x = 0; when x = 2.5 mi, v = 20 mi/h; at t = 7.5 min, x = 3 mi; constant decelerations. The v −t curve is first drawn as shown. (a)

A1 = 2.5 mi

We have

1h  40 + 20  (t1 min)  mi/h × = 2.5 mi  60 min  2  t1 = 5.00 min 

(b)

A2 = 0.5 mi

We have  20 + v2 (7.5 − 5) min ×   2

1h   mi/h × 60 min = 0.5 mi  v2 = 4.00 mi/h 

(c)

We have

afinal = a12 =

(4 − 20) mi/h 5280 ft 1 min 1h × × × (7.5 − 5) min mi 60 s 3600 s afinal = −0.1564 ft/s 2 


PROBLEM 11.68 A temperature sensor is attached to slider AB which moves back and forth through 60 in. The maximum velocities of the slider are 12 in./s to the right and 30 in./s to the left. When the slider is moving to the right, it accelerates and decelerates at a constant rate of 6 in./s2; when moving to the left, the slider accelerates and decelerates at a constant rate of 20 in./s2. Determine the time required for the slider to complete a full cycle, and construct the v – t and x −t curves of its motion.

SOLUTION The v −t curve is first drawn as shown. Then ta =

vright aright

=

12 in./s =2s 6 in./s 2

vleft 30 in./s = aleft 20 in./s = 1.5 s

td =

A1 = 60 in.

Now

[(t1 − 2) s](12 in./s) = 60 in.

or

t1 = 7 s

or

A2 = 60 in.

and or

{[(t2 − 7) − 1.5] s}(30 in./s) = 60 in.

or

t2 = 10.5 s tcycle = t2

Now

tcycle = 10.5 s 

We have xii = xi + (area under v −t curve from ti to tii ) At

t = 5 s:

1 (2) (12) = 12 in. 2 x5 = 12 + (5 − 2)(12)

t = 7 s:

= 48 in. x7 = 60 in.

t = 2 s:

t = 8.5 s: t = 9 s: t = 10.5 s:

x2 =

1 x8.5 = 60 − (1.5)(30) 2 = 37.5 in. x9 = 37.5 − (0.5)(30)

= 22.5 in. x10.5 = 0


PROBLEM 11.69 In a water-tank test involving the launching of a small model boat, the model’s initial horizontal velocity is 6 m/s, and its horizontal acceleration varies linearly from −12 m/s 2 at t = 0 to −2 m/s 2 at t = t1 and then remains equal to −2 m/s 2 until t = 1.4 s. Knowing that v = 1.8 m/s when t = t1 , determine (a) the value of t1 , (b) the velocity and the position of the model at t = 1.4 s.

SOLUTION Given:

v0 = 6 m/s; for 0 < t < t1 ,

for

t1 < t < 1.4 s a = −2 m/s 2 ;

at

t = 0 a = −12 m/s 2 ;

at t = t1

a = −2 m/s 2 , v = 1.8 m/s 2

The a −t and v −t curves are first drawn as shown. The time axis is not drawn to scale. (a)

vt1 = v0 + A1

We have

 12 + 2  1.8 m/s = 6 m/s − (t1 s)  m/s2   2  t1 = 0.6 s 

(b)

v1.4 = vt1 + A2

We have

v1.4 = 1.8 m/s − (1.4 − 0.6) s × 2 m/s 2 v1.4 = 0.20 m/s 

Now x1.4 = A3 + A4 , where A3 is most easily determined using integration. Thus, a=

for 0 < t < t1:

−2 − (−12) 50 t − 12 = t − 12 0.6 3

dv 50 = a = t − 12 dt 3

Now



At t = 0, v = 6 m/s: or



v 6

dv =



t

50  t − 12  dt  0 3 

v=6+

25 2 t − 12t 3

We have

dx 25 = v = 6 − 12t + t 2 dt 3

Then

A3 =



xt1 0

dx =



0.6 0

(6 − 12t +

25 2 t )dt 3

0.6

25   = 6t − 6t 2 + t 3  = 2.04 m 9 0 

Also

 1.8 + 0.2  A4 = (1.4 − 0.6)   = 0.8 m 2  

Then

x1.4 = (2.04 + 0.8) m x1.4 = 2.84 m 

or


PROBLEM 11.70 The acceleration record shown was obtained for a small airplane traveling along a straight course. Knowing that x = 0 and v = 60 m/s when t = 0, determine (a) the velocity and position of the plane at t = 20 s, (b) its average velocity during the interval 6 s < t < 14 s.

SOLUTION

Geometry of “bell-shaped” portion of v −t curve

The parabolic spandrels marked by * are of equal area. Thus, total area of shaded portion of v −t diagram is: = Δx = 6 m

(a)

When t = 20 s:

v20 = 60 m/s  x20 = (60 m/s) (20 s) − (shaded area)

= 1200 m − 6 m (b)

From t = 6 s to t = 14 s:

x20 = 1194 m 

Δt = 8 s

Δ x = (60 m/s)(14 s − 6 s) − (shaded area) = (60 m/s)(8 s) − 6 m = 480 m − 6 m = 474 m vaverage =

Δ x 474 m = Δt 8s

vaverage = 59.25 m/s 


PROBLEM 11.71 In a 400-m race, runner A reaches her maximum velocity v A in 4 s with constant acceleration and maintains that velocity until she reaches the half-way point with a split time of 25 s. Runner B reaches her maximum velocity vB in 5 s with constant acceleration and maintains that velocity until she reaches the half-way point with a split time of 25.2 s. Both runners then run the second half of the race with the same constant deceleration of 0.1 m/s 2. Determine (a) the race times for both runners, (b) the position of the winner relative to the loser when the winner reaches the finish line.

SOLUTION Sketch v −t curves for first 200 m. Runner A:

t1 = 4 s, t2 = 25 − 4 = 21 s A1 =

1 (4)(v A )max = 2(vA ) max 2

A2 = 21(v A )max A1 + A2 = Δ x = 200 m 23(v A ) max = 200

Runner B:

or

t1 = 5 s, A1 =

(v A ) max = 8.6957 m/s

t2 = 25.2 − 5 = 20.2 s

1 (5)(vB ) max = 2.5(vB ) max 2

A2 = 20.2(vB ) max A1 + A2 = Δ x = 200 m 22.7(vB ) max = 200

Sketch v −t curve for second 200 m. A3 = vmaxt3 − t3 =

Runner A:

or

(vB ) max = 8.8106 m/s

Δv = | a |t3 = 0.1t3 1 Δvt3 = 200 2

or

0.05t32 − vmaxt3 + 200 = 0

(

vmax ± (vmax ) 2 − (4)(0.05)(200) = 10 vmax ± (vmax )2 − 40 (2)(0.05)

(vmax ) A = 8.6957,

Reject the larger root. Then total time

(t3 ) A = 146.64 s

and

)

27.279 s

t A = 25 + 27.279 = 52.279 s

t A = 52.2 s 



Runner B: (vmax ) B = 8.8106, (t3 ) B = 149.45 s

and

26.765 s t B = 25.2 + 26.765 = 51.965 s

Reject the larger root. Then total time

t B = 52.0 s 

Velocity of A at t = 51.965 s: v1 = 8.6957 − (0.1)(51.965 − 25) = 5.999 m/s

Velocity of A at t = 51.279 s: v2 = 8.6957 − (0.1)(52.279 − 25) = 5.968 m/s

Over 51.965 s ≤ t ≤ 52.965 s, runner A covers a distance Δ x Δ x = vave (Δt ) =

1 (5.999 + 5.968)(52.279 − 51.965) 2

Δ x = 1.879 m 


PROBLEM 11.72 A car and a truck are both traveling at the constant speed of 35 mi/h; the car is 40 ft behind the truck. The driver of the car wants to pass the truck, i.e., he wishes to place his car at B, 40 ft in front of the truck, and then resume the speed of 35 mi/h. The maximum acceleration of the car is 5 ft/s2 and the maximum deceleration obtained by applying the brakes is 20 ft/s 2 . What is the shortest time in which the driver of the car can complete the passing operation if he does not at any time exceed a speed of 50 mi/h? Draw the v −t curve.

SOLUTION Relative to truck, car must move a distance:

Allowable increase in speed:

Δ x = 16 + 40 + 50 + 40 = 146 ft Δvm = 50 − 35 = 15 mi/h = 22 ft/s

Acceleration Phase:

t1 = 22/5 = 4.4 s

A1 =

1 (22)(4.4) = 48.4 ft 2

Deceleration Phase:

t3 = 22/20 = 1.1 s

A3 =

1 (22)(1.1) = 12.1 ft 2

But: Δ x = A1 + A2 + A3 :

146 ft = 48.4 + (22)t2 + 12.1 ttotal = t1 + t2 + t3 = 4.4 s + 3.89 s + 1.1 s = 9.39 s

t2 = 3.89 s t B = 9.39 s 


PROBLEM 11.73 Solve Problem 11.72, assuming that the driver of the car does not pay any attention to the speed limit while passing and concentrates on reaching position B and resuming a speed of 35 mi/h in the shortest possible time. What is the maximum speed reached? Draw the v −t curve. PROBLEM 11.72 A car and a truck are both traveling at the constant speed of 35 mi/h; the car is 40 ft behind the truck. The driver of the car wants to pass the truck, i.e., he wishes to place his car at B, 40 ft in front of the truck, and then resume the speed of 35 mi/h. The maximum acceleration of the car is 5 ft/s2 and the maximum deceleration obtained by applying the brakes is 20 ft/s 2 . What is the shortest time in which the driver of the car can complete the passing operation if he does not at any time exceed a speed of 50 mi/h? Draw the v −t curve.

SOLUTION Relative to truck, car must move a distance: Δ x = 16 + 40 + 50 + 40 = 146 ft

Δvm = 5t1 = 20t2 ; Δ x = A1 + A2 :

t2 =

146 ft =

1 (Δvm )(t1 + t2 ) 2

146 ft =

1 1   (5t1 )  t1 + t1  2 4  

t12 = 46.72

1 t1 4

t1 = 6.835 s

t2 =

1 t1 = 1.709 4

ttotal = t1 + t2 = 6.835 + 1.709

t B = 8.54 s 

Δvm = 5t1 = 5(6.835) = 34.18 ft/s = 23.3 mi/h Speed vtotal = 35 mi/h, vm = 35 mi/h + 23.3 mi/h

vm = 58.3 mi/h 


PROBLEM 11.74 Car A is traveling on a highway at a constant speed (v A )0 = 60 mi/h, and is 380 ft from the entrance of an access ramp when car B enters the acceleration lane at that point at a speed (vB )0 = 15 mi/h. Car B accelerates uniformly and enters the main traffic lane after traveling 200 ft in 5 s. It then continues to accelerate at the same rate until it reaches a speed of 60 mi/h, which it then maintains. Determine the final distance between the two cars.

SOLUTION Given:

(v A )0 = 60 mi/h, (vB )0 = 1.5 mi/h; at t = 0, ( x A )0 = −380 ft, ( xB ) 0 = 0; at t = 5 s, xB = 200 ft; for 15 mi/h < vB ≤ 60 mi/h, aB = constant; for vB = 60 mi/h, aB = 0

First note

60 mi/h = 88 ft/s 15 mi/h = 22 ft/s

The v −t curves of the two cars are then drawn as shown. Using the coordinate system shown, we have at t = 5 s, xB = 200 ft:

 22 +(vB )5  (5 s)   ft/s = 200 ft 2   (vB )5 = 58 ft/s

or Then, using similar triangles, we have

(88 − 22) ft/s (58 − 22) ft/s = ( = aB ) t1 5s

or

t1 = 9.16 67 s

Finally, at t = t1    22 + 88  ft/s  xB/A = xB − x A =  (9.1667 s)    2    − [−380 ft + (9.1667 s) (88 ft/s)]

xB/A = 77.5 ft 

or


PROBLEM 11.75 An elevator starts from rest and moves upward, accelerating at a rate of 1.2 m/s2 until it reaches a speed of 7.8 m/s, which it then maintains. Two seconds after the elevator begins to move, a man standing 12 m above the initial position of the top of the elevator throws a ball upward with an initial velocity of 20 m/s. Determine when the ball will hit the elevator.

SOLUTION Given:

At t = 0 vE = 0; For 0 < vE ≤ 7.8 m/s, aE = 1.2 m/s 2 ↑; For vE = 7.8 m/s, aE = 0; At t = 2 s, vB = 20m/s ↑

The v −t curves of the ball and the elevator are first drawn as shown. Note that the initial slope of the curve for the elevator is 1.2 m/s2 , while the slope of the curve for the ball is − g (−9.81 m/s 2 ).

The time t1 is the time when vE reaches 7.8 m/s. Thus,

vE = (0) + aE t

or

7.8 m/s = (1.2 m/s 2 )t1

or

t1 = 6.5 s

The time ttop is the time at which the ball reaches the top of its trajectory. Thus, or or

vB = (vB )0 − g (t − 2)

0 = 20 m/s − (9.81 m/s 2 ) (t top − 2) s ttop = 4.0387 s



Using the coordinate system shown, we have 0 < t < t1 :

1  yE = −12 m +  aE t 2  m 2 

At t = ttop :

yB =

and

y E = −12 m +

At and at t = t1 ,

1 (4.0387 − 2) s × (20 m/s) 2 = 20.387 m 1 (1.2 m/s 2 )(4.0387 s) 2 2 = −2.213 m

t = [2 + 2(4.0387 − 2)] s = 6.0774 s, yB = 0 yE = −12 m +

1 (6.5 s) (7.8 m/s) = 13.35 m 2

The ball hits the elevator ( yB = yE ) when ttop ≤ t ≤ t1. For t ≥ ttop :

1  yB = 20.387 m −  g (t − ttop )2  m 2  

Then, when

yB = yE 1 20.387 m − (9.81 m/s 2 ) (t − 4.0387)2 2 1 = −12 m + (1.2 m/s 2 ) (t s)2 2

or Solving

5.505t 2 − 39.6196t + 47.619 = 0 t = 1.525 s and t = 5.67 s

t = 5.67 s 

Since 1.525 s is less than 2 s,


PROBLEM 11.76 Car A is traveling at 40 mi/h when it enters a 30 mi/h speed zone. The driver of car A decelerates at a rate of 16 ft/s2 until reaching a speed of 30 mi/h, which she then maintains. When car B, which was initially 60 ft behind car A and traveling at a constant speed of 45 mi/h, enters the speed zone, its driver decelerates at a rate of 20 ft/s2 until reaching a speed of 28 mi/h. Knowing that the driver of car B maintains a speed of 28 mi/h, determine (a) the closest that car B comes to car A, (b) the time at which car A is 70 ft in front of car B.

SOLUTION (v A )0 = 40 mi/h; For 30 mi/h < v A ≤ 40 mi/h, a A = −16 ft/s 2 ; For v A = 30 mi/h, a A = 0;

Given:

( x A /B )0 = 60 ft; (vB )0 = 45 mi/h; When xB = 0, aB = −20 ft/s 2 ; For vB = 28 mi/h, aB = 0

First note

40 mi/h = 58.667 ft/s 30 mi/h = 44 ft/s 45 mi/h = 66 ft/s 28 mi/h = 41.067 ft/s

At t = 0

The v −t curves of the two cars are as shown. At

t = 0:

Car A enters the speed zone.

t = (tB )1:

Car B enters the speed zone.

t = tA:

Car A reaches its final speed.

t = tmin :

v A = vB

t = (t B )2 :

Car B reaches its final speed.



(a)

(v A )final − (v A )0 tA

aA =

We have

(44 − 58.667) ft/s tA

−16 ft/s 2 =

or

t A = 0.91669 s

or Also

60 ft = (tB )1 (vB )0

or

60 ft = (tB )1 (66 ft/s) aB =

and

−20 ft/s 2 =

or

or

(t B )1 = 0.90909 s

(vB )final − (vB )0 (t B ) 2 − (t B )1 (41.067 − 66) ft/s [(t B )2 − 0.90909] s

Car B will continue to overtake car A while vB > v A . Therefore, ( x A/B ) min will occur when v A = vB , which occurs for (tB )1 < tmin < (tB ) 2

For this time interval v A = 44 ft/s vB = (vB )0 + aB [t − (tB )1 ]

Then

at t = tmin :

44 ft/s = 66 ft/s + (−20 ft/s2 )(tmin − 0.90909) s tmin = 2.00909 s

or

Finally ( x A/B )min = ( x A )tmin − ( xB )tmin   (v ) + (v A )final   = t A  A 0 + (tmin − t A )(v A )final   2       (v ) + (v A )final   − ( xB )0 + (t B )1 (vB )0 + [tmin − (t B )1 ]  B 0  2   

   58.667 + 44  = (0.91669 s)  ft/s + (2.00909 − 0.91669) s × (44 ft/s)   2       66 + 44   −  −60 ft + (0.90909 s)(66 ft/s) + (2.00909 − 0.90909) s ×   ft/s   2    = (47.057 + 48.066) ft − (−60 + 60.000 + 60.500) ft

= 34.623 ft

or ( x A/B ) min = 34.6 ft 



(b)

Since ( x A/B ) ≤ 60 ft for t ≤ tmin , it follows that x A/B = 70 ft for t > (tB ) 2 [Note (tB ) 2  tmin ]. Then, for t > (tB ) 2 x A/B = ( x A/B )min + [(t − tmin )(vA )final ]   + (vB )final   (v ) − [(t B )2 − (tmin )]  A final + [t − (t B )2 ](vB )final   2    

or

70 ft = 34.623 ft + [(t − 2.00909) s × (44 ft/s)]    44 + 41.06  − (2.15574 − 2.00909) s ×   ft/s + (t − 2.15574) s × (41.067) ft/s  2     t = 14.14 s 

or


PROBLEM 11.77 An accelerometer record for the motion of a given part of a mechanism is approximated by an arc of a parabola for 0.2 s and a straight line for the next 0.2 s as shown in the figure. Knowing that v = 0 when t = 0 and x = 0.8 ft when t = 0.4 s, (a) construct the v −t curve for 0 ≤ t ≤ 0.4 s, (b) determine the position of the part at t = 0.3 s and t = 0.2 s.

SOLUTION Divide the area of the a −t curve into the four areas A1, A2 , A3 and A4. 2 (8)(0.2) = 1.0667 ft/s 3 A2 = (16)(0.2) = 3.2 ft/s A1 =

1 (16 + 8)(0.1) = 1.2 ft/s 2 1 A4 = (8)(0.1) = 0.4 ft/s 2 A3 =

Velocities: v0 = 0 v0.2 = v0 + A1 + A2

v0.2 = 4.27 ft/s 

v0.3 = v0.2 + A3

v0.3 = 5.47 ft/s 

v0.4 = v0.3 + A4

v0.4 = 5.87 ft/s 

Sketch the v −t curve and divide its area into A5 , A6 , and A7 as shown. 0.8 0.4  x dx = 0.8 − x =  t vdt

2 (0.4)(0.1) = 0.0267 ft, 3

x0.3 = 0.227 ft 

x0.2 = 0.8 − ( A5 + A6 ) − A7

At t = 0.2 s, With A5 + A6 = and

0.4

x = 0.8 −  t vdt

x0.3 = 0.8 − A5 − (5.47)(0.1)

At t = 0.3 s, With A5 =

or

2 (1.6)(0.2) = 0.2133 ft, 3

A7 = (4.27)(0.2) = 0.8533 ft

x0.2 = 0.8 − 0.2133 − 0.8533

x0.2 = −0.267 ft 


PROBLEM 11.78 A car is traveling at a constant speed of 54 km/h when its driver sees a child run into the road. The driver applies her brakes until the child returns to the sidewalk and then accelerates to resume her original speed of 54 km/h; the acceleration record of the car is shown in the figure. Assuming x = 0 when t = 0, determine (a) the time t1 at which the velocity is again 54 km/h, (b) the position of the car at that time, (c) the average velocity of the car during the interval 1 s ≤ t ≤ t1.

SOLUTION Given:

At

t = 0, x = 0, v = 54 km/h;

For t = t1 ,

v = 54 km/h

First note (a)

54 km/h = 15 m/s vb = va + (area under a −t curve from ta to tb )

We have Then

at

t = 2 s:

v = 15 − (1)(6) = 9 m/s

t = 4.5 s:

1 v = 9 − (2.5)(6) = 1.5 m/s 2

t = t1 :

15 = 1.5 +

1 (t1 − 4.5)(2) 2 t1 = 18.00 s 

or (b)

Using the above values of the velocities, the v −t curve is drawn as shown.



Now

x at t = 18 s x18 = 0 + Σ (area under the v −t curve from t = 0 to t = 18 s)  15 + 9  = (1 s)(15 m/s) + (1 s)   m/s  2  1   + (2.5 s)(1.5 m/s)+ (2.5 s)(7.5 m/s)  3   2   + (13.5 s)(1.5 m/s) + (13.5 s)(13.5 m/s)  3   = [15 + 12 + (3.75 + 6.25) + (20.25 + 121.50)] m

= 178.75 m (c)

First note

or

x18 = 178.8 m 

x1 = 15 m x18 = 178.75 m

Now

vave =

Δ x (178.75 − 15) m = = 9.6324 m/s Δt (18 − 1) s vave = 34.7 km/h 

or


PROBLEM 11.79 An airport shuttle train travels between two terminals that are 1.6 mi apart. To maintain passenger comfort, the acceleration of the train is limited to ±4 ft/s2, and the jerk, or rate of change of acceleration, is limited to ±0.8 ft/s2 per second. If the shuttle has a maximum speed of 20 mi/h, determine (a) the shortest time for the shuttle to travel between the two terminals, (b) the corresponding average velocity of the shuttle.

SOLUTION xmax = 1.6 mi; | amax | = 4 ft/s 2

Given:

 da  = 0.8 ft/s2 /s; vmax = 20 mi/h    dt max

First note (a)

20 mi/h = 29.333 ft/s 1.6 mi = 8448 ft

To obtain tmin , the train must accelerate and decelerate at the maximum rate to maximize the time for which v = vmax . The time Δ t required for the train to have an acceleration of 4 ft/s2 is found from a  da  = max  dt  Δt  max

or or

4 ft/s2 0.8 ft/s2 /s Δt = 5 s Δt =

Now, da    since dt = constant   

after 5 s, the speed of the train is

v5 =

1 (Δt )(amax ) 2

or

v5 =

1 (5 s)(4 ft/s 2 ) = 10 ft/s 2

Then, since v5 < vmax , the train will continue to accelerate at 4 ft/s2 until v = vmax . The a −t curve must then have the shape shown. Note that the magnitude of the slope of each inclined portion of the curve is 0.8 ft/s2/s.



Now

at t = (10 + Δt1 ) s, v = vmax : 1  2  (5 s)(4 ft/s2 )  + (Δt1 )(4 ft/s2 ) = 29.333 ft/s 2 

or

Δt1 = 2.3333 s

Then

at t = 5 s: t = 7.3333 s:

1 (5)(4) = 10 ft/s 2 v = 10 + (2.3333)(4) = 19.3332 ft/s

v =0+

1 t = 12.3333 s: v = 19.3332 + (5)(4) = 29.3332 ft/s 2

Using symmetry, the v −t curve is then drawn as shown.

Noting that A1 = A2 = A3 = A4 and that the area under the v −t curve is equal to xmax , we have   10 + 19.3332   2 (2.3333 s)   ft/s  2     + (10 + Δt2 ) s × (29.3332 ft/s) = 8448 ft

or

Δt2 = 275.67 s

Then

tmin = 4(5 s) + 2(2.3333 s) + 275.67 s

= 300.34 s tmin = 5.01 min 

or (b)

We have

vave =

Δx 1.6 mi 3600 s = × Δt 300.34 s 1h vave = 19.18 mi/h 

or


PROBLEM 11.80 During a manufacturing process, a conveyor belt starts from rest and travels a total of 1.2 ft before temporarily coming to rest. Knowing that the jerk, or rate of change of acceleration, is limited to ±4.8 ft/s2 per second, determine (a) the shortest time required for the belt to move 1.2 ft, (b) the maximum and average values of the velocity of the belt during that time.

SOLUTION Given:

(a)

At

t = 0, x = 0, v = 0; xmax = 1.2 ft;

when

 da  x = xmax , v = 0;   = 4.8 ft/s 2  dt max

Observing that vmax must occur at t = 12 tmin , the a −t curve must have the shape shown. Note that the magnitude of the slope of each portion of the curve is 4.8 ft/s2/s.

We have

at t = Δt : t = 2Δt :

1 1 v = 0 + (Δt )(amax ) = amax Δt 2 2 vmax =

1 1 amax Δt + ( Δt )( amax ) = amax Δt 2 2

Using symmetry, the v −t is then drawn as shown.

Noting that A1 = A2 = A3 = A4 and that the area under the v −t curve is equal to xmax , we have (2Δt )(vmax ) = xmax vmax = amax Δt  2amax Δt 2 = xmax



Now

amax = 4.8 ft/s2 /s so that Δt

2(4.8Δt ft/s3 )Δt 2 = 1.2 ft or Then

Δt = 0.5 s tmin = 4Δt tmin = 2.00 s 

or (b)

We have

vmax = amax Δt = (4.8 ft/s 2 /s × Δt)Δt = 4.8 ft/s 2 /s × (0.5 s) 2

vmax = 1.2 ft/s 

or Also

vave =

Δx 1.2 ft = Δttotal 2.00 s vave = 0.6 ft/s 

or


PROBLEM 11.81 Two seconds are required to bring the piston rod of an air cylinder to rest; the acceleration record of the piston rod during the 2 s is as shown. Determine by approximate means (a) the initial velocity of the piston rod, (b) the distance traveled by the piston rod as it is brought to rest.

SOLUTION a −t curve; at

Given: 1.

t = 2 s, v = 0

The a −t curve is first approximated with a series of rectangles, each of width Δt = 0.25 s. The area (Δt)(aave) of each rectangle is approximately equal to the change in velocity Δv for the specified interval of time. Thus, Δv ≅ aave Δt

where the values of aave and Δv are given in columns 1 and 2, respectively, of the following table. 2.

v(2) = v0 +

Now and approximating the area



2 0



2 0

a dt = 0

a dt under the a −t curve by Σaave Δt ≈ ΣΔv, the initial velocity is then

equal to v0 = −ΣΔv

Finally, using v2 = v1 + Δv12

where Δv12 is the change in velocity between times t1 and t2, the velocity at the end of each 0.25 interval can be computed; see column 3 of the table and the v −t curve. 3.

The v −t curve is then approximated with a series of rectangles, each of width 0.25 s. The area (Δt )(vave ) of each rectangle is approximately equal to the change in position Δ x for the specified interval of time. Thus Δ x ≈ vave Δt

where vave and Δ x are given in columns 4 and 5, respectively, of the table.



4.

With x0 = 0 and noting that x2 = x1 + Δ x12

where Δ x12 is the change in position between times t1 and t2, the position at the end of each 0.25 s interval can be computed; see column 6 of the table and the x−t curve.

(a)

We had found

(b)

At t = 2 s

v0 = 1.914 m/s  x = 0.840 m 


PROBLEM 11.82 The acceleration record shown was obtained during the speed trials of a sports car. Knowing that the car starts from rest, determine by approximate means (a) the velocity of the car at t = 8 s, (b) the distance the car has traveled at t = 20 s.

SOLUTION Given: a −t curve; at 1.

t = 0, x = 0, v = 0

The a −t curve is first approximated with a series of rectangles, each of width Δt = 2 s. The area (Δt )(aave ) of each rectangle is approximately equal to the change in velocity Δv for the specified interval of time. Thus, Δv ≅ aave Δt

where the values of aave and Δv are given in columns 1 and 2, respectively, of the following table. 2.

Noting that v0 = 0 and that v2 = v1 + Δv12

where Δv12 is the change in velocity between times t1 and t2, the velocity at the end of each 2 s interval can be computed; see column 3 of the table and the v −t curve. 3.

The v −t curve is next approximated with a series of rectangles, each of width Δt = 2 s. The area (Δt )(vave ) of each rectangle is approximately equal to the change in position Δx for the specified interval of time. Thus,

Δx ≅ vave Δt

where vave and Δx are given in columns 4 and 5, respectively, of the table. 4.

With x0 = 0 and noting that x2 = x1 + Δ x12

where Δ x12 is the change in position between times t1 and t2, the position at the end of each 2 s interval can be computed; see column 6 of the table and the x −t curve.



(a)

At t = 8 s, v = 32.58 m/s

(b)

At t = 20 s

v = 117.3 km/h 

or

x = 660 m 


PROBLEM 11.83 A training airplane has a velocity of 126 ft/s when it lands on an aircraft carrier. As the arresting gear of the carrier brings the airplane to rest, the velocity and the acceleration of the airplane are recorded; the results are shown (solid curve) in the figure. Determine by approximate means (a) the time required for the airplane to come to rest, (b) the distance traveled in that time.

SOLUTION Given: a −v curve: v0 = 126 ft/s

The given curve is approximated by a series of uniformly accelerated motions (the horizontal dashed lines on the figure). For uniformly accelerated motion v22 = v12 + 2a( x2 − x1 ) v2 = v1 + a(t2 − t1 )

v22 − v12 2a v −v Δt = 2 1 a

Δx =

or

For the five regions shown above, we have Region

v1 , ft/s

v2 , ft/s

a, ft/s 2

Δx, ft

Δt , s

1

126

120

−12.5

59.0

0.480

2

120

100

−33

66.7

0.606

3

100

80

−45.5

39.6

0.440

4

80

40

−54

44.4

0.741

5

40

0

−58

13.8

0.690

223.5

2.957

Σ

(a)

From the table, when v = 0

t = 2.96 s 

(b)

From the table and assuming x0 = 0, when v = 0

x = 224 ft 


PROBLEM 11.84 Shown in the figure is a portion of the experimentally determined v − x curve for a shuttle cart. Determine by approximate means the acceleration of the cart (a) when x = 10 in., (b) when v = 80 in./s.

SOLUTION Given: v − x curve

First note that the slope of the above curve is

dv . dx

a=v

(a)

Now

dv dx

When

x = 10 in., v = 55 in./s

Then

 40 in./s  a = 55 in./s    13.5 in.  a = 163.0 in./s 2 

or (b)

When v = 80 in./s, we have  40 in./s  a = 80 in./s    28 in.  a = 114.3 in./s 2 

or

Note: To use the method of measuring the subnormal outlined at the end of Section 11.8, it is necessary that the same scale be used for the x and v axes (e.g., 1 in. = 50 in., 1 in. = 50 in./s). In the above solution, Δv and Δ x were measured directly, so different scales could be used. PROPRIETARY MATERIAL. © 2013 The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission. 121

PROBLEM 11.85 Using the method of Section 11.8, derive the formula x = x0 + v0t + 12 at 2 for the position coordinate of a particle in uniformly accelerated rectilinear motion.

SOLUTION The a −t curve for uniformly accelerated motion is as shown.

Using Eq. (11.13), we have x = x0 + v0t + (area under a −t curve) (t − t )

 1  = x0 + v0t + (t × a)  t − t   2  1 = x0 + v0 t + at 2 Q.E.D. 2




PROBLEM 11.86 Using the method of Section 11.8 determine the position of the particle of Problem 11.61 when t = 8 s. PROBLEM 11.61 A particle moves in a straight line with the acceleration shown in the figure. Knowing that it starts from the origin with v0 = −14 ft/s, plot the v−t and x−t curves for 0 < t < 15 s and determine (a) the maximum value of the velocity of the particle, (b) the maximum value of its position coordinate.

SOLUTION

x0 = 0 v0 = −14 ft/s

when t = 8s: x = x0 + v0t + Σ A(t1 − t ) = 0 − (14 ft/s)(8 s) + [(3 ft/s2 )(2 s)](7 s) + [(8 ft/s2 )(3 s)](4.5 s) + [(3 ft/s)(3 s)](1.5 s) x8 = −112 ft + 42 ft + 108 ft + 13.5 ft

x8 = 51.5 ft 


PROBLEM 11.87 The acceleration of an object subjected to the pressure wave of a large explosion is defined approximately by the curve shown. The object is initially at rest and is again at rest at time t1. Using the method of section 11.8, determine (a) the time t1, (b) the distance through which the object is moved by the pressure wave.

SOLUTION

(a)

Since v = 0 when t = 0 and when t = t1 the change in v between t = 0 and t = t1 is zero. Thus, area under a−t curve is zero A1 + A2 + A3 = 0

1 1 1 (30)(0.6) + ( −10)(0.2) + ( −10)(t1 − 0.8) = 0 2 2 2 9 − 1 − 5t1 + 4 = 0

(b)

t1 = 2.40 s 

Position when t = t1 = 2.4 s 2 x = x0 + v0t1 + A1 (t1 − 0.2) + A2 (t1 − 0.733) + A3   (t1 − 0.8) 3 1 2 = 0 + 0 + (9)(2.4 − 0.2) + ( −1)(2.4 − 0.733) +  ( −10)(2.4 − 0.8)  (2.4 − 0.8) 2 3 = 19.8 m − 1.667 m − 8.533 m

x = 9.60 m 


PROBLEM 11.88 For the particle of Problem 11.63, draw the a −t curve and determine, using the method of Section 11.8, (a) the position of the particle when t = 52 s, (b) the maximum value of its position coordinate. PROBLEM 11.63 A particle moves in a straight line with the velocity shown in the figure. Knowing that x = −540 m at t = 0, (a) construct the a −t and x −t curves for 0 < t < 50 s, and determine (b) the total distance traveled by the particle when t = 50 s, (c) the two times at which x = 0.

SOLUTION a=

We have where

dv dt

dv dt

is the slope of the v −t curve. Then t=0

from

to t = 10 s:


t = 10 s to t = 26 s: a = t = 26 s to t = 41 s:


t = 41 s to t = 46 s: a = t > 46 s:

−20 − 60 = −5 m/s 2 26 − 10

−5 − (−20) = 3 m/s 2 46 − 41


The a −t curve is then drawn as shown. (a)

From the discussion following Eq. (11.13), we have

x = x0 + v0t1 + ΣA(t1 − t )

where A is the area of a region and t is the distance to its centroid. Then, for t1 = 52 s x = −540 m + (60 m/s)(52 s) + {−[(16 s)(5 m/s 2 )](52 − 18) s + [(5 s)(3 m/s 2 )](52 − 43.5)s} = [−540 + (3120) + (−2720 + 127.5)] m x = −12.50 m 

or



(b)

Noting that xmax occurs when v = 0 ( dx = 0 ), it is seen from the v–t curve that xmax occurs for dt 10 s < t < 26 s. Although similar triangles could be used to determine the time at which x = xmax (see the solution to Problem 11.63), the following method will be used. For

10 s < t1 < 26 s, we have x = −540 + 60t1 1  − [(t1 − 10)(5)]  (t1 − 10)  m 2  5 = −540 + 60t1 − (t1 − 10) 2 2

When x = xmax : or Then

dx = 60 − 5(t1 − 10) = 0 dt

(t1 ) xmax = 22 s 5 xmax = −540 + 60(22) − (22 − 10)2 2 xmax = 420 m 

or


PROBLEM 11.CQ3 Two model rockets are fired simultaneously from a ledge and follow the trajectories shown. Neglecting air resistance, which of the rockets will hit the ground first? (a) A (b) B (c) They hit at the same time. (d) The answer depends on h.

SOLUTION The motion in the vertical direction depends on the initial velocity in the y-direction. Since A has a larger initial velocity in this direction it will take longer to hit the ground. Answer: (b) 


PROBLEM 11.CQ4 Ball A is thrown straight up. Which of the following statements about the ball are true at the highest point in its path? (a) The velocity and acceleration are both zero. (b) The velocity is zero, but the acceleration is not zero. (c) The velocity is not zero, but the acceleration is zero. (d) Neither the velocity nor the acceleration are zero.

SOLUTION At the highest point the velocity is zero. The acceleration is never zero. Answer: (b) 


PROBLEM 11.CQ5 Ball A is thrown straight up with an initial speed v0 and reaches a maximum elevation h before falling back down. When A reaches its maximum elevation, a second ball is thrown straight upward with the same initial speed v0. At what height, y, will the balls cross paths? (a)

y=h

(b)

y > h/2

(c)

y = h/2

(d)

y < h/2

(e)

y=0

SOLUTION When the ball is thrown up in the air it will be constantly slowing down until it reaches its apex, at which point it will have a speed of zero. So, the time it will take to travel the last half of the distance to the apex will be longer than the time it takes for the first half. This same argument can be made for the ball falling from the maximum elevation. It will be speeding up, so the first half of the distance will take longer than the second half. Therefore, the balls should cross above the half-way point. Answer: (b) 


PROBLEM 11.CQ6 Two cars are approaching an intersection at constant speeds as shown. What velocity will car B appear to have to an observer in car A? (a)

(b)

(c)

(d )

(e)

SOLUTION Since vB = vA+ vB/A we can draw the vector triangle and see

v B = v A + v B/A

Answer: (e) 


PROBLEM 11.CQ7 Blocks A and B are released from rest in the positions shown. Neglecting friction between all surfaces, which figure below best indicates the direction α of the acceleration of block B? (a)

(b)

(c)

(d)

(e)

SOLUTION    Since aB = a A + aB/A we get

Answer: (d ) 


PROBLEM 11.89 A ball is thrown so that the motion is defined by the equations x = 5t and y = 2 + 6t − 4.9t 2 , where x and y are expressed in meters and t is expressed in seconds. Determine (a) the velocity at t = 1 s, (b) the horizontal distance the ball travels before hitting the ground.

SOLUTION Units are meters and seconds. Horizontal motion:

vx =

dx =5 dt

Vertical motion:

vy =

dy = 6 − 9.8t dt

(a)

Velocity at t = 1 s.

vx = 5 v y = 6 − 9.8 = −3.8

v = vx2 + v y2 = 52 + 3.82 = 6.28 m/s tan θ =

(b)

vy vx

=

−3.8 5

Horizontal distance:

θ = −37.2°

v = 6.28 m/s

37.2° 

( y = 0) y = 2 + 6t − 4.9t 2 t = 1.4971 s x = (5)(1.4971) = 7.4856 m

x = 7.49 m 


PROBLEM 11.90 The motion of a vibrating particle is defined by the position vector r = 10(1 − e−3t )i + (4e −2t sin15t ) j, where r and t are expressed in millimeters and seconds, respectively. Determine the velocity and acceleration when (a) t = 0, (b) t = 0.5 s.

SOLUTION r = 10(1 − e−3t )i + (4e−2t sin15t ) j

Then

v=

and

a=

(a)

dr = 30e−3t i + [60e−2t cos15t − 8e−2t sin15t ]j dt

dv = −90e −3t i + [−120e −2t cos15t − 900e−2t sin15t − 120e−2t cos15t + 16e−2t sin15t ]j dt = −90e −3t i + [−240e −2t cos15t − 884e −2t sin15t ] j

When t = 0: v = 30i + 60 j mm/s

v = 67.1 mm/s

63.4° 

a = −90i − 240 j mm/s 2

a = 256 mm/s 2

69.4° 

v = 8.29 mm/s

36.2° 

a = 336 mm/s 2

86.6° 

When t = 0.5 s: v = 30e −1.5 i + [60e−1 cos 7.5 − 8e−1 sin 7.5] = 6.694i + 4.8906 j mm/s

a = 90e−1.5 i + [ −240e −1 cos 7.5 − 884e−1 sin 7.5 j] = −20.08i − 335.65 j mm/s 2


PROBLEM 11.91 The motion of a vibrating particle is defined by the position vector r = (4sin π t )i − (cos 2π t ) j, where r is expressed in inches and t in seconds. (a) Determine the velocity and acceleration when t = 1 s. (b) Show that the path of the particle is parabolic.

SOLUTION r = (4sin π t )i − (cos 2π t ) j v = (4π cos π t )i + (2π sin 2π t ) j a = −(4π 2 sin π t )i + (4π 2 cos 2π t ) j

(a)

When t = 1 s: v = (4π cos π )i + (2π sin 2π ) j

v = −(4π in/s)i 

a = −(4π 2 sin π )i − (4π 2 cos π ) j

(b)

a = −(4π 2 in/s 2 ) j 

Path of particle: Since r = xi + y j ;

x = 4sin π t ,

y = − cos 2π t

Recall that cos 2θ = 1 − 2sin 2 θ and write y = − cos 2π t = −(1 − 2sin 2 π t )

But since x = 4sin π t or sin π t =

(1)

1 x, Eq.(1) yields 4

2  1   y = − 1 − 2  x    4   

y=

1 2 x − 1 (Parabola)  8


PROBLEM 11.92 The motion of a particle is defined by the equations x = 10t − 5sin t and y = 10 − 5cos t , where x and y are expresed in feet and t is expressed in seconds. Sketch the path of the particle for the time interval 0 ≤ t ≤ 2π , and determine (a) the magnitudes of the smallest and largest velocities reached by the particle, (b) the corresponding times, positions, and directions of the velocities.

SOLUTION Sketch the path of the particle, i.e., plot of y versus x. Using x = 10t − 5sin t , and y = 10 − 5cos t obtain the values in the table below. Plot as shown. t(s)

x(ft)

y(ft)

0

0.00

5

10.71

10

31.41

15

52.12

10

62.83

5

π 2

π 3

π 2

2π



(a)

Differentiate with respect to t to obtain velocity components. vx =

dx = 10 − 5cos t and v y = 5sin t dt

v 2 = vx2 + v y2 = (10 − 5cos t )2 + 25sin 2 t = 125 − 100cos t d (v ) 2 = 100sin t = 0 t = 0. ± π . ± 2π  ± N π dt

(b)

When t = 2 N π .

cos t = 1.

and

v2 is minimum.

When t = (2 N + 1)π .

cos t = −1.

and

v2 is maximum.

(v 2 )min = 125 − 100 = 25(ft/s) 2

vmin = 5 ft/s 

(v 2 )max = 125 + 100 = 225(ft/s) 2

vmax = 15 ft/s 

When v = vmin . When N = 0,1, 2,

x = 10(2π N ) − 5sin(2π N )

x = 20π N ft 

y = 10 − 5cos(2π N )

y = 5 ft 

vx = 10 − 5cos(2π N )

vx = 5 ft/s 

v y = 5sin(2π N )

tan θ =

vy vx

vy = 0 

θ =0 

= 0,



t = (2 N + 1)π s 

When v = vmax . x = 10[2π ( N − 1)] − 5sin[2π ( N + 1)]

x = 20π ( N + 1) ft 

y = 10 − 5cos[2π ( N + 1)]

y = 15 ft 

vx = 10 − 5cos[2π ( N + 1)]

vx = 15 ft/s 

v y = 5sin[2π ( N + 1)]

tan θ =

vy vx

vy = 0 

θ =0 

= 0,


PROBLEM 11.93 The damped motion of a vibrating particle is defined by the position vector r = x1[1 − 1/(t + 1)]i + ( y1e−π t/2 cos 2π t ) j, where t is expressed in seconds. For x1 = 30 mm and y1 = 20 mm, determine the position, the velocity, and the acceleration of the particle when (a) t = 0, (b) t = 1.5 s.

SOLUTION We have

1   −π t/2 r = 30 1 − cos 2π t ) j  i + 20(e  t +1

Then

v=

= 30

1  π  i + 20  − e −π t/2 cos 2π t − 2π e−π t/2 sin 2π t  j 2 (t + 1)  2 

= 30

 1 1  i − 20π e −π t/ 2  cos 2π t + 2 sin 2π t   j 2 2 (t + 1)   

a=

and

dr dt

dv dt

 π  2 1  i − 20π  − e−π t/2  cos 2π t + 2 sin 2π t  + e−π t/2 (−π sin 2π t + 4 cos 2π t )  j 3 (t + 1) 2   2  60 =− i + 10π 2 e−π t/2 (4 sin 2π t − 7.5 cos 2π t ) j (t + 1)3 = −30

(a)

At t = 0:

 1 r = 30 1 −  i + 20(1) j  1 r = 20 mm 

or  1 1  v = 30   i − 20π (1)  + 0   j 1   2

v = 43.4 mm/s

or a=−

46.3° 

60 i + 10π 2 (1)(0 − 7.5) j (1) a = 743 mm/s 2

or

85.4° 



(b)

At t = 1.5 s:

1   −0.75π r = 30 1 − (cos 3π ) j  i + 20e 2.5   = (18 mm)i + ( −1.8956 mm) j

or

r = 18.10 mm

6.01° 

v = 5.65 mm/s

31.8° 

a = 70.3 mm/s 2

86.9° 

30 1  i − 20π e−0.75π  cos 3π + 0  j 2 (2.5) 2   = (4.80 mm/s)i + (2.9778 mm/s) j

v=

or a=−

60 i + 10π 2 e−0.75π (0 − 7.5 cos 3π ) j (2.5)3

= (−3.84 mm/s 2 )i + (70.1582 mm/s 2 ) j

or


PROBLEM 11.94 The motion of a particle is defined by the position vector r = A(cos t + t sin t )i + A(sin t − t cos t ) j, where t is expressed in seconds. Determine the values of t for which the position vector and the acceleration are (a) perpendicular, (b) parallel.

SOLUTION We have

r = A(cos t + t sin t )i + A(sin t − t cos t ) j

Then

v=

and

a=

(a)

dr = A( − sin t + sin t + t cos t )i dt + A(cos t − cos t + t sin t ) j = A(t cos t )i + A(t sin t ) j dv = A(cos t − t sin t )i + A(sin t + t cos t ) j dt

When r and a are perpendicular, r ⋅ a = 0 A[(cos t + t sin t )i + (sin t − t cos t ) j] ⋅ A[(cos t − t sin t )i + (sin t + t cos t ) j] = 0

(b)

or

(cos t + t sin t )(cos t − t sin t ) + (sin t − t cos t )(sin t + t cos t ) = 0

or

(cos 2 t − t 2 sin 2 t ) + (sin 2 t − t 2 cos 2 t ) = 0

or

1− t2 = 0

or

t =1 s 

or

t=0 

When r and a are parallel, r × a = 0 A[(cos t + t sin t )i + (sin t − t cos t ) j] × A[(cos t − t sin t )i + (sin t + t cos t ) j] = 0

or Expanding

[(cos t + t sin t )(sin t + t cos t ) − (sin t − t cos t )(cos t − t sin t )]k = 0 (sin t cos t + t + t 2 sin t cos t ) − (sin t cos t − t + t 2 sin t cos t ) = 0 2t = 0

or


PROBLEM 11.95 The three-dimensional motion of a particle is defined by the position vector r = (Rt cos ωnt)i + ctj + (Rt sin ωnt)k. Determine the magnitudes of the velocity and acceleration of the particle. (The space curve described by the particle is a conic helix.)

SOLUTION We have

r = ( Rt cos ωn t )i + ctj + ( Rt sin ωn t )k

Then

v=

and

a=

dr = R(cos ωn t − ωn t sin ωn t )i + cj + R(sin ωn t + ωn t cos ωn t )k dt

dv dt = R(−ωn sin ωn t − ωn sin ωn t − ωn2t cos ωn t )i + R(ωn cos ωn t + ωn cos ωn t − ωn2 t sin ωn t )k = R(−2ωn sin ωn t − ωn2 t cos ωn t )i + R(2ωn cos ωn t − ωn2 t sin ωn t )k

Now

v 2 = vx2 + v 2y + vz2 = [ R(cos ωn t − ωn t sin ωn t )]2 + (c)2 + [ R(sin ωn t + ωn t cos ωn t )]2

(

)

= R 2  cos 2 ωn t − 2ωn t sin ωn t cos ωn t + ωn2 t 2 sin 2 ωn t  + sin 2 ωn t + 2ωn t sin ωn t cos ωn t + ωn2 t 2 cos 2 ωn t  + c 2 

(

(

)

)

= R 2 1 + ωn2 t 2 + c 2

(

Also,

)

v = R 2 1 + ωn2 t 2 + c 2 

or a 2 = ax2 + a 2y + az2

(

) cos ω t − ω t sin ω t )  

2

=  R −2ωn sin ωn t − ωn2 t cos ωn t  + (0) 2  

( (

2

2 +  R 2ωn n n n  = R 2  4ωn2 sin 2 ωn t + 4ωn3t sin ωn t cos ωn t + ωn4t 2 cos 2 ωn t  + 4ωn2 cos 2 ωn t − 4ωn3t sin ωn t cos ωn t + ωn4 t 2 sin 2 ωn t  

(

(

= R 2 4ωn2 + ωn4 t 2

)

)

)

a = Rωn 4 + ωn2 t 2 

or


PROBLEM 11.96 The three-dimensional motion of a particle is defined by the position vector r = ( At cos t )i + ( A t 2 + 1) j + ( Bt sin t )k , where r and t are expressed in feet and seconds, respectively. Show that the curve described by the particle lies on the hyperboloid (y/A)2 − (x/A)2 − (z/B)2 = 1. For A = 3 and B = 1, determine (a) the magnitudes of the velocity and acceleration when t = 0, (b) the smallest nonzero value of t for which the position vector and the velocity are perpendicular to each other.

SOLUTION We have

r = ( At cos t )i + ( A t 2 + 1) j + ( Bt sin t )k

or

x = At cos t cos t =

Then

x At

y = A t 2 + 1 z = Bt sin t sin t =

2

 y t2 =   −1  A

z Bt 2

2

2

2

2

x z t2 =   +    A  B

or 2

Then

 y x z  A  −1 =  A  +  B       

or

 y  x  z  A  −  A  −  B  =1      

2

(a)

2

 x   z  cos 2 t + sin 2 t = 1    +   = 1  At   Bt 

Now

2

2



Q.E.D.

With A = 3 and B = 1, we have v=

dr t j + (sin t + t cos t )k = 3(cos t − t sin t )i + 3 2 dt t +1

( ) t

and

t 2 + 1 − t t 2 +1 dv a= j = 3( − sin t − sin t − t cos t )i + 3 dt (t 2 + 1) + (cos t + cos t − t sin t )k 1 j + (2 cos t − t sin t )k = −3(2 sin t + t cos t )i + 3 2 (t + 1)3/2



At t = 0:

v = 3(1 − 0)i + (0) j + (0)k v = vx2 + v y2 + vz2 v = 3 ft/s 

or and Then

a = −3(0)i + 3(1) j + (2 − 0)k a 2 = (0) 2 + (3) 2 + (2) 2 = 13 a = 3.61 ft/s 2 

or (b)

If r and v are perpendicular, r ⋅ v = 0 t   [(3t cos t )i + (3 t 2 + 1) j + (t sin t )k ] ⋅ [3(cos t − t sin t )i +  3  j + (sin t + t cos t )k ] = 0 2  t +1 

or Expanding

t   (3t cos t )[3(cos t − t sin t )] + (3 t 2 + 1)  3  + (t sin t )(sin t + t cos t ) = 0 2 t 1 +   (9t cos 2 t − 9t 2 sin t cos t ) + (9t ) + (t sin 2 t + t 2 sin t cos t ) = 0 10 + 8 cos 2 t − 8t sin t cos t = 0

or (with t ≠ 0)

7 + 2 cos 2t − 2t sin 2t = 0

or Using “trial and error” or numerical methods, the smallest root is

t = 3.82 s 

Note: The next root is t = 4.38 s. 


PROBLEM 11.97 An airplane used to drop water on brushfires is flying horizontally in a straight line at 180 mi/h at an altitude of 300 ft. Determine the distance d at which the pilot should release the water so that it will hit the fire at B.

SOLUTION First note

v0 = 180 km/h = 264 ft/s

Place origin of coordinates at Point A. Vertical motion. (Uniformly accelerated motion) y = 0 + (0)t −

At B: or

1 2 gt 2

1 −300 ft = − (32.2 ft/s 2 )t 2 2 t B = 4.31666 s

Horizontal motion. (Uniform) x = 0 + (v x ) 0 t

At B:

d = (264 ft/s)(4.31666 s) d = 1140 ft 

or


PROBLEM 11.98 A helicopter is flying with a constant horizontal velocity of 180 km/h and is directly above Point A when a loose part begins to fall. The part lands 6.5 s later at Point B on an inclined surface. Determine (a) the distance d between Points A and B, (b) the initial height h.

SOLUTION Place origin of coordinates at Point A. Horizontal motion:

(vx )0 = 180 km/h = 50 m/s x = x0 + (vx )0 t = 0 + 50t m

At Point B where t B = 6.5 s, (a)

Distance AB. 325 cos10°

From geometry

d=

Vertical motion:

y = y0 + ( v y ) 0 t −

At Point B (b)

xB = (50)(6.5) = 325 m

d = 330 m  1 2 gt 2

1 − xB tan10° = h + 0 − (9.81)(6.5) 2 2 h = 149.9 m 

Initial height.


PROBLEM 11.99 A baseball pitching machine “throws” baseballs with a horizontal velocity v0. Knowing that height h varies between 788 mm and 1068 mm, determine (a) the range of values of v0, (b) the values of α corresponding to h = 788 mm and h = 1068 mm.

SOLUTION

(a)

y0 = 1.5 m, (v y )0 = 0

Vertical motion: t =

2( y0 − y ) g

or

tB =

2( y0 − h) g

When h = 788 mm = 0.788 m,

tB =

(2)(1.5 − 0.788) = 0.3810 s 9.81

When h = 1068 mm = 1.068 m,

tB =

(2)(1.5 − 1.068) = 0.2968 s 9.81

y = y0 + (v y )0 t −

At Point B,

Horizontal motion:

1 2 gt 2

or y =h

x0 = 0, (vx )0 = v0 , x = v0t

or

v0 =

x x = B t tB



v0 =

12.2 = 32.02 m/s 0.3810

and

v0 =

12.2 = 41.11 m/s 0.2968

32.02 m/s ≤ v0 ≤ 41.11 m/s

or

Vertical motion:

v y = (v y )0 − gt = − gt

Horizontal motion:

vx = v0

With xB = 12.2 m,

(b)

we get

tan α = −

115.3 km/h ≤ v0 ≤ 148.0 km/h 

(v y ) B dy gt =− = B dx (vx ) B v0

For h = 0.788 m,

tan α =

(9.81)(0.3810) = 0.11673, 32.02

α = 6.66° 

For h = 1.068 m,

tan α =

(9.81)(0.2968) = 0.07082, 41.11

α = 4.05° 


PROBLEM 11.100 While delivering newspapers, a girl throws a newspaper with a horizontal velocity v0. Determine the range of values of v0 if the newspaper is to land between Points B and C.

SOLUTION Vertical motion. (Uniformly accelerated motion) y = 0 + (0)t −

1 2 gt 2

Horizontal motion. (Uniform) x = 0 + (vx )0 t = v0 t

At B:

y:

t B = 0.455016 s

or Then

x:

y:

or

1 −2 ft = − (32.2 ft/s 2 )t 2 2 tC = 0.352454 s

or Then

7 ft = (v0 ) B (0.455016 s) (v0 ) B = 15.38 ft/s

or At C:

1 1 −3 ft = − (32.2 ft/s 2 )t 2 3 2

x:

1 12 ft = (v0 )C (0.352454 s) 3 (v0 )C = 35.0 ft/s 15.38 ft/s < v0 < 35.0 ft/s 


PROBLEM 11.101 Water flows from a drain spout with an initial velocity of 2.5 ft/s at an angle of 15° with the horizontal. Determine the range of values of the distance d for which the water will enter the trough BC.

SOLUTION First note

(vx )0 = (2.5 ft/s) cos 15° = 2.4148 ft/s (v y )0 = −(2.5 ft/s) sin 15° = −0.64705 ft/s

Vertical motion. (Uniformly accelerated motion) y = 0 + (v y ) 0 t −

1 2 gt 2

At the top of the trough 1 −8.8 ft = (−0.64705 ft/s) t − (32.2 ft/s 2 ) t 2 2

or

t BC = 0.719491 s

(the other root is negative)


In time t BC

xBC = (2.4148 ft/s)(0.719491 s) = 1.737 ft

Thus, the trough must be placed so that xB < 1.737 ft or xC ≥ 1.737 ft 0 < d < 1.737 ft 

Since the trough is 2 ft wide, it then follows that


PROBLEM 11.102 Milk is poured into a glass of height 140 mm and inside diameter 66 mm. If the initial velocity of the milk is 1.2 m/s at an angle of 40° with the horizontal, determine the range of values of the height h for which the milk will enter the glass.

SOLUTION First note (vx )0 = (1.2 m/s) cos 40° = 0.91925 m/s (v y )0 = −(1.2 m/s) sin 40° = −0.77135 m/s


Vertical motion. (Uniformly accelerated motion) y = y0 + ( v y ) 0 t −

1 2 gt 2

Milk enters glass at B. x:

0.08 m = (0.91925 m/s) t or tB = 0.087028 s

y : 0.140 m = hB + (−0.77135 m/s)(0.087028 s) 1 − (9.81 m/s 2 )(0.087028 s) 2 2

or

hB = 0.244 m

Milk enters glass at C. x : 0.146 m = (0.91925 m/s) t or tC = 0.158825 s y : 0.140 m = hC + (−0.77135 m/s)(0.158825 s) 1 − (9.81 m/s 2 )(0.158825 s) 2 2

or

hC = 0.386 m 0.244 m < h < 0.386 m 




PROBLEM 11.103 A volleyball player serves the ball with an initial velocity v0 of magnitude 13.40 m/s at an angle of 20° with the horizontal. Determine (a) if the ball will clear the top of the net, (b) how far from the net the ball will land.

SOLUTION First note

(vx )0 = (13.40 m/s) cos 20° = 12.5919 m/s (v y )0 = (13.40 m/s) sin 20° = 4.5831 m/s

(a)


At C

9 m = (12.5919 m/s) t or tC = 0.71475 s

Vertical motion. (Uniformly accelerated motion) y = y0 + ( v y ) 0 t −

At C:

1 2 gt 2

yC = 2.1 m + (4.5831 m/s)(0.71475 s) 1 − (9.81 m/s 2 )(0.71475 s)2 2 = 2.87 m yC > 2.43 m (height of net)  ball clears net 

(b)

At B, y = 0:

1 0 = 2.1 m + (4.5831 m/s)t − (9.81 m/s 2 )t 2 2

Solving

t B = 1.271175 s (the other root is negative)

Then

d = (vx )0 t B = (12.5919 m/s)(1.271175 s) = 16.01 m b = (16.01 − 9.00) m = 7.01 m from the net 

The ball lands


PROBLEM 11.104 A golfer hits a golf ball with an initial velocity of 160 ft/s at an angle of 25° with the horizontal. Knowing that the fairway slopes downward at an average angle of 5°, determine the distance d between the golfer and Point B where the ball first lands.

SOLUTION (vx )0 = (160 ft/s) cos 25°

First note

(v y )0 = (160 ft/s) sin 25° xB = d cos 5°

and at B Now

yB = − d sin 5°

Horizontal motion. (Uniform) x = 0 + (v x ) 0 t d cos 5° = (160 cos 25°)t or t B =

At B

cos 5° d 160 cos 25°

Vertical motion. (Uniformly accelerated motion) y = 0 + (v y )0 t −

1 2 gt 2

( g = 32.2 ft/s 2 ) 1 2 gt B 2

At B:

− d sin 5° = (160 sin 25°)t B −

Substituting for t B

 cos 5°  1  cos 5°  2 − d sin 5° = (160 sin 25°)  d − g   d 2  160 cos 25°   160 cos 25° 

2

or

2 (160 cos 25°) 2 (tan 5° + tan 25°) 32.2 cos 5° = 726.06 ft

d=

d = 242 yd 

or


PROBLEM 11.105 A homeowner uses a snowblower to clear his driveway. Knowing that the snow is discharged at an average angle of 40° with the horizontal, determine the initial velocity v0 of the snow.

SOLUTION First note

(vx )0 = v0 cos 40° (v y )0 = v0 sin 40°


At B:

14 = (v0 cos 40°) t or t B =

14 v0 cos 40°


At B:

1 2 gt 2

1.5 = (v0 sin 40°) t B −

( g = 32.2 ft/s 2 )

1 2 gt B 2

Substituting for t B   1   14 14 1.5 = (v0 sin 40°)  − g   v0 cos 40°  2  v0 cos 40° 

or

v02

=

1 2

2

(32.2)(196)/ cos 2 40° −1.5 + 14 tan 40° v0 = 22.9 ft/s 

or


PROBLEM 11.106 At halftime of a football game souvenir balls are thrown to the spectators with a velocity v0. Determine the range of values of v0 if the balls are to land between Points B and C.

SOLUTION The motion is projectile motion. Place the origin of the xy-coordinate system at ground level just below Point A. The coordinates of Point A are x0 = 0, y0 = 2m. The components of initial velocity are (vx )0 = v0 cos 40° m/s and (v y )0 = v0 sin 40°. Horizontal motion:

x = x0 + (vx )0 t = (v0 cos 40°)t

Vertical motion:

y = y0 + ( v y ) 0 t =

From (1), Then

1 2 gt 2 1 = 2 + (v0 sin 40°) = − (9.81)t 2 2

v0 t =

(2)

x cos 40°

(3)

y = 2 + x tan 40° − 4.905t 2 t2 =

Point B:

(1)

2 + x tan 40° − y 4.905

(4)

x = 8 + 10 cos 35° = 16.1915 m y = 1.5 + 10sin 35° = 7.2358 m 16.1915 v0 t = = 21.1365 m cos 40° 2 + 16.1915 tan 40° − 7.2358 t2 = 4.905 21.1365 v0 = 1.3048

t = 1.3048 s v0 = 16.199 m/s



Point C:

x = 8 + (10 + 7) cos 35° = 21.9256 m y = 1.5 + (10 + 7)sin 35° = 11.2508 m v0 t =

21.9256 = 28.622 m cos 40°

t2 =

2 + 21.9256 tan 40° − 11.2508 4.905

v0 =

28.622 1.3656

t = 1.3656 s v0 = 20.96 m/s 16.20 m/s < v0 < 21.0 m/s 

Range of values of v0.


PROBLEM 11.107 A basketball player shoots when she is 16 ft from the backboard. Knowing that the ball has an initial velocity v0 at an angle of 30° with the horizontal, determine the value of v0 when d is equal to (a) 9 in., (b) 17 in.

SOLUTION First note

(vx )0 = v0 cos 30°

Horizontal motion. (Uniform)

(v y )0 = v0 sin 30°

x = 0 + (v x ) 0 t

(16 − d ) = (v0 cos 30°) t or t B =

At B:

16 − d v0 cos 30°

y = 0 + (v y )0 t −

Vertical motion. (Uniformly accelerated motion)

1 2 gt 2

1 2 gt B 2

At B:

3.2 = (v0 sin 30°) t B −

Substituting for tB

 16 − d  1  16 − d  3.2 = (v0 sin 30°)  − g   v0 cos 30°  2  v0 cos 30° 

or

v02 =

(a)

d = 9 in.:

v02

d = 17 in.:

v02

(b)

=

=

( g = 32.2 ft/s 2 )

2

2 g (16 − d ) 2 3 

1 3

(16 − d ) − 3.2  

2(32.2) (16 − 129 )

3 

1 3

(16 − 129 ) − 3.2

2(32.2) (16 − 17 12 )

3 

1 3

2

v0 = 29.8 ft/s 

2

(16 − 1217 ) − 3.2

v0 = 29.6 ft/s 


PROBLEM 11.108 A tennis player serves the ball at a height h = 2.5 m with an initial velocity of v0 at an angle of 5° with the horizontal. Determine the range for which of v0 for which the ball will land in the service area which extends to 6.4 m beyond the net.

SOLUTION The motion is projectile motion. Place the origin of the xy-coordinate system at ground level just below the point where the racket impacts the ball. The coordinates of this impact point are x0 = 0, y0 = h = 2.5 m. The components of initial velocity are (vx )0 = v0 cos 5° and (v y )0 = v0 sin 5°. Horizontal motion:

x = x0 + (vx )0 t = (v0 cos 5°)t

Vertical motion:

y = y0 + ( v y ) 0 t =

(1)

1 2 gt 2

1 = 2.5 − (v0 sin 5°)t = − (9.81)t 2 2

From (1), Then

v0 t =

x cos 5°

(2) (3)

y = 2.5 − x tan 5° − 4.905t 2 t2 =

2.5 − x tan 5° − y 4.905

(4)

At the minimum speed the ball just clears the net. x = 12.2 m,

y = 0.914 m

12.2 = 12.2466 m cos 5° 2.5 − 12.2 tan 5° − 0.914 t2 = 4.905 12.2466 v0 = 0.32517

v0 t =

t = 0.32517 s v0 = 37.66 m/s



At the maximum speed the ball lands 6.4 m beyond the net. x = 12.2 + 6.4 = 18.6 m y=0 18.6 v0 t = = 18.6710 m cos 5° 2.5 − 18.6 tan 5° − 0 t2 = t = 0.42181 s 4.905 18.6710 v0 = v0 = 44.26 m/s 0.42181 37.7 m/s < v0 < 44.3 m/s 

Range for v0.


PROBLEM 11.109 The nozzle at A discharges cooling water with an initial velocity v0 at an angle of 6° with the horizontal onto a grinding wheel 350 mm in diameter. Determine the range of values of the initial velocity for which the water will land on the grinding wheel between Points B and C.

SOLUTION First note (vx )0 = v0 cos 6° (v y )0 = −v0 sin 6°

Horizontal motion. (Uniform) x = x0 + (vx )0 t

Vertical motion. (Uniformly accelerated motion) y = y0 + (v y )0 t −

1 2 gt 2

( g = 9.81 m/s 2 )

x = (0.175 m) sin 10° y = (0.175 m) cos 10°

At Point B:

x : 0.175 sin 10° = −0.020 + (v0 cos 6°)t tB =

or

0.050388 v0 cos 6°

y : 0.175 cos 10° = 0.205 + (−v0 sin 6°)t B −

1 2 gtB 2

Substituting for t B  0.050388  1  0.050388  −0.032659 = (−v0 sin 6°)   − (9.81)    v0 cos 6°  2  v0 cos 6° 

2



v02 =

or

1 2

(9.81)(0.050388)2

cos 2 6°(0.032659 − 0.050388 tan 6°)

(v0 ) B = 0.678 m/s

or

x = (0.175 m) cos 30° y = (0.175 m) sin 30°

At Point C:

x : 0.175 cos 30° = −0.020 + (v0 cos 6°)t tC =

or

0.171554 v0 cos 6°

y : 0.175 sin 30° = 0.205 + (−v0 sin 6°)tC −

1 2 gtC 2

Substituting for tC  0.171554  1  0.171554  −0.117500 = (−v0 sin 6°)   − (9.81)    v0 cos 6°  2  v0 cos 6° 

or or

v02 =

1 2

2

(9.81)(0.171554) 2

cos 2 6°(0.117500 − 0.171554 tan 6°)

(v0 )C = 1.211 m/s 0.678 m/s < v0 < 1.211 m/s 




PROBLEM 11.110 While holding one of its ends, a worker lobs a coil of rope over the lowest limb of a tree. If he throws the rope with an initial velocity v0 at an angle of 65° with the horizontal, determine the range of values of v0 for which the rope will go over only the lowest limb.

SOLUTION First note (vx )0 = v0 cos 65° (v y )0 = v0 sin 65°


At either B or C, x = 5 m s = (v0 cos 65°)t B,C

or

t B ,C =

5 (v0 cos 65°)


1 2 gt 2

( g = 9.81 m/s 2 )

At the tree limbs, t = tB ,C   1   5 5 yB ,C = (v0 sin 65°)  − g   v0 cos 65°  2  v0 cos 65° 

2



or

v02 = =

1 2

(9.81)(25)

2

cos 65°(5 tan 65° − yB , C ) 686.566 5 tan 65° − yB, C

At Point B:

v02 =

686.566 5 tan 65° − 5

or

(v0 ) B = 10.95 m/s

At Point C:

v02 =

686.566 5 tan 65° − 5.9

or

(v0 )C = 11.93 m/s 10.95 m/s < v0 < 11.93 m/s 




PROBLEM 11.111 The pitcher in a softball game throws a ball with an initial velocity v0 of 72 km/h at an angle α with the horizontal. If the height of the ball at Point B is 0.68 m, determine (a) the angle α, (b) the angle θ that the velocity of the ball at Point B forms with the horizontal.

SOLUTION First note v0 = 72 km/h = 20 m/s (vx )0 = v0 cos α = (20 m/s) cos α

and

(v y )0 = v0 sin α = (20 m/s) sin α

(a)

Horizontal motion. (Uniform) x = 0 + (vx )0 t = (20 cos α ) t

At Point B:

14 = (20 cos α )t or t B =

7 10 cos α


At Point B:

1 2 1 gt = (20 sin α )t − gt 2 2 2

0.08 = (20 sin α )t B −

( g = 9.81 m/s 2 )

1 2 gt B 2

Substituting for t B   1   7 7 0.08 = (20 sin α )  − g   10 cos α  2  10 cos α 

or Now

8 = 1400 tan α −

2

1 49 g 2 cos 2 α

1 = sec2 α = 1 + tan 2 α 2 cos α



Then or Solving

8 = 1400 tan α − 24.5 g (1 + tan 2 α ) 240.345 tan 2 α − 1400 tan α + 248.345 = 0

α = 10.3786° and α = 79.949°

Rejecting the second root because it is not physically reasonable, we have

α = 10.38°  (b)

We have

vx = (vx )0 = 20 cos α

and

v y = (v y )0 − gt = 20 sin α − gt

At Point B:

(v y ) B = 20 sin α − gt B = 20 sin α −

7g 10 cos α

Noting that at Point B, v y < 0, we have tan θ = = =

|(v y ) B | vx 7g 10 cos α

− 20 sin α

20 cos α 7 9.81 200 cos 10.3786°

− sin 10.3786°

cos 10.3786°

θ = 9.74° 

or


PROBLEM 11.112 A model rocket is launched from Point A with an initial velocity v0 of 75 m/s. If the rocket’s descent parachute does not deploy and the rocket lands a distance d = 100 m from A, determine (a) the angle α that v0 forms with the vertical, (b) the maximum height above Point A reached by the rocket, and (c) the duration of the flight.

SOLUTION Set the origin at Point A.

x0 = 0,

y0 = 0

Horizontal motion:

x = v0 t sin α

Vertical motion:

y = v0t cos α − cos α = sin 2 α + cos 2 α =

sin α =

x v0t

1 2 gt 2

1  1  y + gt 2   v0 t  2 

(2)

2 1  2  1 2  x y gt + +     =1 2 (v0 t )2    

x 2 + y 2 + gyt 2 +

1 24 g t = v02t 2 4

1 24 g t − v02 − gy t 2 + ( x 2 + y 2 ) = 0 4

(

At Point B,

(1)

)

(3)

x 2 + y 2 = 100 m, x = 100 cos 30° m y = −100 sin 30° = −50 m 1 (9.81)2 t 4 − [752 − (9.81)(−50)] t 2 + 1002 = 0 4 24.0590 t 4 − 6115.5 t 2 + 10000 = 0 t 2 = 252.54 s 2 t = 15.8916 s

and 1.6458 s 2 and 1.2829 s



Restrictions on α :

0 < α < 120° tan α =

x 2

y + gt 1 2

=

100 cos 30° = 0.0729 −50 + (4.905)(15.8916) 2

α = 4.1669° 100 cos 30° = −2.0655 −50 + (4.905)(1.2829)2 α = 115.8331°

and

Use α = 4.1669° corresponding to the steeper possible trajectory. (a)

Angle α .

(b)

Maximum height.

α = 4.17°  v y = 0 at

y = ymax

v y = v0 cos α − gt = 0 t=

v0 cos α g

ymax = v0 t cos α − =

(c)

Duration of the flight.

v 2 cos 2 α 1 gt = 0 2 2g

(75) 2 cos 2 4.1669° (2)(9.81)

ymax = 285 m 

(time to reach B) t = 15.89 s 




PROBLEM 11.113 The initial velocity v0 of a hockey puck is 105 mi/h. Determine (a) the largest value (less than 45°) of the angle α for which the puck will enter the net, (b) the corresponding time required for the puck to reach the net.

SOLUTION First note

v0 = 105 mi/h = 154 ft/s (vx )0 = v0 cos α = (154 ft/s) cos α

and

(v y )0 = v0 sin α = (154 ft/s) sin α

(a)

Horizontal motion. (Uniform) x = 0 + (vx )0 t = (154 cos α )t

At the front of the net, Then or

x = 16 ft 16 = (154 cos α )t tenter =

8 77 cos α

Vertical motion. (Uniformly accelerated motion) 1 2 gt 2 1 = (154 sin α ) t − gt 2 2

y = 0 + (v y )0 t −

( g = 32.2 ft/s 2 )

At the front of the net, yfront = (154 sin α ) tenter −

1 2 gtenter 2

  1   8 8 = (154 sin α )  − g   77 cos α  2  77 cos α  32 g = 16 tan α − 5929 cos 2 α

2



Now Then or

1 = sec2 α = 1 + tan 2 α cos 2 α yfront = 16 tan α − tan 2 α −

32 g (1 + tan 2 α ) 5929

 5929  5929 tan α + 1 + yfront  = 0 2g 32 g   5929 2g

(

 ±  − 5929 2g 

)

2

(

)

1/2

 − 4 1 + 5929 y 32 g front   2

Then

tan α =

or

 5929 2  5929 5929  ±  − tan α = yfront    − 1 + 4 × 32.2  4 × 32.2   32 × 32.2  

or

tan α = 46.0326 ± [(46.0326) 2 − (1 + 5.7541 yfront )]1/2

1/2

Now 0 < yfront < 4 ft so that the positive root will yield values of α > 45° for all values of yfront. When the negative root is selected, α increases as yfront is increased. Therefore, for α max , set yfront = yC = 4 ft

(b)

Then

tan α = 46.0326 − [(46.0326) 2 − (1 + 5.7541 + 4)]1/ 2

or

α max = 14.6604°

We had found

α max = 14.66° 

8 77 cos α 8 = 77 cos 14.6604°

tenter =

tenter = 0.1074 s 

or


PROBLEM 11.114 A worker uses high-pressure water to clean the inside of a long drainpipe. If the water is discharged with an initial velocity v0 of 11.5 m/s, determine (a) the distance d to the farthest Point B on the top of the pipe that the worker can wash from his position at A, (b) the corresponding angle α.

SOLUTION First note

(vx )0 = v0 cos α = (11.5 m/s) cos α (v y )0 = v0 sin α = (11.5 m/s) sin α

By observation, d max occurs when

ymax = 1.1 m.

Vertical motion. (Uniformly accelerated motion) v y = (v y )0 − gt = (11.5 sin α ) − gt y = ymax

When Then

at

1 2 gt 2 1 = (11.5 sin α ) t − gt 2 2

y = 0 + (v y ) 0 t −

B , (v y ) B = 0

(v y ) B = 0 = (11.5 sin α ) − gt 11.5 sin α g

( g = 9.81 m/s 2 )

or

tB =

and

yB = (11.5 sin α ) t B −

1 2 gt B 2

Substituting for t B and noting yB = 1.1 m  11.5 sin α 1.1 = (11.5 sin α )  g  1 = (11.5) 2 sin 2 α 2g

or

sin 2 α =

2.2 × 9.81 11.52

 1  11.5 sin α  − g  g  2  

2

α = 23.8265°



(a)

Horizontal motion. (Uniform) x = 0 + (vx )0 t = (11.5 cos α ) t

At Point B: where Then

x = d max tB =

and t = t B

11.5 sin 23.8265° = 0.47356 s 9.81

d max = (11.5)(cos 23.8265°)(0.47356) d max = 4.98 m 

or (b)

α = 23.8° 

From above


PROBLEM 11.115 An oscillating garden sprinkler which discharges water with an initial velocity v0 of 8 m/s is used to water a vegetable garden. Determine the distance d to the farthest Point B that will be watered and the corresponding angle α when (a) the vegetables are just beginning to grow, (b) the height h of the corn is 1.8 m.

SOLUTION First note

(vx )0 = v0 cos α = (8 m/s) cos α (v y )0 = v0 sin α = (8 m/s) sin α

Horizontal motion. (Uniform) x = 0 + (vx )0 t = (8 cos α ) t

At Point B: or

x = d : d = (8 cos α ) t d tB = 8 cos α

Vertical motion. (Uniformly accelerated motion) 1 2 gt 2 1 = (8 sin α ) t − gt 2 ( g = 9.81 m/s 2 ) 2 1 0 = (8 sin α ) t B − gt B2 2 y = 0 + (v y )0 t −

At Point B: Simplifying and substituting for t B

0 = 8 sin α − d=

or (a)

1  d  g  2  8 cos α 

64 sin 2α g

(1)

When h = 0, the water can follow any physically possible trajectory. It then follows from Eq. (1) that d is maximum when 2α = 90°

α = 45° 

or Then

d=

64 sin (2 × 45°) 9.81 d max = 6.52 m 

or



(b)

Based on Eq. (1) and the results of Part a, it can be concluded that d increases in value as α increases in value from 0 to 45° and then d decreases as α is further increased. Thus, d max occurs for the value of α closest to 45° and for which the water just passes over the first row of corn plants. At this row, xcom = 1.5 m tcorn =

so that

1.5 8 cos α

Also, with ycorn = h, we have h = (8 sin α ) tcorn −

1 2 gtcorn 2

Substituting for tcorn and noting h = 1.8 m,  1.5  1  1.5  1.8 = (8 sin α )  − g   8 cos α  2  8 cos α 

or Now Then or

1.8 = 1.5 tan α −

2

2.25 g 128 cos 2 α

1 = sec2 α = 1 + tan 2 α cos 2 α 1.8 = 1.5 tan α −

2.25(9.81) (1 + tan 2 α ) 128

0.172441 tan 2 α − 1.5 tan α + 1.972441 = 0

α = 58.229° and α = 81.965°

Solving

From the above discussion, it follows that d = d max when

α = 58.2°  Finally, using Eq. (1) d=

64 sin (2 × 58.229°) 9.81 d max = 5.84 m 

or


PROBLEM 11.116* A mountain climber plans to jump from A to B over a crevasse. Determine the smallest value of the climber’s initial velocity v0 and the corresponding value of angle α so that he lands at B.

SOLUTION First note

(vx )0 = v0 cos α (v y )0 = v0 sin α

Horizontal motion. (Uniform) x = 0 + (vx )0 t = (v0 cos α ) t

At Point B: or

1.8 = (v0 cos α )t tB =

1.8 v0 cos α

Vertical motion. (Uniformly accelerated motion) 1 2 gt 2 1 = (v0 sin α ) t − gt 2 2

y = 0 + (v y )0 t −

At Point B:

−1.4 = (v0 sin α ) t B −

( g = 9.81 m/s 2 )

1 2 gt B 2

Substituting for t B  1.8 −1.4 = (v0 sin α )   v0 cos α

or

 1  1.8  − g   2  v0 cos α 

v02 =

1.62 g cos α (1.8 tan α + 1.4)

=

1.62 g 0.9 sin 2α + 1.4 cos 2 α

2

2



Now minimize v02 with respect to α. We have

dv02 −(1.8 cos 2α − 2.8 cos α sin α ) = 1.62 g =0 dα (0.9 sin 2α + 1.4 cos 2 α ) 2 1.8 cos 2α − 1.4 sin 2α = 0

or

tan 2α =

or

18 14

α = 26.0625° and α = 206.06°

or

Rejecting the second value because it is not physically possible, we have

α = 26.1°  Finally,

v02 =

1.62 × 9.81 cos 26.0625°(1.8 tan 26.0625° + 1.4) 2

(v0 )min = 2.94 m/s 

or


PROBLEM 11.117 The velocities of skiers A and B are as shown. Determine the velocity of A with respect to B.

SOLUTION We have

vA = vB + vA/B

The graphical representation of this equation is then as shown. Then

v 2A/B = 302 + 452 − 2(30)(45) cos 15°

or

vA /B = 17.80450 ft/s

and or

30 17.80450 = sin α sin 15°

α = 25.8554° α + 25° = 50.8554°

vA /B = 17.8 ft/s

50.9° 

Alternative solution. vA /B = v A − vB

= 30 cos 10° i − 30 sin 10° j − (45 cos 25° i − 45° sin 25° j) = 11.2396i + 13.8084 j = 5.05 m/s = 17.8 ft/s

50.9°


PROBLEM 11.118 The three blocks shown move with constant velocities. Find the velocity of each block, knowing that the relative velocity of A with respect to C is 300 mm/s upward and that the relative velocity of B with respect to A is 200 mm/s downward.

SOLUTION From the diagram Cable 1:

y A + yD = constant

Then

v A + vD = 0

Cable 2:

(1)

( yB − yD ) + ( yC − yD ) = constant vB + vC − 2vD = 0

Then

(2)

Combining Eqs. (1) and (2) to eliminate vD , 2v A + vB + vC = 0

(3)

Now

v A/C = v A − vC = −300 mm/s

(4)

and

vB/A = vB − v A = 200 mm/s

(5)

(3) + (4) − (5) 

Then

(2v A + vB + vC ) + (v A − vC ) − (vB − v A ) = (−300) − (200) v A = 125 mm/s 

or vB − (−125) = 200

and using Eq. (5)

v B = 75 mm/s 

or −125 − vC = −300

Eq. (4)

vC = 175 mm/s 

or


PROBLEM 11.119 Three seconds after automobile B passes through the intersection shown, automobile A passes through the same intersection. Knowing that the speed of each automobile is constant, determine (a) the relative velocity of B with respect to A, (b) the change in position of B with respect to A during a 4-s interval, (c) the distance between the two automobiles 2 s after A has passed through the intersection.

SOLUTION

v A = 45 mi/h = 66 ft/s vB = 30 mi/h = 44 ft/s

Law of cosines vB2/A = 662 + 442 − 2(66)(44) cos110° vB/A = 90.99 ft/s

vB = v A + vB/A

Law of sines sin β sin110° = 66 90.99

β = 42.97°

α = 90° − β = 90° − 42.97° = 47.03° v B/A = 91.0 ft/s

(a)

Relative velocity:

(b)

Change in position for Δt = 4 s. ΔrB/A = vB/A Δt = (91.0 ft/s)(4 s)

(c)

rB/A = 364 ft

47.0° 

47.0° 

Distance between autos 2 seconds after auto A has passed intersection. Auto A travels for 2 s. v A = 66 ft/s

20°

rA = v At = (66 ft/s)(2 s) = 132 ft rA = 132 ft

20°



v B = 44 ft/s

Auto B

rB = v B t = (44 ft/s)(5 s) = 220 ft rB = rA + rB/A

Law of cosines rB2/A = (132) 2 + (220) 2 − 2(132)(220) cos110° rB/A = 292.7 ft

Distance between autos = 293 ft 


PROBLEM 11.120 Shore-based radar indicates that a ferry leaves its slip with a velocity v = 18 km/h 70°, while instruments aboard the ferry indicate a speed of 18.4 km/h and a heading of 30° west of south relative to the river. Determine the velocity of the river.

SOLUTION

We have

v F = v R + v F/R

or

v F = v F/R + v R

The graphical representation of the second equation is then as shown. We have

vR2 = 182 + 18.42 − 2(18)(18.4) cos 10°

or

vR = 3.1974 km/h

and or

18 3.1974 = sin α sin 10°

α = 77.84°

Noting that v R = 3.20 km/h

17.8° 

Alternatively one could use vector algebra.


PROBLEM 11.121 Airplanes A and B are flying at the same altitude and are tracking the eye of hurricane C. The relative velocity of C with respect to A is vC/A = 350 km/h 75°, and the relative velocity of C with respect to B is vC/B = 400 km/h 40°. Determine (a) the relative velocity of B with respect to A, (b) the velocity of A if ground-based radar indicates that the hurricane is moving at a speed of 30 km/h due north, (c) the change in position of C with respect to B during a 15-min interval.

SOLUTION (a)

We have

v C = v A + v C/ A

and

v C = v B + v C/ B

Then

v A + vC/A = v B + v C/B

or

v B − v A = v C/A − vC/B

Now

v B − v A = v B/A

so that

v B/A = vC/A − vC/B

or

vC/A = vC/B + v B/A

The graphical representation of the last equation is then as shown. We have

vB2/A = 3502 + 4002 − 2(350)(400) cos 65°

or

vB/A = 405.175 km/h

and or

400 405.175 = sin α sin 65°

α = 63.474° 75° − α = 11.526°

v B/A = 405 km/h

11.53° 



(b)

We have

v C = v A + v C/ A

or

v A = (30 km/h) j − (350 km/h)(− cos 75°i − sin 75° j ) v A = (90.587 km/h)i + (368.07 km/h) j v A = 379 km/h

or (c)

76.17° 

Noting that the velocities of B and C are constant, we have rB = (rB )0 + v B t

Now

rC = (rC )0 + v C t

rC/B = rC − rB = [(rC )0 − (rB )0 ] + ( vC − v B )t

= [(rC )0 − (rB )0 ] + v C/B t

Then

ΔrC/B = (rC/B )t2 − (rC/B )t1 = v C/B (t2 − t1 ) = vC/B Δt

For Δt = 15 min:

1  ΔrC/B = (400 km/h)  h  = 100 km 4 

ΔrC/B = 100 km

40° 


PROBLEM 11.122 Pin P moves at a constant speed of 150 mm/s in a counterclockwise sense along a circular slot which has been milled in the slider block A shown. Knowing that the block moves downward at a constant speed 100 mm/s determine the velocity of pin P when (a) θ = 30°, (b) θ = 120°.

SOLUTION v P = v A + v P/A v P = 100 ms (− j) + 150(cos θ i + sin θ j)mm/s

(a)

For θ = 30°

v P = −100 mm/s ( j) + 150( − cos(30°)i + sin(30°) j)mm/s v P = (−75i + 29.9038 j)mm/s v P = 80.7 mm/s

(b)

For θ = 120°

21.7° 

v P = −100 mm/s ( j) + 150(− cos(120°)i + sin(120°) j)mm/s v P = (−129.9038i + −175 j) mm/s v P = 218 mm/s

53.4° 

Alternative Solution (a)

For θ = 30°, vP /A = 7.5 in./s

30°

vP = v A + vP/A

Law of cosines vP2 = (150) 2 + (100) 2 − 2(100)(150) cos 30° vP = 80.7418 mm/s

Law of sines sin β sin 30° = 150 80.7418

β = 111.7° vP = 80.7 mm/s

21.7° 



(b)

For θ = 120°, vP/A = 150 mm/s

30°

Law of cosines vP2 = (150) 2 + (100) 2 − 2(100)(150) cos120° vP = 217.9449 mm/s

Law of sines sin β sin120° = β = 36.6° 150 217.9449 α = 90 − β = 90° − 36.6 = 53.4°

vP = 218 mm/s

53.4° 


PROBLEM 11.123 Knowing that at the instant shown assembly A has a velocity of 9 in./s and an acceleration of 15 in./s2 both directed downward, determine (a) the velocity of block B, (b) the acceleration of block B.

SOLUTION Length of cable = constant L = x A + 2 xB/A = constant v A + 2vB/A = 0

(1)

a A + 2aB/A = 0

(2)

a A = 15 in./s 2

Data:

v A = 9 in./s

Eqs. (1) and (2)

a A = −2aB/A

v A = −2vB/A

15 = −2aB/A

9 = −2vB/A

aB/A = −7.5 in./s 2 a B/A = 7.5 in./s 2

(a)

vB/A = −4.5 in./s

40°

v B/A = −4.5 in./s

Velocity of B.

v B = v A + v B/A

Law of cosines:

vB2 = (9) 2 + (4.5) 2 − 2(9)(4.5) cos 50°

40°

vB = 7.013 in./s

Law of sines:

sin β sin 50° = 4.5 7.013

β = 29.44°

α = 90° − β = 90° − 29.44° = 60.56° v B = 7.01 in./s

60.6° 



(b)

Acceleration of B. a B may be found by using analysis similar to that used above for vB . An alternate method is a B = a A + a B/A a B = 15 in./s 2 ↓ +7.5 in./s 2

40°

= −15 j − (7.5 cos 40°)i + (7.5 sin 40°) j = −15 j − 5.745i + 4.821j a B = −5.745i − 10.179 j a B = 11.69 in./s 2

60.6° 


PROBLEM 11.124 Knowing that at the instant shown block A has a velocity of 8 in./s and an acceleration of 6 in./s2 both directed down the incline, determine (a) the velocity of block B, (b) the acceleration of block B.

SOLUTION From the diagram

2 x A + xB/A = constant

Then

2v A + vB/A = 0 | vB/A | = 16 in./s

or

2a A + aB/A = 0

and

| aB/A | = 12 in./s 2

or

Note that v B/A and a B/A must be parallel to the top surface of block A. (a)

We have

v B = v A + v B/A

The graphical representation of this equation is then as shown. Note that because A is moving downward, B must be moving upward relative to A. We have

vB2 = 82 + 162 − 2(8)(16) cos 15°

or

vB = 8.5278 in./s

and or

8 8.5278 = sin α sin 15°

α = 14.05° v B = 8.53 in./s

(b)

54.1° 

The same technique that was used to determine v B can be used to determine a B . An alternative method is as follows. We have

a B = a A + a B/A

= (6i ) + 12(− cos 15°i + sin 15° j)* = −(5.5911 in./s 2 )i + (3.1058 in./s 2 ) j a B = 6.40 in./s 2

or

54.1° 

* Note the orientation of the coordinate axes on the sketch of the system. PROPRIETARY MATERIAL. © 2013 The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission. 186

PROBLEM 11.125 A boat is moving to the right with a constant deceleration of 0.3 m/s2 when a boy standing on the deck D throws a ball with an initial velocity relative to the deck which is vertical. The ball rises to a maximum height of 8 m above the release point and the boy must step forward a distance d to catch it at the same height as the release point. Determine (a) the distance d, (b) the relative velocity of the ball with respect to the deck when the ball is caught.

SOLUTION Horizontal motion of the ball:

vx = (vx )0 ,

Vertical motion of the ball:

v y = (v y )0 − gt yB = (v y )0 t −

At maximum height,

xball = (vx )0 t

1 2 gt , (v y ) 2 − (v y )02 = −2 gy 2 vy = 0 and y = ymax

(v y )2 = 2 gymax = (2)(9.81)(8) = 156.96 m 2 /s 2 (v y )0 = 12.528 m/s

At time of catch,

tcatch = 2.554 s

or Motion of the deck:

1 (9.81)t 2 2 and v y = 12.528 m/s

y = 0 = 12.528 −

vx = (vx )0 + aDt ,

xdeck = (vx )0 t +

1 a Dt 2 2

Motion of the ball relative to the deck: (vB/D ) x = (vx )0 − [(vx )0 + aDt ] = −aDt 1 1   xB/D = (vx )0 t − (vx )0 t + aDt 2  = − aDt 2 2 2   (vB/D ) y = (v y )0 − gt , yB/D = yB

(a) (b)

At time of catch,

1 d = xD/B = − (− 0.3)(2.554) 2 2 (vB/D ) x = −(− 0.3)(2.554) = + 0.766 m/s (vB/D ) y = 12.528 m/s

d = 0.979 m  or 0.766 m/s v B/D = 12.55 m/s

86.5° 


PROBLEM 11.126 The assembly of rod A and wedge B starts from rest and moves to the right with a constant acceleration of 2 mm/s2. Determine (a) the acceleration of wedge C, (b) the velocity of wedge C when t = 10 s.

SOLUTION (a)

We have

a C = a B + a C/ B

The graphical representation of this equation is then as shown. First note

α = 180° − (20° + 105°) = 55°

Then

aC 2 = sin 20° sin 55° aC = 0.83506 mm/s 2

(b)

aC = 0.835 mm/s 2

75° 

vC = 8.35 mm/s

75° 

For uniformly accelerated motion vC = 0 + aC t

At t = 10 s:

vC = (0.83506 mm/s 2 )(10 s) = 8.3506 mm/s

or


PROBLEM 11.127 Determine the required velocity of the belt B if the relative velocity with which the sand hits belt B is to be (a) vertical, (b) as small as possible.

SOLUTION A grain of sand will undergo projectile motion. vsx = vsx = constant = −5 ft/s 0

vs y = 2 gh = (2)(32.2 ft/s 2 )(3 ft) = 13.90 ft/s ↓

y-direction.

v S/B = v S − v B

Relative velocity. (a)

(1)

If vS/B is vertical, −vS /B j = −5i − 13.9 j − ( −vB cos 15°i + vB sin 15° j) = −5i − 13.9 j + vB cos 15°i − vB sin 15° j

Equate components.

i : 0 = −5 + vB cos 15°

vB =

5 = 5.176 ft/s cos 15° v B = 5.18 ft/s

(b)

15° 

vS/C is as small as possible, so make vS/B ⊥ to vB into (1). −vS/B sin 15°i − vS/B cos 15° j = − 5i − 13.9 j + vB cos 15°i − vB sin 15° j

Equate components and transpose terms. (sin 15°) vS/B + (cos 15°) vB = 5 (cos 15°) vS/B − (sin 15°) vB = 13.90

Solving,

vS/B = 14.72 ft/s

vB = 1.232 ft/s v B = 1.232 ft/s

15° 


PROBLEM 11.128 Conveyor belt A, which forms a 20° angle with the horizontal, moves at a constant speed of 4 ft/s and is used to load an airplane. Knowing that a worker tosses duffel bag B with an initial velocity of 2.5 ft/s at an angle of 30° with the horizontal, determine the velocity of the bag relative to the belt as it lands on the belt.

SOLUTION First determine the velocity of the bag as it lands on the belt. Now [(vB ) x ]0 = (vB )0 cos 30° = (2.5 ft/s) cos 30° [(vB ) y ]0 = (vB )0 sin 30° = (2.5 ft/s) sin 30°

Horizontal motion. (Uniform) x = 0 + [(vB ) x ]0 t

(vB ) x = [(vB ) x ]0

= (2.5 cos 30°) t

= 2.5 cos 30°

Vertical motion. (Uniformly accelerated motion) y = y0 + [(vB ) y ]0 t −

1 2 gt 2

= 1.5 + (2.5 sin 30°) t −

(vB ) y = [(vB ) y ]0 − gt

1 2 gt 2

= 2.5 sin 30° − gt

The equation of the line collinear with the top surface of the belt is y = x tan 20°

Thus, when the bag reaches the belt 1.5 + (2.5 sin 30°) t −

1 2 gt = [(2.5 cos 30°) t ] tan 20° 2

or

1 (32.2) t 2 + 2.5(cos 30° tan 20° − sin 30°) t − 1.5 = 0 2

or

16.1t 2 − 0.46198t − 1.5 = 0

Solving

t = 0.31992 s and t = −0.29122 s (Reject)



The velocity v B of the bag as it lands on the belt is then v B = (2.5 cos 30°)i + [2.5 sin 30° − 32.2(0.319 92)] j

= (2.1651 ft/s)i − (9.0514 ft/s) j

Finally or

v B = v A + v B/A v B/A = (2.1651i − 9.0514 j) − 4(cos 20°i + sin 20° j)

= −(1.59367 ft/s)i − (10.4195 ft/s) j v B/A = 10.54 ft/s

or

81.3° 


PROBLEM 11.129 During a rainstorm the paths of the raindrops appear to form an angle of 30° with the vertical and to be directed to the left when observed from a side window of a train moving at a speed of 15 km/h. A short time later, after the speed of the train has increased to 24 km/h, the angle between the vertical and the paths of the drops appears to be 45°. If the train were stopped, at what angle and with what velocity would the drops be observed to fall?

SOLUTION vrain = vtrain + vrain/train

Case :

vT = 15 km/h

;

vR / T

30°

Case :

vT = 24 km/h

;

vR / T

45°

Case :

(vR ) y tan 30 = 15 − (vR ) x

(1)

Case :

(vR ) y tan 45° = 24 − (vR ) x

(2)

Substract (1) from (2)

(vR ) y (tan 45° − tan 30°) = 9 (vR ) y = 21.294 km/h

Eq. (2):

21.294 tan 45° = 25 − (vR ) x (vR ) x = 2.706 km/h



3.706 21.294 β = 7.24° 21.294 = 21.47 km/h = 5.96 m/s vR = cos 7.24°

tan β =

vR = 5.96 m/s

82.8° 

Alternate solution Alternate, vector equation

v R = vT + v R / T

For first case,

v R = 15i + vR / T −1 (− sin 30°i − cos 30° j)

For second case,

v R = 24i + vR /T − 2 (− sin 45°i − cos 45° j)

Set equal 15i + vR /T −1 (− sin 30°i − cos 30° j) = 24i + vR /T − 2 (− sin 45°i − cos 45° j)

Separate into components: i:

15 − vR /T −1 sin 30° = 24 − vR /T − 2 sin 45° −vR /T −1 sin 30° + vR /T − 2 sin 45° = 9

(3)

−vR /T −1 cos 30° = −vR / T − 2 cos 45°

j:

vR / T −1 cos 30° + vR /T − 2 cos 45° = 0

(4)

Solving Eqs. (3) and (4) simultaneously, vR /T −1 = 24.5885 km/h

vR /T − 2 = 30.1146 km/h

Substitute v R /T − 2 back into equation for v R . v R = 24i + 30.1146(− sin 45°i − cos 45° j) v R = 2.71i − 21.29 j

 −21.29   = −82.7585°  2.71 

θ = tan −1 

v R = 21.4654 km/hr = 5.96 m/s v R = 5.96 m/s

82.8° 


PROBLEM 11.130 As observed from a ship moving due east at 9 km/h, the wind appears to blow from the south. After the ship has changed course and speed, and as it is moving north at 6 km/h, the wind appears to blow from the southwest. Assuming that the wind velocity is constant during the period of observation, determine the magnitude and direction of the true wind velocity.

SOLUTION v wind = v ship + v wind/ship v w = v s + v w/s

Case  v s = 9 km/h →; v w /s

Case  v s = 6 km/h ↑ ; v w /s 15 = 1.6667 9 α = 59.0°

tan α =

vw = 92 + 152 = 17.49 km/h

v w = 17.49 km/h

59.0° 


PROBLEM 11.131 When a small boat travels north at 5 km/h, a flag mounted on its stern forms an angle θ = 50° with the centerline of the boat as shown. A short time later, when the boat travels east at 20 km/h, angle θ is again 50°. Determine the speed and the direction of the wind.

SOLUTION We have

vW = v B + vW/B

Using this equation, the two cases are then graphically represented as shown.

With vW now defined, the above diagram is redrawn for the two cases for clarity.

Noting that

θ = 180° − (50° + 90° + α ) = 40° − α We have

vW 5 = sin 50° sin (40° − α )

φ = 180° − (50° + α ) = 130° − α

vW 20 = sin 50° sin (130° − α )



Therefore or or

5 20 = sin (40° − α ) sin (130° − α ) sin 130° cos α − cos 130° sin α = 4(sin 40° cos α − cos 40° sin α ) tan α =

sin 130° − 4 sin 40° cos 130° − 4 cos 40°

or

α = 25.964°

Then

vW =

5 sin 50° = 15.79 km/h sin (40° − 25.964°)

vW = 15.79 km/h

26.0° 


PROBLEM 11.132 As part of a department store display, a model train D runs on a slight incline between the store’s up and down escalators. When the train and shoppers pass Point A, the train appears to a shopper on the up escalator B to move downward at an angle of 22° with the horizontal, and to a shopper on the down escalator C to move upward at an angle of 23° with the horizontal and to travel to the left. Knowing that the speed of the escalators is 3 ft/s, determine the speed and the direction of the train.

SOLUTION v D = v B + v D/B

We have

v D = vC + v D/C

The graphical representations of these equations are then as shown. Then

vD 3 = sin 8° sin (22° + α )

Equating the expressions for

vD 3 = sin 7° sin (23° − α )

vD 3

sin 8° sin 7° = sin (22° + α ) sin (23° − α )

or

or or Then

sin 8° (sin 23° cos α − cos 23° sin α ) = sin 7° (sin 22° cos α + cos 22° sin α ) tan α =

sin 8° sin 23° − sin 7° sin 22° sin 8° cos 23° + sin 7° cos 22°

α = 2.0728° vD =

3 sin 8° = 1.024 ft/s sin (22° + 2.0728°)

v D = 1.024 ft/s

2.07° 



Alternate solution using components. v B = (3 ft/s)

30° = (2.5981 ft/s)i + (1.5 ft/s) j

vC = (3 ft/s)

30° = (2.5981 ft/s)i − (1.5 ft/s) j

v D/B = u1

22° = −(u1 cos 22°)i − (u1 sin 22°) j

v D/C = u2

23° = −(u2 cos 23°)i + (u2 sin 23°) j

v D = vD

α = −(vD cos α )i + (vD sin α ) j

v D = v B + v D/B = vC + v D/C 2.5981i + 1.5 j − (u1 cos 22°)i − (u1 sin 22°) j = 2.5981i − 1.5 j − (u2 cos 23°)i + (u2 sin 23°) j

Separate into components, transpose, and change signs. u1 cos 22° − u2 cos 23° = 0 u1 sin 22° + u1 sin 23° = 3

Solving for u1 and u2 ,

u1 = 3.9054 ft/s

u2 = 3.9337 ft/s

v D = 2.5981i + 1.5 j − (3.9054 cos 22°)i − (3.9054 sin 22°) j = −(1.0229 ft/s)i + (0.0370 ft/s) j

or

v D = 2.5981i − 1.5 j − (3.9337 cos 23°)i + (3.9337 sin 23°) j = −(1.0229 ft/s)i + (0.0370 ft/s) j

v D = 1.024 ft/s

2.07° 


PROBLEM 11.CQ8 The Ferris wheel is rotating with a constant angular velocity ω. What is the direction of the acceleration of Point A? (a) (b) (c) (d ) (e) The acceleration is zero.

SOLUTION The tangential acceleration is zero since the speed is constant, so there will only be normal acceleration pointed upwards. Answer: (b) 


PROBLEM 11.CQ9 A racecar travels around the track shown at a constant speed. At which point will the racecar have the largest acceleration? (a) A (b) B (c) C (d ) The acceleration will be zero at all the points.

SOLUTION The tangential acceleration is zero since the speed is constant, so there will only be normal acceleration. The normal acceleration will be maximum where the radius of curvature is a minimum, that is at Point A. Answer: (a) 


PROBLEM 11.CQ10 A child walks across merry-go-round A with a constant speed u relative to A. The merry-go-round undergoes fixed axis rotation about its center with a constant angular velocity ω counterclockwise.When the child is at the center of A, as shown, what is the direction of his acceleration when viewed from above. (a) (b) (c) (d ) (e) The acceleration is zero.

SOLUTION Polar coordinates are most natural for this problem, that is,  a = ( r − rθ 2 )er + (rθ + 2rθ)eθ

(1)

r = 0, θ = 0, r = 0, θ = ω, r = -u. When we substitute From the information given, we know  these values into (1), we will only have a term in the −θ direction. Answer: (d ) 


PROBLEM 11.133 Determine the smallest radius that should be used for a highway if the normal component of the acceleration of a car traveling at 72 km/h is not to exceed 0.8 m/s 2 .

SOLUTION an =

v2

an = 0.8 m/s 2

ρ

v = 72 km/h = 20 m/s 0.8 m/s 2 =

(20 m/s) 2

ρ

ρ = 500 m  


PROBLEM 11.134 Determine the maximum speed that the cars of the roller-coaster can reach along the circular portion AB of the track if ρ is 25 m and the normal component of their acceleration cannot exceed 3 g.

SOLUTION We have

an =

v2

ρ

Then

(vmax ) 2AB = (3 × 9.81 m/s 2 )(25 m)

or

(vmax ) AB = 27.124 m/s (vmax ) AB = 97.6 km/h 

or


PROBLEM 11.135 A bull-roarer is a piece of wood that produces a roaring sound when attached to the end of a string and whirled around in a circle. Determine the magnitude of the normal acceleration of a bull-roarer when it is spun in a circle of radius 0.9 m at a speed of 20 m/s.

SOLUTION an =

v2

ρ

=

(20 m/s) 2 = 444.4 m/s 2 0.9 m an = 444 m/s 2 




PROBLEM 11.136 To test its performance, an automobile is driven around a circular test track of diameter d. Determine (a) the value of d if when the speed of the automobile is 45 mi/h, the normal component of the acceleration is 11 ft/s 2 , (b) the speed of the automobile if d = 600 ft and the normal component of the acceleration is measured to be 0.6 g.

SOLUTION (a)

First note Now

v = 45 mi/h = 66 ft/s an =

ρ=

v2

ρ (66 ft/s) 2 = 396 ft 11 ft/s 2

d = 2ρ

(b)

d = 792 ft 

v2

We have

an =

Then

1  v 2 = (0.6 × 32.2 ft/s 2 )  × 600 ft  2 

ρ

v = 76.131 ft/s v = 51.9 mi/h 


PROBLEM 11.137 An outdoor track is 420 ft in diameter. A runner increases her speed at a constant rate from 14 to 24 ft/s over a distance of 95 ft. Determine the magnitude of the total acceleration of the runner 2 s after she begins to increase her speed.

SOLUTION We have uniformly accelerated motion v22 = v12 + 2at Δs12

Substituting or

(24 ft/s)2 = (14 ft/s)2 + 2at (95 ft) at = 2 ft/s 2

Also

v = v1 + at t

At t = 2 s:

v = 14 ft/s + (2 ft/s 2 )(2 s) = 18 ft/s v2

Now

an =

At t = 2 s:

an =

Finally

a 2 = at2 + an2

At t = 2 s:

a 2 = 22 + 1.542862

ρ (18 ft/s) 2 = 1.54286 ft/s 2 210 ft

a = 2.53 ft/s 2 

or


PROBLEM 11.138 A robot arm moves so that P travels in a circle about Point B, which is not moving. Knowing that P starts from rest, and its speed increases at a constant rate of 10 mm/s2, determine (a) the magnitude of the acceleration when t = 4 s, (b) the time for the magnitude of the acceleration to be 80 mm/s2.

SOLUTION Tangential acceleration: Speed:

at = 10 mm/s 2 v = at t v2

an =

where

ρ = 0.8 m = 800 mm

(a)

When t = 4 s

ρ

v = (10)(4) = 40 mm/s an =

Acceleration:

ρ

=

at2 t 2

Normal acceleration:

(40)2 = 2 mm/s 2 800

a = at2 + an2 = (10) 2 + (2) 2 a = 10.20 mm/s 2 

(b)

Time when a = 80 mm/s 2 a 2 = an2 + at2 2

 (10) 2 t 2  2 (80) =   + 10 800   2

t 4 = 403200 s 4 t = 25.2 s 


PROBLEM 11.139 A monorail train starts from rest on a curve of radius 400 m and accelerates at the constant rate at . If the maximum total acceleration of the train must not exceed 1.5 m/s 2 , determine (a) the shortest distance in which the train can reach a speed of 72 km/h, (b) the corresponding constant rate of acceleration at .

SOLUTION When v = 72 km/h = 20 m/s and ρ = 400 m, an =

v2

ρ

=

(20)2 = 1.000 m/s 2 400

a = an2 + at2

But

at = a 2 − an2 = (1.5) 2 − (1.000) 2 = ± 1.11803 m/s 2

Since the train is accelerating, reject the negative value. (a)

Distance to reach the speed. v0 = 0

Let

x0 = 0 v12 = v02 + 2at ( x1 − x0 ) = 2at x1 x1 =

(b)

v12 (20) 2 = 2at (2)(1.11803)

x1 = 178.9 m 

Corresponding tangential acceleration. at = 1.118 m/s 2 




PROBLEM 11.140 A motorist starts from rest at Point A on a circular entrance ramp when t = 0, increases the speed of her automobile at a constant rate and enters the highway at Point B. Knowing that her speed continues to increase at the same rate until it reaches 100 km/h at Point C, determine (a) the speed at Point B, (b) the magnitude of the total acceleration when t = 20 s.

SOLUTION v0 = 0

Speeds:

v1 = 100 km/h = 27.78 m/s s =

Distance:

π 2

(150) + 100 = 335.6 m v12 = v02 + 2at s

Tangential component of acceleration: at =

At Point B,

v12 − v02 (27.78) 2 − 0 = = 1.1495 m/s 2 2s (2)(335.6)

vB2 = v02 + 2at sB

where

sB =

π 2

(150) = 235.6 m

vB2 = 0 + (2)(1.1495)(235.6) = 541.69 m 2 /s 2 vB = 23.27 m/s

(a)

At t = 20 s,

vB = 83.8 km/h 

v = v0 + at t = 0 + (1.1495)(20) = 22.99 m/s

ρ = 150 m

Since v < vB , the car is still on the curve.

(b)

Normal component of acceleration:

an =

Magnitude of total acceleration:

|a| =

v2

ρ

=

(22.99)2 = 3.524 m/s 2 150

at2 + an2 =

(1.1495) 2 + (3.524) 2

| a | = 3.71 m/s 2 


PROBLEM 11.141 Racecar A is traveling on a straight portion of the track while racecar B is traveling on a circular portion of the track. At the instant shown, the speed of A is increasing at the rate of 10 m/s2, and the speed of B is decreasing at the rate of 6 m/s2. For the position shown, determine (a) the velocity of B relative to A, (b) the acceleration of B relative to A.

SOLUTION v A = 240 km/h = 66.67 m/s

Speeds:

vB = 200 km/h = 55.56 m/s

vA = 66.67 m/s

Velocities:

vB = 55.56 m/s

(a)

50°

vB/A = vB − vA

Relative velocity:

vB/A = (55.56 cos 50°) ← + 55.56sin 50° ↓ + 66.67 = 30.96 → + 42.56 = 52.63 m/s

53.96° vB /A = 189.5 km/h

Tangential accelerations:

(aA )t = 10 m/s 2 (aB )t = 6 m/s 2 an =

Normal accelerations:

ρ

( ρ = ∞)

Car B:

( ρ = 300 m)

Acceleration of B relative to A:

50°

v2

Car A:

(aB ) n =

(b)

54.0° 

(aA ) n = 0

(55.56)2 = 10.288 300

(aB ) n = 10.288 m/s 2

40°

aB/A = aB − aA a B/A = (a B )t + (a B ) n − (a A )t − (a A ) n =6

50° + 10.288

40° + 10 → + 0

= (6cos 50° + 10.288cos 40° + 10) + (6sin 50° − 10.288sin 40°) = 21.738 → + 2.017

a B/A = 21.8 m/s 2

5.3° 


PROBLEM 11.142 At a given instant in an airplane race, airplane A is flying horizontally in a straight line, and its speed is being increased at the rate of 8 m/s 2 . Airplane B is flying at the same altitude as airplane A and, as it rounds a pylon, is following a circular path of 300-m radius. Knowing that at the given instant the speed of B is being decreased at the rate of 3 m/s 2 , determine, for the positions shown, (a) the velocity of B relative to A, (b) the acceleration of B relative to A.

SOLUTION First note

v A = 450 km/h vB = 540 km/h = 150 m/s

(a)

v B = v A + v B/A

We have

The graphical representation of this equation is then as shown. We have

vB2 /A = 4502 + 5402 − 2(450)(540) cos 60° vB/A = 501.10 km/h

and

540 501.10 = sin α sin 60°

α = 68.9° v B/A = 501 km/h

(b)

First note Now

a A = 8 m/s 2 ( aB ) n =

v 2B

ρB

=

Then

60°

(150 m/s) 2 300 m

(a B ) n = 75 m/s 2



(a B )t = 3 m/s 2

68.9° 

30° 

a B = (a B )t + (a B )n = 3(− cos 60° i + sin 60° j) + 75(−cos 30° i − sin 30° j) = −(66.452 m/s 2 )i − (34.902 m/s 2 ) j

Finally

a B = a A + a B/A

a B/A = ( −66.452i − 34.902 j) − (8i ) = −(74.452 m/s 2 )i − (34.902 m/s 2 ) j

a B/A = 82.2 m/s 2

25.1° 


PROBLEM 11.143 From a photograph of a homeowner using a snowblower, it is determined that the radius of curvature of the trajectory of the snow was 30 ft as the snow left the discharge chute at A. Determine (a) the discharge velocity v A of the snow, (b) the radius of curvature of the trajectory at its maximum height.

SOLUTION (a) The acceleration vector is 32.2 ft/s . At Point A, tangential and normal components of a are as shown in the sketch. an = a cos 40° = 32.2 cos 40° = 24.67 ft/s 2 v A2 = ρ A (a A ) n = (30)(24.67) = 740.0 ft 2 /s 2 v A = 27.2 ft/s

40° 

vx = 27.20 cos 40° = 20.84 ft/s

(b)

At maximum height,

v = vx = 20.84 ft/s an = g = 32.2 ft/s 2 ,

ρ =

v2 (20.84) 2 = 32.2 an

ρ = 13.48 ft 


PROBLEM 11.144 A basketball is bounced on the ground at Point A and rebounds with a velocity v A of magnitude 2.5 m/s as shown. Determine the radius of curvature of the trajectory described by the ball (a) at Point A, (b) at the highest point of the trajectory.

SOLUTION (a)

(a A ) n =

We have

ρA =

or

v 2A

ρA (2.5 m/s) 2 (9.81 m/s 2 ) sin 15°

ρ A = 2.46 m 

or (b)

( aB ) n =

We have

vB2

ρB

where Point B is the highest point of the trajectory, so that vB = (v A ) x = v A sin 15°



ρB =

Then

[(2.5 m/s) sin 15°]2 = 0.0427 m 9.81 m/s 2

ρ B = 42.7 mm 

or


PROBLEM 11.145 A golfer hits a golf ball from Point A with an initial velocity of 50 m/s at an angle of 25° with the horizontal. Determine the radius of curvature of the trajectory described by the ball (a) at Point A, (b) at the highest point of the trajectory.

SOLUTION (a)

We have or

(a A ) n =

ρA =

v 2A

ρA (50 m/s)2 (9.81 m/s 2 ) cos 25°

ρ A = 281 m 

or (b)

We have

( aB ) n =

vB2

ρB

where Point B is the highest point of the trajectory, so that vB = (v A ) x = v A cos 25°

Then

ρB =

[(50 m/s) cos 25°]2 9.81 m/s 2

ρ B = 209 m 

or


PROBLEM 11.146 Three children are throwing snowballs at each other. Child A throws a snowball with a horizontal velocity v0. If the snowball just passes over the head of child B and hits child C, determine the radius of curvature of the trajectory described by the snowball (a) at Point B, (b) at Point C.

SOLUTION The motion is projectile motion. Place the origin at Point A. Horizontal motion:

v x = v0

x = v0 t

Vertical motion:

y0 = 0,

(v y ) = 0

v y = − gt

1 y = − gt 2 2

2h , g

t=

where h is the vertical distance fallen.

| v y| = 2 gh

Speed:

v 2 = vx2 + v 2y = v02 + 2 gh

Direction of velocity. cos θ =

v0 v

Direction of normal acceleration. an = g cos θ =

Radius of curvature: At Point B,

ρ=

gv0 v 2 = v ρ

v3 gv0

hB = 1 m; xB = 7 m tB =

(2)(1 m) = 0.45152 s 9.81 m/s 2

xB = v0t B

v0 =

xB 7m = = 15.504 m/s t B 0.45152 s

vB2 = (15.504) 2 + (2)(9.81)(1) = 259.97 m 2 /s 2 PROPRIETARY MATERIAL. © 2013 The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission. 215


(a)

Radius of curvature at Point B.

ρB = At Point C

(259.97 m 2 /s 2 )3/ 2 (9.81 m/s 2 )(15.504 m/s)

ρ B = 27.6 m 

hC = 1 m + 2 m = 3 m

vC2 = (15.504) 2 + (2)(9.81)(3) = 299.23 m 2 /s 2

(b)

Radius of curvature at Point C.

ρC =

(299.23 m 2 /s 2 )3/2 (9.81 m/s 2 )(15.504 m/s)

ρC = 34.0 m 


PROBLEM 11.147 Coal is discharged from the tailgate A of a dump truck with an initial velocity v A = 2 m/s 50° . Determine the radius of curvature of the trajectory described by the coal (a) at Point A, (b) at the point of the trajectory 1 m below Point A.

SOLUTION a A = g = 9.81 m/s 2

(a) At Point A.

Sketch tangential and normal components of acceleration at A. (a A ) n = g cos 50°

ρA =

vA2 (2) 2 = (a A ) n 9.81cos 50°

ρ A = 0.634 m 

(b) At Point B, 1 meter below Point A. Horizontal motion: (vB ) x = (v A ) x = 2 cos 50° = 1.286 m/s Vertical motion:

(vB ) 2y = (v A ) 2y + 2a y ( yB − y A ) = (2 cos 40°)2 + (2)(−9.81)(−1) = 21.97 m 2 /s 2 (vB ) y = 4.687 m/s

tan θ =

(vB ) y (vB ) x

=

4.687 , 1.286

or

θ = 74.6°

aB = g cos 74.6°

(vB ) 2x + (vB ) 2y vB 2 ρB = = (aB )n g cos 74.6° =

(1.286) 2 + 21.97 9.81cos 74.6°

ρ B = 9.07 m 


PROBLEM 11.148 From measurements of a photograph, it has been found that as the stream of water shown left the nozzle at A, it had a radius of curvature of 25 m. Determine (a) the initial velocity vA of the stream, (b) the radius of curvature of the stream as it reaches its maximum height at B.

SOLUTION (a)

We have

(a A ) n =

v 2A

ρA

or

4  v A2 =  (9.81 m/s 2 )  (25 m) 5  

or

v A = 14.0071 m/s vA = 14.01 m/s

(b)

We have Where

Then

( aB ) n =

vB2

ρB

vB = ( v A ) x =

ρB

36.9° 

4 vA 5

( 4 × 14.0071 m/s ) = 5

2

9.81 m/s 2

ρ B = 12.80 m 

or


PROBLEM 11.149 A child throws a ball from Point A with an initial velocity v A of 20 m/s at an angle of 25° with the horizontal. Determine the velocity of the ball at the points of the trajectory described by the ball where the radius of curvature is equal to three-quarters of its value at A.

SOLUTION Assume that Points B and C are the points of interest, where yB = yC and vB = vC . Now

(a A ) n =

v 2A

ρA

or

ρA =

v 2A g cos 25°

Then

ρB =

v A2 3 3 ρA = 4 4 g cos 25° vB2

We have

( aB ) n =

where

(aB ) n = g cos θ

so that

v A2 vB2 3 = 4 g cos 25° g cos θ

or

vB2 =

ρB

3 cos θ 2 vA 4 cos 25°

(1)

Noting that the horizontal motion is uniform, we have ( v A ) x = ( vB ) x

where Then or

(v A ) x = v A cos 25°

(vB ) x = vB cos θ

v A cos 25° = vB cos θ cos θ =

vA cos 25° vB


PROBLEM 11.149 (Continued) Substituting for cos θ in Eq. (1), we have

or

vB2 =

 v A2 3  vA  cos 25°  4  vB  cos 25°

vB3 =

3 3 vA 4

3 vB = 3 v A = 18.17 m/s 4 4 cos 25° 3 θ = ± 4.04°

cos θ = 3

and

v B = 18.17 m/s

4.04° 

v B = 18.17 m/s

4.04° 


PROBLEM 11.150 A projectile is fired from Point A with an initial velocity v 0 . (a) Show that the radius of curvature of the trajectory of the projectile reaches its minimum value at the highest Point B of the trajectory. (b) Denoting by θ the angle formed by the trajectory and the horizontal at a given Point C, show that the radius of curvature of the trajectory at C is ρ = ρ min /cos3 θ .

SOLUTION For the arbitrary Point C, we have (aC ) n =

ρC =

or

vC2

ρC vC2 g cos θ

Noting that the horizontal motion is uniform, we have (v A ) x = (vC ) x (v A ) x = v0 cos α

where

v0 cos α = vC cos θ

Then

cos α v0 cos θ

or

vC =

so that

1 ρC = g cos θ

(a)

2

 cos α  v 2 cos 2 α v0  = 0  g cos3 θ  cos θ 

In the expression for ρC , v0 , α , and g are constants, so that ρC is minimum where cos θ is maximum. By observation, this occurs at Point B where θ = 0.

ρ min = ρ B = (b)

(vC ) x = vC cos θ

ρC =

1 cos3 θ

ρC =

ρ min cos3 θ

v02 cos 2 α g

 v02 cos 2 α  g 

Q.E.D.



Q.E.D.



  


PROBLEM 11.151* Determine the radius of curvature of the path described by the particle of Problem 11.95 when t = 0. PROBLEM 11.95 The three-dimensional motion of a particle is defined by the position vector r = (Rt cos ωnt)i + ctj + (Rt sin ωnt)k. Determine the magnitudes of the velocity and acceleration of the particle. (The space curve described by the particle is a conic helix.)

SOLUTION We have

v=

dr = R(cos ωn t − ωn t sin ωn t )i + cj + R (sin ωn t + ωn t cos ωn t )k dt

and

a=

dv = R − ωn sin ωn t − ωn sin ωn t − ωn2t cos ωn t i dt

(

)

(

)

+ R ωn cos ωn t + ωn cos ωn t − ωn2 t sin ωn t k

or Now

a = ωn R [−(2 sin ωn t + ωn t cos ωn t )i + (2 cos ωn t − ωn t sin ωn t ) k ]

v 2 = R 2 (cos ωn t − ωn t sin ωn t )2 + c 2 + R 2 (sin ωn t + ωn t cos ωn t )2

(

)

= R 2 1 + ωn2 t 2 + c 2

Then and

Now At t = 0:

(

)

1/2

v =  R 2 1 + ωn2 t 2 + c 2   

R 2ωn2 t dv = 1/ 2 dt  2 R 1 + ωn2 t 2 + c 2   

(

2

a =

at2

+

an2

)

2 2  dv   v  =   +    dt   ρ 

2

dv =0 dt a = ωn R(2 k ) or a = 2ωn R v2 = R2 + c2

Then, with we have or

dv = 0, dt a= 2ωn R =

v2

ρ R 2 + c2

ρ=

ρ

R 2 + c2  2ωn R


PROBLEM 11.152* Determine the radius of curvature of the path described by the particle of Problem 11.96 when t = 0, A = 3, and B = 1.

SOLUTION With

A = 3,

B =1

we have

r = (3t cos t )i + 3 t 2 + 1 j + (t sin t )k

Now

v=

and

  2 dv   t t + 1 − a= = 3(− sin t − sin t − t cos t )i + 3   dt 2 t + 1 

)

(

 3t  dr = 3(cos t − t sin t )i +  2 j + (sin t + t cos t )k  t + 1  dt   t t2 + 1

  j  

+ (cos t + cos t − t sin t )k = − 3(2sin t + t cos t )i + 3

1 j 2 t + ( 1)1/2

+ (2 cos t − t sin t )k v 2 = 9 (cos t − t sin t )2 + 9

Then

t2 + (sin t + t cos t )2 t2 + 1

Expanding and simplifying yields v 2 = t 4 + 19t 2 + 1 + 8(cos 2 t + t 4 sin 2 t ) − 8(t 3 + t )sin 2t

Then

v = [t 4 + 19t 2 + 1 + 8(cos 2 t + t 4 sin 2 t ) − 8(t 3 + t )sin 2t ]1/ 2

and

dv 4t 3 + 38t + 8(−2 cos t sin t + 4t 3sin 2 t + 2t 4 sin t cos t ) − 8[(3t 2 + 1)sin 2t + 2(t 3 + t ) cos 2t ] = dt 2[t 4 + 19t 2 + 1 + 8(cos 2 t + t 4 sin 2 t ) − 8(t 3 + t ) sin 2t ]1/ 2

Now

2 2  dv   v  a 2 = at2 + an2 =   +    dt   ρ 

2



At t = 0:

a = 3j + 2k

or

a = 13 ft/s 2 dv =0 dt v 2 = 9 (ft/s) 2

Then, with

dv = 0, dt

we have

a=

or

ρ=

v2

ρ 9 ft 2 /s 2

ρ = 2.50 ft 

13 ft/s 2


PROBLEM 11.153 A satellite will travel indefinitely in a circular orbit around a planet if the normal component of the acceleration of the satellite is equal to g ( R /r ) 2 , where g is the acceleration of gravity at the surface of the planet, R is the radius of the planet, and r is the distance from the center of the planet to the satellite. Knowing that the diameter of the sun is 1.39 Gm and that the acceleration of gravity at its surface is 274 m/s 2 , determine the radius of the orbit of the indicated planet around the sun assuming that the orbit is circular. Earth: (umean )orbit = 107 Mm/h.

SOLUTION g = 274 m/s 2 ,

For the sun, R=

and

Given that an =

1 1 D =   (1.39 × 109 ) = 0.695 × 109 m 2 2

gR 2 v2 and that for a circular orbit = a n r r2 r =

Eliminating an and solving for r,

v = 107 × 106 m/h = 29.72 × 103 m/s

For the planet Earth, Then

gR 2 v2

r =

(274)(0.695 × 109 ) 2 = 149.8 × 109 m (29.72 × 103 ) 2

r = 149.8 Gm 


PROBLEM 11.154 A satellite will travel indefinitely in a circular orbit around a planet if the normal component of the acceleration of the satellite is equal to g ( R /r ) 2 , where g is the acceleration of gravity at the surface of the planet, R is the radius of the planet, and r is the distance from the center of the planet to the satellite. Knowing that the diameter of the sun is 1.39 Gm and that the acceleration of gravity at its surface is 274 m/s 2 , determine the radius of the orbit of the indicated planet around the sun assuming that the orbit is circular. Saturn: (umean )orbit = 34.7 Mm/h.

SOLUTION g = 274 m/s 2

For the sun, R=

and

Given that an =

1 1 D =   (1.39 × 109 ) = 0.695 × 109 m 2 2

gR 2 v2 .and that for a circular orbit: = a n r r2 gR 2 v2

Eliminating an and solving for r,

r =

For the planet Saturn,

v = 34.7 × 106 m/h = 9.639 × 103 m/s

Then,

r =

(274)(0.695 × 109 ) 2 = 1.425 × 1012 m (9.639 × 103 )2

r = 1425 Gm 


PROBLEM 11.155 Determine the speed of a satellite relative to the indicated planet if the satellite is to travel indefinitely in a circular orbit 100 mi above the surface of the planet. (See information given in Problems 11.153–11.154). Venus: g = 29.20 ft/s 2 , R = 3761 mi.

SOLUTION From Problems 11.153 and 11.154,

an =

gR 2 r2

For a circular orbit,

an =

v2 r

v= R

Eliminating an and solving for v, For Venus,

g r

g = 29.20 ft/s 2 R = 3761 mi = 19.858 × 106 ft. r = 3761 + 100 = 3861 mi = 20.386 × 106 ft

Then,

v = 19.858 × 106

29.20 = 23.766 × 103 ft/s 20.386 × 106 v = 16200 mi/h 


PROBLEM 11.156 Determine the speed of a satellite relative to the indicated planet if the satellite is to travel indefinitely in a circular orbit 100 mi above the surface of the planet. (See information given in Problems 11.153–11.154). Mars: g = 12.17 ft/s 2 , R = 2102 mi.


an =

gR 2 r2


an =

v2 r

v= R

Eliminating an and solving for v, For Mars,

g r

g = 12.17 ft/s 2 R = 2102 mi = 11.0986 × 106 ft r = 2102 + 100 = 2202 mi = 11.6266 × 106 ft

Then,

v = 11.0986 × 106

12.17 = 11.35 × 103 ft/s 11.6266 × 106 v = 7740 mi/h 


PROBLEM 11.157 Determine the speed of a satellite relative to the indicated planet if the satellite is to travel indefinitely in a circular orbit 100 mi above the surface of the planet. (See information given in Problems 11.153–11.154). Jupiter: g = 75.35 ft/s 2 , R = 44, 432 mi.


an =

gR 2 r2


an =

v2 r

v= R

Eliminating an and solving for v, For Jupiter,

g r

g = 75.35 ft/s 2 R = 44432 mi = 234.60 × 106 ft r = 44432 + 100 = 44532 mi = 235.13 × 106 ft

Then,

v = (234.60 × 106 )

75.35 = 132.8 × 103 ft/s 6 235.13 × 10 v = 90600 mi/h 


PROBLEM 11.158 A satellite is traveling in a circular orbit around Mars at an altitude of 300 km. After the altitude of the satellite is adjusted, it is found that the time of one orbit has increased by 10 percent. Knowing that the radius of Mars is 3382 km, determine the new altitude of the satellite. (See information given in Problems 11.153–11.155.)

SOLUTION an = g

We have Then

g

R2 r2

and an =

v2 r

R2 v2 = r r2 v=R

g r

where

r =R+h

The circumference s of a circular orbit is equal to s = 2π r

Assuming that the speed of the satellite in each orbit is constant, we have s = vtorbit

Substituting for s and v 2π r = R torbit =

=

Now

g torbit r

2π r 3/2 R g 2π ( R + h)3/2 R g

(torbit ) 2 = 1.1(torbit )1 2π ( R + h2 )3/2 2π ( R + h1 )3/2 = 1.1 R R g g h2 = (1.1)2/3 ( R + h1 ) − R

= (1.1)2/3 (3382 + 300) km − (3382 km) h2 = 542 km 


PROBLEM 11.159 Knowing that the radius of the earth is 6370 km, determine the time of one orbit of the Hubble Space Telescope, knowing that the telescope travels in a circular orbit 590 km above the surface of the earth. (See information given in Problems 11.153–11.155.)

SOLUTION an = g

We have Then or

g

R2 r2

and an =

v2 r

R2 v2 = r r2 v=R

g r

where

r =R+h

The circumference s of the circular orbit is equal to s = 2π r

Assuming that the speed of the telescope is constant, we have s = vtorbit

Substituting for s and v 2π r = R

or

torbit = =

g torbit r

2π r 3/2 R g 2π [(6370 + 590) km]3/2 1h × −3 2 1/2 6370 km [9.81 × 10 km/s ] 3600 s torbit = 1.606 h 

or


PROBLEM 11.160 Satellites A and B are traveling in the same plane in circular orbits around the earth at altitudes of 120 and 200 mi, respectively. If at t = 0 the satellites are aligned as shown and knowing that the radius of the earth is R = 3960 mi, determine when the satellites will next be radially aligned. (See information given in Problems 11.153–11.155.)

SOLUTION an = g

We have Then

g

R2 r2

R2 v2 = r r2

and an = or

v2 r

v=R

g r

r =R+h

where The circumference s of a circular orbit is equal to

s = 2π r

Assuming that the speeds of the satellites are constant, we have s = vT

Substituting for s and v 2π r = R

or Now

T=

g T r

2π r 3/ 2 2π ( R + h)3/2 = R g R g

hB > hA  (T ) B > (T ) A

Next let time TC be the time at which the satellites are next radially aligned. Then, if in time TC satellite B completes N orbits, satellite A must complete ( N + 1) orbits. Thus, TC = N (T ) B = ( N + 1)(T ) A

or

 2π ( R + hB )3/2   2π ( R + hA )3/2  ( 1) N = N +    g g  R   R 



N=

or

=

Then

( R + hA )3/2 ( R + hB )

3/ 2

− ( R + hA )

1

(

3960 + 200 3960 +120

)

3/2

TC = N (T ) B = N

−1

3/2

=

1

(

R + hB R + hA

)

3/2

−1

= 33.835 orbits

2π ( R + hB )3/2 R g

[(3960 + 200) mi] × 1 h 2π 3960 mi 32.2 ft/s 2 × 1 mi 1/2 3600s 5280 ft 3/2

= 33.835

(

)

TC = 51.2 h 

or Alternative solution

From above, we have (T ) B > (T ) A . Thus, when the satellites are next radially aligned, the angles θ A and θ B swept out by radial lines drawn to the satellites must differ by 2π . That is,

θ A = θ B + 2π For a circular orbit

s = rθ

From above

s = vt and v = R

Then

θ=

At time TC : or

R g ( R + hA )

3/2

TC = TC = =

g r

R g R g s vt 1  g = = R t  t = 3/2 t =  r r r r  r ( R + h)3/2 R g ( R + hB )3/ 2

TC + 2π 2π

 1 1 R g −  ( R + hA )3/ 2 ( R + hB )3/ 2  2π

(

1 mi (3960 mi) 32.2 ft/s 2 × 5280 ft

× ×

1 [(3960 + 120) mi ]3/ 2

1 −

)

1/2

1

[(3960 + 200) mi ]3/ 2

1h 3600 s TC = 51.2 h 

or


PROBLEM 11.161 The oscillation of rod OA about O is defined by the relation θ = (3/π )(sin π t ), where θ and t are expressed in radians and seconds, respectively. Collar B slides along the rod so that its distance from O is r = 6(1 − e−2t ) where r and t are expressed in inches and seconds, respectively. When t = 1 s, determine (a) the velocity of the collar, (b) the acceleration of the collar, (c) the acceleration of the collar relative to the rod.

SOLUTION Calculate the derivatives with respect to time. 3

r = 6 − 6e −2t in.

θ=

r = 12e −2t in/s

θ = 3cosπ t rad/s

r = −24e−2t in/s 2

θ = −3π sin π t rad/s2

π

sin π t rad

At t = 1 s,

(a)

3

r = 6 − 6e −2 = 5.1880 in.

θ=

r = 12e −2 = 1.6240 in/s

θ = 3cos π = −3 rad/s

 r = −24e−2 = −3.2480 in/s 2

θ = −3π sin π = 0

π

sin π = 0

Velocity of the collar. v = rer + rθeθ = 1.6240 e r + (5.1880)(−3)eθ v = (1.624 in/s)er + (15.56 in/s)eθ 

(b)

Acceleration of the collar. a = (r − rθ 2 )e r + (rθ + 2rθ)eθ = [ −3.2480 − (5.1880)( −3) 2 ]er + (5.1880)(0) + (2)(1.6240)(−3)]eθ (−49.9 in/s 2 )er + (−9.74 in/s 2 )eθ 

(c)

Acceleration of the collar relative to the rod. a B /OA = (−3.25 in/s 2 )er 

a B /OA =  re r


PROBLEM 11.162 The rotation of rod OA about O is defined by the relation θ = t 3 − 4t , where θ and t are expressed in radians and seconds, respectively. Collar B slides along the rod so that its distance from O is r = 2.5t 3 − 5t 2 , where r and t are expressed in inches and seconds, respectively. When t = 1 s, determine (a) the velocity of the collar, (b) the acceleration of the collar, (c) the radius of curvature of the path of the collar.

SOLUTION Calculate the derivatives with respect to time. r = 2.5t 3 − 5t 2

θ = t 3 − 4t

r = 7.5t 2 − 10t

θ = 3t 2 − 4

 r = 15t − 10

θ = 6t

At t = 1 s,

(a)

r = 2.5 − 5 = −2.5 in.

θ = 1 − 4 = −3 rad

r = 7.5 − 10 = −2.5 in./s

θ = 3 − 4 = −1 rad/s

r = 15 − 10 = 5 in./s 2

θ = 6 rad/s 2

Velocity of the collar. v = rer + rθeθ = −2.5er + (−2.5)( −1)eθ v = ( −2.50 in./s)er + (2.50 in./s)eθ  v = (2.50) 2 + (2.50) 2 = 3.5355 in./s

Unit vector tangent to the path. et =

(b)

v = −0.70711e r + 0.70711eθ v

Acceleration of the collar. a = (r − rθ 2 )e r + (rθ + 2rθ)eθ = [5 − (−2.5)(−1)2 ]er + [(−2.5)(6) + (2)(−2.5)(−1)]eθ

a = (7.50 in/s 2 )er + (−10.00 in/s 2 )eθ 


PROBLEM 11.162 (Continued) Magnitude:

a = (7.50) 2 + (10.00) 2 = 12.50 in./s 2 at = aet

Tangential component:

at = (7.50)(−0.70711) + (−10.00)(0.70711) = −12.374 in./s 2

Normal component: (c)

an = a 2 − at2 = 1.7674 in./s 2

Radius of curvature of path. an =

ρ=

v2

ρ v 2 (3.5355 in./s) 2 = an 1.7674 in./s 2

ρ = 7.07 in. 


PROBLEM 11.163 The path of particle P is the ellipse defined by the relations r = 2/(2 − cos π t ) and θ = π t , where r is expressed in meters, t is in seconds, and θ is in radians. Determine the velocity and the acceleration of the particle when (a) t = 0, (b) t = 0.5 s.

SOLUTION We have

r=

2 2 − cos π t

θ = πt

Then

r =

−2π sin π t (2 − cos π t )2

θ = π

and

r = −2π

π cos π t (2 − cos π t ) − sin π t (2π sin π t ) (2 − cos π t )3

= −2π 2

(a)

At t = 0:

Now

2cos π t − 1 − sin 2 π t (2 − cos π t )3

r=2m

θ =0

r = 0

θ = π rad/s

r = −2π 2 m/s 2

θ = 0

v = rer + rθeθ = (2)(π )eθ v = (2π m/s)eθ 

or and

θ = 0

a = (r − rθ 2 )e r + (rθ + 2rθ)eθ = [ −2π 2 − (2)(π )2 ]er

a = −(4π 2 m/s 2 )er 

or (b)

At t = 0.5 s:

θ=

r =1 m r =

−2π π = − m/s 2 2 (2)

r = −2π 2

−1 − 1 π 2 = m/s 2 3 2 (2)

π rad 2

θ = π rad/s θ = 0



Now

 π v = rer + rθeθ =  −  e r + (1)(π )eθ  2 π  v = −  m/s  er + (π m/s)eθ  2  

or and

a = (r − rθ 2 )e r + (rθ + 2rθ)eθ

π 2    π  = − (1)(π ) 2  e r +  2  −  (π )  eθ   2   2 

π2  a = −  m/s 2  er − (π 2 m/s 2 )eθ   2 

or


PROBLEM 11.164 The two-dimensional motion of a particle is defined by the relations r = 2a cos θ and θ = bt 2 /2, where a and b are constants. Determine (a) the magnitudes of the velocity and acceleration at any instant, (b) the radius of curvature of the path. What conclusion can you draw regarding the path of the particle?

SOLUTION (a)

We have

r = 2a cos θ

1 θ = bt 2 2

Then

r = −2aθ sin θ

θ = bt

and

 r = −2a (θ sin θ + θ 2 cos θ )

θ = b

Substituting for θ and θ r = −2abt sin θ  r = −2ab(sin θ + bt 2 cos θ )

Now Then

vθ = rθ = 2abt cos θ

vr = r = −2abt sin θ

v = vr2 + vθ2 = 2abt[( − sin θ ) 2 + (cos θ ) 2 ]1/2 v = 2abt 

or Also

ar = r − rθ 2 = −2ab(sin θ + bt 2 cos θ ) − 2ab 2 t 2 cos θ = −2ab(sin θ + 2bt 2 cos θ )

and

aθ = rθ + 2rθ = 2ab cos θ − 4ab 2 t 2 sin θ = −2ab(cos θ − 2bt 2 sin θ )

Then

a = ar2 + aB2 = 2ab[(sin θ + 2bt 2 cos θ ) 2 + (cos θ − 2bt 2 sin θ ) 2 ]1/2 a = 2ab 1 + 4b 2 t 4 

or (b)

2

at2

+

an2

2 2  dv   v  =   +    dt   ρ 

Now

a =

Then

dv d = (2abt ) = 2ab dt dt

2



so that or

(

2ab 1 + 4b 2 t 4

)

2

= (2ab) 2 + an2

4a 2b 2 (1 + 4b 2t 4 ) = 4a 2 b 2 + an2

or

an = 4ab 2 t 2

Finally

an =

v2 (2abt ) 2 ρ= ρ 4ab 2 t 2

ρ =a 

or Since the radius of curvature is a constant, the path is a circle of radius a.




PROBLEM 11.165 As rod OA rotates, pin P moves along the parabola BCD. Knowing that the equation of this parabola is r = 2b /(1 + cos θ ) and that θ = kt , determine the velocity and acceleration of P when (a) θ = 0, (b) θ = 90°.

SOLUTION 2b θ = kt 1 + cos kt 2bk sin kt θ = k θ = 0 r = (1 + cos kt ) 2 2bk [(1 + cos kt )2 k cos kt + (sin kt )2(1 + cos kt )( k sin kt )] r = (1 + cos kt ) 4 r=

(a)

When θ = kt = 0: 2bk 1 [(2)2 k (1) + 0] = bk 2 4 2 (2)

r =b

r = 0

 r=

θ =0

θ = k

θ = 0

vθ = rθ = bk

vr = r = 0

v = bk eθ 

1 1  ar =  r − rθ 2 = bk 2 − bk 2 = − bk 2  2 2   aθ = rθ + 2rθ = b(0) + 2(0) = 0 

(b)

1 a = − bk 2 er  2

When θ = kt = 90°: r = 2b

r = 2bk

θ = 90°

θ = k

vr = r = 2bk

2bk [0 + 2k ] = 4bk 2 19 θ = 0 r =

vθ = rθ = 2bk

v = 2bk er + 2bk eθ 

ar = r − rθ 2 = 4bk 2 − 2bk 2 = 2bk 2 a = rθ + 2rθ = 2b(0) + 2(2bk )k = 4bk 2 θ

a = 2bk 2 e r + 4bk 2 eθ 


PROBLEM 11.166 The pin at B is free to slide along the circular slot DE and along the rotating rod OC. Assuming that the rod OC rotates at a constant rate θ, (a) show that the acceleration of pin B is of constant magnitude, (b) determine the direction of the acceleration of pin B.

SOLUTION From the sketch: r = 2b cos θ r = −2b sin θ θ

Since θ = constant, θ = 0 r = −2b cos θ θ 2 ar = r − rθ 2 = −2b cos θ θ 2 − (2b cos θ )θ 2 a = −4b cos θ θ 2 r

aθ = rθ + 2rθ = (2b cos θ )(0) + 2(−2b sin θ )θ 2 a = −4b sin θ θ 2 θ

a = ar2 + aθ2 = 4bθ 2 (− cos θ ) 2 + (− sin θ )2 a = 4bθ 2

Since both b and θ are constant, we find that a = constant 

γ = tan −1

 −4b sin θ θ 2 aθ = tan −1  2 ar  −4b cos θ θ

  

γ = tan −1 (tan θ ) γ =θ Thus, a is directed toward A 


PROBLEM 11.167 To study the performance of a racecar, a high-speed camera is positioned at Point A. The camera is mounted on a mechanism which permits it to record the motion of the car as the car travels on straightway BC. Determine (a) the speed of the car in terms of b, θ , and θ, (b) the magnitude of the acceleration in terms of b, θ , θ, and θ.

SOLUTION (a)

We have

r=

b cos θ

Then

r =

bθ sin θ cos 2 θ

We have

v 2 = vr2 + vθ2 = (r)2 + (rθ) 2 2

or

 bθ sin θ   bθ 2 =   +   2  cos θ   cos θ   b2θ 2 b2θ 2  sin 2θ = + 1  = 2  2 4 cos θ  cos θ  cos θ bθ v=± cos 2θ

For the position of the car shown, θ is decreasing; thus, the negative root is chosen.

v=−

bθ  cos 2θ

v=−

bθ  cos 2θ

Alternative solution. From the diagram or

r = −v sin θ bθ sin θ = −v sin θ cos 2θ

or   


PROBLEM 11.167 (Continued) (b)

For rectilinear motion

a=

dv dt

Using the answer from Part a v=−

Then

a=

bθ cos 2θ

d  bθ  − dt  cos 2θ

= −b

  

θ cos 2θ − θ(−2θ cos θ sin θ ) cos 4θ a=−

or

b (θ + 2θ 2 tan θ )  cos 2θ

Alternative solution From above Then

Now where

and

b bθ sin θ r = cos θ cos 2 θ (θ sin θ + θ 2 cos θ )(cos 2θ ) − (θ sin θ )(−2θ cos θ sin θ ) r = b cos 4θ  θ sin θ θ 2 (1 + sin 2 θ )  = b +  2 cos3θ  cos θ  r=

a 2 = ar2 + aθ2  θ sin θ θ 2 (1 + sin 2 θ )  bθ 2 + ar = r − rθ 2 = b  − 2 cos 2θ  cos θ  cos θ b   2θ 2 sin 2 θ  = + θ sin θ   cos θ  cos 2θ  b sin θ  ar = (θ + 2θ 2 tan θ ) 2 cos θ aθ = rθ + 2rθ = =

Then

bθ bθ 2 sin θ +2 cos θ cos 2θ

b cos θ  (θ + 2θ tan θ ) cos 2θ

a=±

b (θ + 2θ 2 tan θ )[(sin θ ) 2 + (cos θ )2 ]1/ 2 2 cos θ

For the position of the car shown, θ is negative; for a to be positive, the negative root is chosen. b a=− (θ + 2θ 2 tan θ )  cos 2 θ PROPRIETARY MATERIAL. © 2013 The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission. 244

PROBLEM 11.168 After taking off, a helicopter climbs in a straight line at a constant angle β . Its flight is tracked by radar from Point A. Determine the speed of the helicopter in terms of d, β , θ , and θ.

SOLUTION From the diagram r d = sin (180° − β ) sin ( β − θ ) or

d sin β = r (sin β cos θ − cos β sin θ ) tan β tan β cos θ − sin θ

or

r=d

Then

r = d tan β

−(− tan β sin θ − cos θ )  θ (tan β cos θ − sin θ )2

tan β sin θ + cos θ = dθ tan β (tan β cos θ − sin θ ) 2

From the diagram vr = v cos ( β − θ )

where

vr = r

Then dθ tan β

tan β sin θ + cos θ = v(cos β cos θ + sin β sin θ ) (tan β cos θ − sin θ )2 = v cos β (tan β sin θ + cos θ ) v=

or

dθ tan β sec β  (tan β cosθ − sin θ )2

Alternative solution. We have

v 2 = vr2 + vθ2 = (r) 2 + ( rθ) 2


PROBLEM 11.168 (Continued) Using the expressions for r and r from above  tan β sin θ + cos θ  v =  dθ tan β  (tan β cos θ − sin θ ) 2  

2

1/ 2

or

 (tan β sin θ + cos θ )2  dθ tan β v=± + 1  2 (tan β cos θ − sin θ )  (tan β cos θ − sin θ )  1/ 2

  tan 2 β + 1 dθ tan β =±  2 (tan β cos θ − sin θ )  (tan β cos θ − sin θ ) 

Note that as θ increases, the helicopter moves in the indicated direction. Thus, the positive root is chosen. v=

dθ tan β sec β  (tan β cos θ − sin θ )2


PROBLEM 11.169 At the bottom of a loop in the vertical plane, an airplane has a horizontal velocity of 315 mi/h and is speeding up at a rate of 10 ft/s2. The radius of curvature of the loop is 1 mi. The plane is being tracked by radar at O. What are r , θ and θ for this the recorded values of r,  instant?

SOLUTION Geometry. The polar coordinates are r = (2400) 2 + (1800)2 = 3000 ft

Velocity Analysis.

 1800   = 36.87°  2400 

θ = tan −1 

v = 315 mi/h = 462 ft/s

vr = 462 cos θ = 369.6 ft/s vθ = −462sin θ = −277.2 ft/s vr = r vθ = rθ

θ =

r = 370 ft/s 

vθ 277.2 =− r 3000

θ = −0.0924 rad/s  Acceleration analysis.

at = 10 ft/s 2 an =

v2

ρ

=

(462) 2 = 40.425 ft/s 2 5280



ar = at cos θ + an sin θ = 10 cos 36.87° + 40.425 sin 36.87° = 32.255 ft/s 2 aθ = − at sin θ + an cos θ = −10 sin 36.87° + 40.425 cos 36.87° = 26.34 ft/s 2 ar =  r − rθ 2  r = ar + rθ 2  r = 32.255 + (3000)( −0.0924) 2

 r = 57.9 ft/s 2 

aθ = rθ + 2rθ a 2rθ θ = θ − r r 26.34 (2)(369.6)(−0.0924) = − 3000 3000

θ = 0.0315 rad/s 2 


PROBLEM 11.170 Pin C is attached to rod BC and slides freely in the slot of rod OA which rotates at the constant rate ω. At the instant when β = 60°, r and θ. Express your answers in terms determine (a) r and θ, (b)  of d and ω.

SOLUTION

Looking at d and β as polar coordinates with d = 0, vβ = d β = d ω,

vd = d = 0

aβ = d β + 2d β = 0,

r = d 3 for angles shown.

Geometry analysis: (a)

Velocity analysis:

ad = d − d β 2 = −d ω 2

Sketch the directions of v, er and eθ. vr = r = v ⋅ er = d ω cos120° 1 r = − d ω  2 vθ = rθ = v ⋅ eθ = d ω cos 30°

θ = (b)

Acceleration analysis:

3 dω cos 30° dω 2 = r d 3

θ =

1 ω 2

Sketch the directions of a, er and eθ. ar = a ⋅ e r = a cos150° = −

3 dω 2 2

3  r − rθ2 = − d ω2 2  r =−

3 3 1  dω 2 + rθ 2 = − dω 2 + d 3  ω  2 2 2 

2

r = −

3 dω 2  4

1 aθ = a ⋅ eθ = d ω 2 cos120° = − d ω 2 2 a = rθ + 2rθ θ

θ =

1 (aθ − 2rθ) = r

1 3d

 1  1  1   2  − d ω − (2)  − d ω  ω   2 2   2   

θ = 0 


PROBLEM 11.171 For the racecar of Problem 11.167, it was found that it took 0.5 s for the car to travel from the position θ = 60° to the position θ = 35°. Knowing that b = 25 m, determine the average speed of the car during the 0.5-s interval. PROBLEM 11.167 To study the performance of a racecar, a high-speed camera is positioned at Point A. The camera is mounted on a mechanism which permits it to record the motion of the car as the car travels on straightway BC. Determine (a) the speed of the car in terms of b, θ , and θ, (b) the magnitude of the acceleration in terms of b, θ , θ, and θ.

SOLUTION From the diagram: Δr12 = 25 tan 60° − 25 tan 35° = 25.796 m

Now

vave = =

Δr12 Δt12 25.796 m 0.5 s

= 51.592 m/s vave = 185.7 km/h 

or


PROBLEM 11.172 For the helicopter of Problem 11.168, it was found that when the helicopter was at B, the distance and the angle of elevation of the helicopter were r = 3000 ft and θ = 20°, respectively. Four seconds later, the radar station sighted the helicopter at r = 3320 ft and θ = 23.1°. Determine the average speed and the angle of climb β of the helicopter during the 4-s interval. PROBLEM 11.168 After taking off, a helicopter climbs in a straight line at a constant angle β . Its flight is tracked by radar from Point A. Determine the speed of the helicopter in terms of d, β , θ , and θ.

SOLUTION We have

r0 = 3000 ft r4 = 3320 ft

θ0 = 20° θ 4 = 23.1°

From the diagram: Δr 2 = 30002 + 33202 − 2(3000)(3320) cos (23.1° − 20°)

or

Δr = 362.70 ft

Now

vave =

Δr Δt 362.70 ft = 4s = 90.675 ft/s vave = 61.8 mi/h 

or Also, or

Δr cos β = r4 cos θ 4 − r0 cos θ0 cos β =

3320 cos 23.1° − 3000 cos 20° 362.70

β = 49.7° 

or


PROBLEM 11.173 A particle moves along the spiral shown; determine the magnitude of the velocity of the particle in terms of b, θ , and θ.

SOLUTION Hyperbolic spiral.

r=

b

θ

dr b dθ b =− 2 = − 2 θ dt θ dt θ b  b vr = r = − 2 θ vθ = rθ = θ r =

θ

θ

2

 1  1 v = vr2 + vθ2 = bθ  − 2  +    θ  θ  bθ = 2 1+θ 2

2

θ

v=

b

θ2

1 + θ 2 θ 


PROBLEM 11.174 A particle moves along the spiral shown; determine the magnitude of the velocity of the particle in terms of b, θ , and θ.

SOLUTION Logarithmic spiral.

r = ebθ r =

dr dθ = bebθ = bebθ θ dt dt

vr = r = bebθ θ vθ = rθ = ebθ θ v=

vr2

+ vθ = e θ b 2 + 1 bθ

2

v = ebθ 1 + b 2 θ 


PROBLEM 11.175 A particle moves along the spiral shown. Knowing that θ is constant and denoting this constant by ω , determine the magnitude of the acceleration of the particle in terms of b, θ , and ω.

SOLUTION b

Hyperbolic spiral.

r=

From Problem 11.173

r = −

θ

 r =−

b

θ2

θ

b  2b  2 θ + 3θ 2

θ

θ

b 2b b ar =  r − rθ 2 = − 2 θ + 3 θ 2 − θ 2

θ

θ

θ

b b b  b  aθ = rθ + 2rθ = θ + 2  − 2 θ θ = θ − 2 2 θ 2 θ θ θ  θ 

Since θ = ω = constant, θ = 0, and we write: b 2 bω 2 2 − ω ω = 3 (2 − θ 2 ) θ θ3 θ 2 b bω aθ = −2 2 ω 2 = − 3 (2θ ) θ θ ar = +

2b

a = ar2 + aθ2 =

bω 2

θ

3

(2 − θ 2 ) 2 + (2θ ) 2 =

bω 2

θ

3

4 − 4θ 2 + θ 4 + 4θ 2 a=

bω 2

θ3

4 +θ 4 


PROBLEM 11.176 A particle moves along the spiral shown. Knowing that θ is constant and denoting this constant by ω , determine the magnitude of the acceleration of the particle in terms of b, θ , and ω.

SOLUTION Logarithmic spiral.

r = ebθ

dr = bebθ θ dt r = bebθ θ + b 2 ebθ θ 2 = bebθ (θ + bθ 2 ) r =

ar = r − rθ 2 = bebθ (θ + bθ 2 ) − ebθ θ 2 a = rθ + 2rθ = ebθ θ + 2(bebθ θ)θ θ

Since θ = ω = constant, θ = θ , and we write ar = bebθ (bω 2 ) − ebθ ω 2 = ebθ (b 2 − 1)ω 2 aθ = 2bebθ ω 2 a = ar2 + aθ2 = ebθ ω 2 (b2 − 1)2 + (2b) 2 = ebθ ω 2 b4 − 2b 2 + 1 + 4b 2 = ebθ ω 2 b 4 + 2b 2 + 1 = ebθ ω 2 (b2 + 1) 2 = ebθ ω 2 (b2 + 1) a = (1 + b 2 )ω 2 ebθ 


PROBLEM 11.177 The motion of a particle on the surface of a right circular cylinder is defined by the relations R = A, θ = 2π t , and z = B sin 2π nt , where A and B are constants and n is an integer. Determine the magnitudes of the velocity and acceleration of the particle at any time t.

SOLUTION R=A R = 0

θ = 2π t

z = B sin 2π nt

θ = 2π

z = 2π n B cos 2π nt

 = 0 R

θ = 0

 z = −4π 2 n 2 B sin 2π nt

Velocity (Eq. 11.49) v = R e R + Rθeθ + zk v=

+ A(2π )eθ + 2π n B cos 2π nt k v = 2π A2 + n 2 B 2 cos 2 2π nt 

Acceleration (Eq. 11.50)  - Rθ 2 )e + ( Rθ + 2 Rθ)e +  a = (R zk R θ

a = −4π 2 Aek − 4π 2 n 2 B sin 2π nt k a = 4π 2 A2 + n4 B 2 sin 2 2π nt 


PROBLEM 11.178 Show that r = hφ sin θ knowing that at the instant shown, step AB of the step exerciser is rotating counterclockwise at a constant rate φ.

SOLUTION From the diagram r 2 = d 2 + h 2 − 2dh cos φ

Then Now or

2rr = 2dhφ sin φ r d = sin φ sin θ r=

d sin φ sin θ

Substituting for r in the expression for r  d sin φ    sin θ  r = dhφ sin φ  

or

r = hφ sin θ

Q.E.D.



Alternative solution. First note

α = 180° − (φ + θ )

Now

v = v r + vθ = re r + rθeθ

With B as the origin vP = dφ

( d = constant  d = 0)



With O as the origin

(vP )r = r

where

(vP )r = vP sin α

Then Now or substituting

r = dφ sin α h d = sin α sin θ d sin α = h sin θ r = hφ sin θ

Q.E.D.


PROBLEM 11.179 The three-dimensional motion of a particle is defined by the relations R = A(1 − e −t ), θ = 2π t , and z = B(1 − e− t ). Determine the magnitudes of the velocity and acceleration when (a) t = 0, (b) t = ∞.

SOLUTION R = A(1 − e −t ) R = Ae −t  = − Ae−t R

θ = 2π t

z = B (1 − e−t )

θ = 2π

z = Be−t

θ = 0

 z = − Be−t

Velocity (Eq. 11.49) v = R e R + Rθeθ + zk v = Ae −t e R + 2π A(1 − e −t )eθ + Be−t k

(a)

When

t = 0: e−t = e0 = 1;

(b)

When

t = ∞ : e− t = e −∞ = 0

v = A2 + B 2 

v = Ae R + Bk v = 2π Aeθ

v = 2π A 

Acceleration (Eq. 11.50)  − Rθ1 )e + ( Rθ + 2 Rθ)e +  a = (R zk R θ = [ − Ae−t − A(1 − e −t )4π 2 ]e R + [0 + 2 Ae−t (2π )]eθ − Be −t k

(a)

When

t = 0: e −t = e0 = 1

a = − Ae R + 4π Aeθ − B k a = A2 + (4π A) 2 + B 2

(b)

When

a = (1 + 16π 2 ) A2 + B 2 

t = ∞ : e −t = e−∞ = 0

a = −4π 2 Ae R

a = 4π 2 A 


PROBLEM 11.180* For the conic helix of Problem 11.95, determine the angle that the osculating plane forms with the y axis. PROBLEM 11.95 The three-dimensional motion of a particle is defined by the position vector r = (Rt cos ωnt)i + ctj + (Rt sin ωnt)k. Determine the magnitudes of the velocity and acceleration of the particle. (The space curve described by the particle is a conic helix.)

SOLUTION First note that the vectors v and a lie in the osculating plane. Now

r = ( Rt cos ωn t )i + ctj + ( Rt sin ωn t )k

Then

v=

dr = R(cos ωn t − ωn t sin ωn t )i + cj + R(sin ωn t + ωn t cos ωn t )k dt

and

a=

dv dt

(

)

= R −ωn sin ωn t − ωn sin ωn t − ωn2 t cos ωn t i +R

(

ωn cos ωn t + ωn cos ωn t − ωn2 t sin ωn t

)k

= ωn R[−(2sin ωn t + ωn t cos ωn t )i + (2cos ωn t − ωn t sin ωn t )k ]

It then follows that the vector ( v × a) is perpendicular to the osculating plane.

i j k ( v × a) = ωn R R(cos ωn t − ωn t sin ωn t ) c R(sin ωn t + ωn t cos ωn t ) −(2sin ωn t + ωn t cos ωn t ) 0 (2cos ωn t − ωn t sin ωn t ) = ωn R{c(2 cos ωn t − ωn t sin ωn t )i + R[ −(sin ωn t + ωn t cos ωn t )(2sin ωn t + ωn t cos ωn t ) − (cos ωn t − ωn t sin ωn t )(2 cos ωn t − ωn t sin ωn t )] j + c(2sin ωn t + ωn t cos ωn t )k

(

)

= ωn R c(2 cos ωn t − ωn t sin ωn t )i − R 2 + ωn2 t 2 j + c(2sin ωn t + ωn t cos ωn t )k   



The angle α formed by the vector ( v × a) and the y axis is found from cos α =

Where

( v × a) ⋅ j | ( v × a) || j |

|j| =1

(

( v × a) ⋅ j = −ωn R 2 2 + ωn2 t 2

)

(

|( v × a) | = ωn R c 2 (2 cos ωn t − ωn t sin ωn t ) 2 + R 2 2 + ωn2 t 2 

)

2

1/ 2

+ c 2 (2sin ωn t + ωn t cos ωn t )2 

(

)

2 1/2

(

) 

= ωn R c 2 4 + ωn2 t 2 + R 2 2 + ωn2t 2 

Then

( ) ω R c ( 4 + ω t ) + R ( 2 + ω t )    −R ( 2 + ω t ) = c 4 + ω t + R 2 + ω t  ) ( )   ( −ωn R 2 2 + ωn2t 2

cos α =

2

n

2 2 n

2

2 2 n

2

1/2

2 2 n

2

2 2 n

2

2 2 n

2 1/2

The angle β that the osculating plane forms with y axis (see the above diagram) is equal to

β = α − 90° Then

cos α = cos ( β + 90°) = −sin β − sin β =

Then

tan β =

( ) ) + R ( 2 + ω t ) 

− R 2 + ωn2t 2

(

c 2 4 + ω 2t 2 n 

(

R 2 + ωn2 t 2 c

2

2 2 n

2

1/ 2

)

4 + ωn2 t 2

(

 R 2 + ωn2t 2 β = tan   c 4 + ω 2t 2 n  −1

or

)    


PROBLEM 11.181* Determine the direction of the binormal of the path described by the particle of Problem 11.96 when (a) t = 0, (b) t = π /2 s.

SOLUTION Given:

)

(

r = ( At cos t )i + A t 2 + 1 j + ( Bt sin t )k r − ft, t − s;

A = 3, B − 1

First note that eb is given by eb =

v×a |v ×a |

)

(

Now

r = (3t cos t )i + 3 t 2 + 1 j + (t sin t )k

Then

v=

dr dt 3t

= 3(cos t − t sin t )i +

j + (sin t + t cos t )k

(

t

)

2 dv t 2 +1 a= = 3( − sin t − sin t − t cos t )i + 3 t + 12− t j dt t +1 + (cos t + cos t − t sin t )k 3 = −3(2sin t + t cos t )i + 2 j + (2 cos t − t sin t )k (t + 1)3/2

and

(a)

t2 +1

At t = 0:

v = (3 ft/s)i a = (3 ft/s 2 ) j + (2 ft/s 2 )k

Then

and Then

v × a = 3i × (3j + 2k ) = 3(−2 j + 3k ) | v × a | = 3 (−2) 2 + (3) 2 = 3 13 eb =

3( −2 j + 3k ) 3 13

cos θ x = 0

or

θ x = 90°

=

cos θ y = −

1 13 2

(−2 j + 3k )

13

θ y = 123.7°

cos θ z =

3 13

θ z = 33.7°





(b)

At t =

Then

π 2

s:

  3π   3π v = − ft/s  i +  ft/s  j + (1 ft/s)k    2   π2 +4    π 24  a = −(6 ft/s 2 )i +  2 ft/s 2  j −  ft/s 2  k 3/2   (π + 4)  2 i 3π v×a = − 2 −6

j 3π 2 (π + 4)1/ 2 24 (π 2 + 4)3/ 2

k 1 −

π 2

   3π 2 24 3π 2 6 i = − + − +   2 1/2 4 (π 2 + 4)3/ 2    2(π + 4)   36π 18π k + − 2 + 2 3/2 1/2  (π + 4)   (π + 4) = −4.43984i − 13.40220 j + 12.99459k

and

Then

or

  j 

| v × a | = [(−4.43984) 2 + ( −13.40220)2 + (12.99459) 2 ]1/ 2 = 19.18829 1 (−4.43984i − 13.40220 j + 12.99459k ) 19.1829 4.43984 13.40220 12.99459 cos θ x = − cos θ y = − cos θ z = 19.18829 19.18829 19.18829

eb =

θ x = 103.4°

θ y = 134.3°

θ z = 47.4°




PROBLEM 11.182 The motion of a particle is defined by the relation x = 2t 3 − 15t 2 + 24t + 4, where x and t are expressed in meters and seconds, respectively. Determine (a) when the velocity is zero, (b) the position and the total distance traveled when the acceleration is zero.

SOLUTION x = 2t 3 − 15t 2 + 24t + 4

dx = 6t 2 − 30t + 24 dt dv a= = 12t − 30 dt v=

so

(a)

0 = 6t 2 − 30t + 24 = 6 (t 2 − 5t + 4)

Times when v = 0.

(t − 4)(t − 1) = 0

(b)

t = 1.00 s, t = 4.00 s 

Position and distance traveled when a = 0. a = 12t − 30 = 0

t = 2.5 s

x2 = 2(2.5)3 − 15(2.5) 2 + 24(2.5) + 4

so

x = 1.50 m 

Final position For

0 ≤ t ≤ 1 s, v > 0.

For

1 s ≤ t ≤ 2.5 s, v ≤ 0.

At

t = 0,

At

t = 1 s, x1 = (2)(1)3 − (15)(1) 2 + (24)(1) + 4 = 15 m

x0 = 4 m.

Distance traveled over interval: x1 − x0 = 11 m For

1 s ≤ t ≤ 2.5 s,

v≤0

Distance traveled over interval | x2 − x1 | = |1.5 − 15 | = 13.5 m

Total distance:

d = 11 + 13.5

d = 24.5 m 


PROBLEM 11.183 A particle starting from rest at x = 1 m is accelerated so that its velocity doubles in magnitude between x = 2 m and x = 8 m. Knowing that the acceleration of the particle is defined by the relation a = k[ x − (A/x)], determine the values of the constants A and k if the particle has a velocity of 29 m/s when x = 16 m.

SOLUTION We have

v



When x = 1 ft, v = 0:

v 0

dv A  = a = kx−  dx x 

vdv =



x

1

A  k  x −  dx x 

1 2 1 v = k  x 2 − A ln 2 2

or

x

 x 1

1 1 = k  x 2 − A ln x −  2 2

1 2 1 1 3  v2 = k  (2) 2 − A ln 2 −  = k  − A ln 2  2 2 2 2    

At x = 2 ft:

1 2 1 1 v8 = k  (8) 2 − A ln 8 −  = k (31.5 − A ln 8) 2 2 2

x = 8 ft:

Now

1 2 v 2 8 1 2 v 2 2

v8 = 2: v2

= (2) 2 =

k (31.5 − A ln 8) k ( 32 − A ln 2 )

6 − 4 A ln 2 = 31.5 − A ln 8

1 25.5 = A(ln 8 − 4 ln 2) = A(ln 8 − ln 24 ) = A ln   2 A=

When x = 16 m, v = 29 m/s:

25.5 ln 12

A = −36.8 m 2 

1 1 25.5 1 (29) 2 = k  (16) 2 − ln(16) −  1 2 2 2 ln ( 2 )   1  420.5k = k 128 + 102 −  = 230.5k 2  = 230.5k

k = 1.832 s −2 


PROBLEM 11.184 A particle moves in a straight line with the acceleration shown in the figure. Knowing that the particle starts from the origin with v0 = −2 m/s, (a) construct the v −t and x −t curves for 0 < t < 18 s, (b) determine the position and the velocity of the particle and the total distance traveled when t = 18 s.

SOLUTION Compute areas under a − t curve. A1 = (−0.75)(8) = −6 m/s A2 = (2)(4) = 8 m/s A3 = (6)(6) = 36 m/s v0 = −2 m/s v8 = v0 + A1 = −8 m/s v12 = v8 + A2 = 0 v18 = v12 + A3

v18 = 36 m/s 

Sketch v −t curve using straight line portions over the constant acceleration periods. Compute areas under the v −t curve. 1 (−2 − 8)(8) = −40 m 2 1 A5 = (−8)(4) = −16 m 2 1 A6 = (36)(6) = 108 m 2

A4 =

x0 = 0 x8 = x0 + A4 = −40 m x12 = x8 + A5 = −56 m x18 = x12 + A6

Total distance traveled = 56 + 108

x18 = 52 m  d = 164 m 


PROBLEM 11.185 The velocities of commuter trains A and B are as shown. Knowing that the speed of each train is constant and that B reaches the crossing 10 min after A passed through the same crossing, determine (a) the relative velocity of B with respect to A, (b) the distance between the fronts of the engines 3 min after A passed through the crossing.

SOLUTION (a)

We have

v B = v A + v B/A

The graphical representation of this equation is then as shown. Then

vB2 /A = 662 + 482 − 2(66)(48) cos 155°

or

vB/A = 111.366 km/h

and or

48 111.366 = sin α sin 155°

α = 10.50° v B/A = 111.4 km/h

(b)

10.50° 

First note that at t = 3 min, A is (66 km/h) ( 603 ) = 3.3 km west of the crossing. 7 at t = 3 min, B is (48 km/h) ( 60 ) = 5.6 km southwest of the crossing.

Now

rB = rA + rB/A

Then at t = 3 min, we have rB2/A = 3.32 + 5.62 − 2(3.3)(5.6) cos 25° rB/A = 2.96 km 

or


PROBLEM 11.186 Slider block B starts from rest and moves to the right with a constant acceleration of 1 ft/s 2. Determine (a) the relative acceleration of portion C of the cable with respect to slider block A, (b) the velocity of portion C of the cable after 2 s.

SOLUTION Let d be the distance between the left and right supports. Constraint of entire cable: xB + ( xB − x A ) + 2(d − x A ) = constant 2vB − 3v A = 0 aA =

and

2 2 aB = (1) = 0.667 ft/s 2 3 3

Constraint of Point C:

2aB − 3a A = 0 a A = 0.667 ft/s 2

or

2(d − x A ) + yC/ A = constant

−2v A + vC/ A = 0

− 2a A + aC/ A = 0

and

aC/ A = 2a A = 2(0.667) = 1.333 ft/s 2

(a)

aC/ A = 1.333 ft/s 2



Velocity vectors after 2s: v A = (0.667)(2) = 1.333 ft/s vC/ A = (1.333)(2) = 2.666 ft/s v C = v A + v C/ A

Sketch the vector addition. vC2 = v A2 + vC2 / A = (1.333) 2 + (2.666) 2 = 8.8889(ft/s) 2 vC = 2.981 ft/s tan θ =

vC/ A vA

=

2.666 = 2, 1.333

θ = 63.4°

vC = 2.98 ft/s

(b)

63.4° 


PROBLEM 11.187 Collar A starts from rest at t = 0 and moves downward with a constant acceleration of 7 in./s 2 . Collar B moves upward with a constant acceleration, and its initial velocity is 8 in./s. Knowing that collar B moves through 20 in. between t = 0 and t = 2 s, determine (a) the accelerations of collar B and block C, (b) the time at which the velocity of block C is zero, (c) the distance through which block C will have moved at that time.

SOLUTION 

From the diagram



− y A + ( yC − y A ) + 2 yC + ( yC − yB ) = constant

Then

−2v A − vB + 4vC = 0

(1)

and

−2a A − aB + 4aC = 0

(2)

(v A )0 = 0

Given:

(a A ) = 7 in./s 2 ( v B )0 = 8 in./s

a B = constant

 

At t = 2 s (a)

y − (yB )0 = 20 in. y B = ( y B ) 0 + ( vB ) 0 t +

We have At t = 2 s:

−20 in. = (−8 in./s)(2 s) + aB = −4 in./s 2

1 aB t 2 2 1 aB (2 s) 2 2

or

a B = 2 in./s 2 

Then, substituting into Eq. (2) −2(7 in./s 2 ) − (−2 in./s 2 ) + 4aC = 0 aC = 3 in./s 2

or

aC = 3 in./s 2 



(b)

Substituting into Eq. (1) at t = 0 −2(0) − (−8 in./s) + 4(vC )0 = 0 or (vC )0 = −2 in./s  vC = (vC )0 + aC t

Now

(c)

When vC = 0:

0 = (−2 in./s) + (3 in./s 2 )t

or

t=

2 3

t = 0.667 s 

yC = ( yC )0 + (vC )0 t +

We have At t =

2 s 3

s:

2 yC − ( yC )0 = ( −2 in./s)  3 = −0.667 in.

1 aC t 2 2

 1 2 s  + (3 in./s 2 )   2 3 or

 s 

2

y C − (y C )0 = 0.667 in. 


PROBLEM 11.188 A golfer hits a ball with an initial velocity of magnitude v0 at an angle α with the horizontal. Knowing that the ball must clear the tops of two trees and land as close as possible to the flag, determine v0 and the distance d when the golfer uses (a) a six-iron with α = 31°, (b) a five-iron with α = 27°.

SOLUTION The horizontal and vertical motions are x t cos α 1 2 1 y = (v0 sin α )t − gt = x tan α − gt 2 2 2 x = (v0 cos α )t

v0 =

or

(1)

2( x tan α − y) g

or

t2 =

At the landing Point C:

yC = 0,

And

xC = (v0 cos α )t =

(2) t =

2v0 sin α g

2v02 sin α cos α g

(3)

(a) α = 31° To clear tree A:

x A = 30 m, y A = 12 m

From (2),

t A2 =

From (1),

(v0 ) A =

To clear tree B:

2(30 tan 31° − 12) = 1.22851 s 2 , 9.81

t A = 1.1084 s

30 = 31.58 m/s 1.1084cos 31°

xB = 100 m,

yB = 14 m

From (2),

(t B ) 2 =

2(100 tan 31° − 14) = 9.3957 s 2 , 9.81

From (1),

(v0 ) B =

100 = 38.06 m/s 3.0652 cos 31°

The larger value governs,

v0 = 38.06 m/s

From (3),

xC =

t B = 3.0652 s

v0 = 38.1 m/s 

(2)(38.06) 2 sin 31° cos 31° = 130.38 m 9.81

d = xC − 110

d = 20.4 m 



(b)

α = 27° By a similar calculation,

t A = 0.81846 s, t B = 2.7447 s,

(v0 ) A = 41.138 m/s, (v0 ) B = 40.890 m/s,

v0 = 41.138 m/s

v0 = 41.1 m/s 

xC = 139.56 m,

d = 29.6 m 


PROBLEM 11.189 As the truck shown begins to back up with a constant acceleration of 4 ft/s 2 , the outer section B of its boom starts to retract with a constant acceleration of 1.6 ft/s 2 relative to the truck. Determine (a) the acceleration of section B, (b) the velocity of section B when t = 2 s.

SOLUTION For the truck,

a A = 4 ft/s 2

For the boom,

a B/ A = 1.6 ft/s 2

50°

(a) a B = a A + a B/ A Sketch the vector addition. By law of cosines: aB2 = a A2 + aB2/ A − 2a AaB/ A cos 50° = 42 + 1.62 − 2(4)(1.6) cos 50° aB = 3.214 ft/s 2

Law of sines:

sin α =

aB/ A sin 50° aB

α = 22.4°,

=

1.6 sin 50° = 0.38131 3.214

a B = 3.21 ft/s 2

22.4° 

v B = (vB )0 + aBt = 0 + (3.214)(2)

(b)

v B = 6.43 ft/s 2

22.4° 


PROBLEM 11.190 A motorist traveling along a straight portion of a highway is decreasing the speed of his automobile at a constant rate before exiting from the highway onto a circular exit ramp with a radius of 560-ft. He continues to decelerate at the same constant rate so that 10 s after entering the ramp, his speed has decreased to 20 mi/h, a speed which he then maintains. Knowing that at this constant speed the total acceleration of the automobile is equal to one-quarter of its value prior to entering the ramp, determine the maximum value of the total acceleration of the automobile.

SOLUTION First note

v10 = 20 mi/h =

88 ft/s 3

While the car is on the straight portion of the highway. a = astraight = at

and for the circular exit ramp a = at2 + an2

where

an =

v2

ρ

By observation, amax occurs when v is maximum, which is at t = 0 when the car first enters the ramp. For uniformly decelerated motion v = v0 + at t

and at t = 10 s:

v = constant  a = an = a=

Then or

astraight

2 v10

ρ

1 ast. 4

88 v 2 ( 3 ft/s ) 1 = at  at = 10 = ρ 4 560 ft

2

at = −6.1460 ft/s 2

(The car is decelerating; hence the minus sign.)



Then at t = 10 s: or Then at t = 0:

88 ft/s = v0 + (−6.1460 ft/s 2 )(10 s) 3 v0 = 90.793 ft/s amax =

at2

 v2 + 0 ρ 

  

2

1/2

2   (90.793 ft/s) 2    2 2 = (−6.1460 ft/s ) +    560 ft    

amax = 15.95 ft/s 2 

or


PROBLEM 11.191 Sand is discharged at A from a conveyor belt and falls onto the top of a stockpile at B. Knowing that the conveyor belt forms an angle α = 25° with the horizontal, determine (a) the speed v0 of the belt, (b) the radius of curvature of the trajectory described by the sand at Point B.

SOLUTION The motion is projectile motion. Place the origin at Point A. Then x0 = 0 and y0 = 0. The coordinates of Point B are xB = 30 ft and yB = −18 ft. Horizontal motion:

Vertical motion:

vx = v0 cos 25°

(1)

x = v0 t cos 25°

(2)

v y = v0 sin 25° − gt y = v0t sin 25° −

(3)

1 2 gt 2

(4)

At Point B, Eq. (2) gives v0 t B =

xB 30 = = 33.101 ft cos 25° cos 25°

Substituting into Eq. (4), 1 −18 = (33.101)(sin 25°) − (32.2)t B2 2 tB = 1.40958 s

(a)

Speed of the belt.

v0 =

v0t B 33.101 = = 23.483 1.40958 tB





v0 = 23.4 ft/s 

Eqs. (1) and (3) give vx = 23.483cos 25° = 21.283 ft/s v y = (23.483) sin 25° − (32.2)(1.40958) = −35.464 ft/s tan θ

−v y vx

= 1.66632

θ = 59.03°

v = 41.36 ft/s



Components of acceleration. a = 32.2 ft/s 2

at = 32.2sin θ

an = 32.2cos θ = 32.2 cos 59.03° = 16.57 ft/s 2

(b)

Radius of curvature at B.

an =

ρ=

v2

ρ v 2 (41.36) 2 = an 16.57

ρ = 103.2 ft 


PROBLEM 11.192 The end Point B of a boom is originally 5 m from fixed Point A when the driver starts to retract the boom with a constant radial acceleration of  r = −1.0 m/s 2 and lower it with a constant angular acceleration θ = −0.5 rad/s 2 . At t = 2 s, determine (a) the velocity of Point B, (b) the acceleration of Point B, (c) the radius of curvature of the path.

SOLUTION r0 = 5 m, r0 = 0,  r = −1.0 m/s 2

Radial motion.

1 2  rt = 5 + 0 − 0.5t 2 2 r = r0 +  rt = 0 − 1.0t r = r0 + r0 t +

r = 5 − (0.5)(2)2 = 3 m r = (−1.0)(2) = −2 m/s

At t = 2 s,

θ 0 = 60° =

Angular motion.

π 3

rad, θ0 = 0, θ = −0.5 rad/s 2

1 π + 0 − 0.25t 2 2 3 θ = θ0 + θt = 0 − 0.5t

θ = θ0 + θ0 + θt 2 =

θ=

At t = 2 s,

π

+ 0 − (0.25)(2) 2 = 0.047198 rad = 2.70°

3  θ = −(0.5)(2) = −1.0 rad/s

Unit vectors er and eθ .

(a)

Velocity of Point B at t = 2 s. v B = re r + rθeθ = −(2 m/s)e r + (3 m)(−1.0 rad/s)eθ



 v B = (−2.00 m/s)er + ( −3.00 m/s)eθ 

 tan α = v=

vθ −3.0 = = 1.5 vr −2.0 vr2

2

α = 56.31° 2



2

+ vθ = (−2) + (−3) = 3.6055 m/s



Direction of velocity. et =

v −2er − 3eθ = = −0.55470er − 0.83205eθ 3.6055 v

θ + α = 2.70 + 56.31° = 59.01°  (b)

v B = 3.61 m/s



59.0° 

Acceleration of Point B at t = 2 s. a B = ( r − rθ 2 )er + ( rθ + 2rθ)eθ

= [−1.0 − (3)(−1) 2 ]e r + [(3)(−0.5) + (2)(−1.0)(−0.5)]eθ a B = ( −4.00 m/s 2 )er + (2.50 m/s 2 )eθ 

tan β =

aθ 2.50 = = −0.625 ar −4.00

β = −32.00°

a = ar2 + aθ2 = ( −4) 2 + (2.5) 2 = 4.7170 m/s 2

θ + β = 2.70° − 32.00° = −29.30° a B = 4.72 m/s 2

Tangential component:

29.3° 

at = (a ⋅ et )et at = (−4er + 2.5eθ ) ⋅ (−0.55470er − 0.83205eθ )et

= [(−4)(−0.55470) + (2.5)( −0.83205)]et = (0.138675 m/s 2 )et = 0.1389 m/s 2

Normal component:

59.0°

a n = a − at

a n = −4er + 2.5eθ − (0.138675)(−0.55470e r − 0.83205eθ )

= (−3.9231 m/s 2 )er + (2.6154 m/s 2 )eθ an = (3.9231) 2 + (2.6154) 2 = 4.7149 m/s 2

(c)

Radius of curvature of the path. an =

ρ=

v2

ρ v 2 (3.6055 m/s)2 = an 4.7149 m/s

ρ = 2.76 m 


PROBLEM 11.193 A telemetry system is used to quantify kinematic values of a ski jumper immediately before she leaves the ramp. According to the system r = 500 ft, r = −105 ft/s,  r = −10 ft/s 2 , θ = 25°, θ = 0.07 rad/s, θ = 0.06 rad/s 2 . Determine (a) the velocity of the skier immediately before she leaves the jump, (b) the acceleration of the skier at this instant, (c) the distance of the jump d neglecting lift and air resistance.

SOLUTION (a)

Velocity of the skier.

(r = 500 ft, θ = 25°) v = vr e r + vθ eθ = re r + rθeθ

= ( −105 ft/s)e r + (500 ft)(0.07 rad/s)eθ v = ( −105 ft/s)e r + (35 ft/s)eθ 

Direction of velocity:

v = ( −105cos 25° − 35cos 65°)i + (35sin 65° − 105sin 25°) j = ( −109.95 ft/s)i + ( −12.654 ft/s) j v y −12.654 α = 6.565° tan α = = vx −109.95

v = (105) 2 + (35)2 = 110.68 ft/s v = 110.7 ft/s

6.57° 



(b)

Acceleration of the skier. a = ar er + aθ eθ = ( r − rθ 2 )er + (rθ + 2rθ)eθ

ar = −10 − (500)(0.07)2 = −12.45 ft/s 2 aθ = (500)(0.06) + (2)(−105)(0.07) = 15.30 ft/s 2 a = (−12.45 ft/s 2 )er + (15.30 ft/s 2 )eθ 

a = (−12.45)(i cos 25° + j sin 25°) + (15.30)(−i cos 65° + j sin 65°)

= (−17.750 ft/s 2 )i + (8.6049 ft/s 2 ) j ay 8.6049 tan β = = β = −25.9° ax −17.750 a = (12.45) 2 + (15.30)2 = 19.725 ft/s 2 a = 19.73 ft/s 2

(c)

25.9° 

Distance of the jump d. Projectile motion. Place the origin of the xy-coordinate system at the end of the ramp with the x-coordinate horizontal and positive to the left and the y-coordinate vertical and positive downward. Horizontal motion: (Uniform motion)







 Vertical motion:





x0 = 0 x0 = 109.95 ft/s





x = x0 + x0 t = 109.95t 

(Uniformly accelerated motion) y0 = 0 y 0 = 12.654 ft/s  y = 32.2 ft/s



(from Part a )

(from Part a) 

2

y = y0 + y 0 t +

1 2  yt = 12.654t − 16.1t 2  2

At the landing point, x = d cos 30° y = 10 + d sin 30° or

(1) y − 10 = d sin 30°

(2)



Multiply Eq. (1) by sin 30° and Eq. (2) by cos 30° and subtract x sin 30° − ( y − 10) cos 30° = 0 (109.95t )sin 30° − (12.654t + 16.1t 2 − 10) cos 30° = 0 −13.943t 2 + 44.016t + 8.6603 = 0 t = −0.1858 s and 3.3427 s

Reject the negative root. x = (109.95 ft/s)(3.3427 s) = 367.53 ft d=

x cos 30°

d = 424 ft 


CHAPTER 11 PROBLEM 11.CQ1

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