KINEMATICS SESSION 1& 2 AIM Basic definitions and their explanations BASIC DEFINITIONS AND THEIR EXPLANATIONS Distance: The actual path length of the particle from initial to final positions of journey. Symbol: “S” SI unit: m The distance travelled by the particle is independent of direction of motion and it is scalar quantity. Displacement: The shortest distance between initial and final positions of journey. Displacement is a vector quantity and its magnitude is the shortest distance between the initial and final positions of journey. Displacement and distance can be evaluated not only between initial and final positions but also between any two points of journey. Consider A and B are two positions of particle in its journey. y
A
= = =
1
→Final position vector → Initial position vector → Displacement vector Hence displacement is “final position vector” minus “initial position vector”. Displacement can have zero magnitude, but distance travelled can never be zero. If r =a1 ̂+b1 ̂+c1 and r = a2 ̂+b2 ̂+c2 then r = r − r = (a2 ̂+b2 ̂+c2 ) – (a1 ̂+b1 ̂+c1 ) = (a2–a1) ̂+ (b2–b1) ̂+ (c2–c1) Speed: The distance travelled per unit time Symbol: “v” SI unit: ms-1 Speed is the scalar quantity Velocity: The displacement of the particle per unit time. Symbol: “v”: SI unit: ms-1 Velocity is a vector quantity; its direction is same as that of displacement. Let us consider an example to discuss their concepts: 2
A
1
B 3
Let a particle move from point A to point B as shown: Along path 1, distance travelled is 100m, time taken is 10s Along path 2, distance is 200m, time taken is 15s Along path 3, distance travelled is 150m, time taken is 12s
2
Along path 1: distance =s=100m:speed = Velocity = ̅ =
ms–1 = 10ms–1;
ms–1 (from A to B) ⟹ ̅ = 10ms–1 from A to B
(displacement is same distance along path 1) Along path 2: displacement = 100 m; time = 15 s speed =
=
ms–1; Velocity = ̅ =
=
ms–1 (from A to B)
Along path 3: displacement = 100 m; distance = 150m; time = 12s Speed = = ms–1; velocity = = ms–1 (from A to B) Note: The direction of displacement and velocity is same for all the three paths i.e. imagine a vector drawn from A to B. The direction of vector gives the direction of displacement and velocity. Average Speed: - When any object travels with non-uniform speeds during different time intervals or travelling different segments of path, the ratio of total distance covered to the total time taken is called 'Average Speed'. e.g. Average Speed= (
(
)
)
If an object travels with speeds v1, v2, v3........vN, during time intervals t1, t2, t3 ...tN, travelling for distances s1, s2, s3..........sN, its average speed can be discussed in following three ways. When Distances & Speeds are When Distances & Time intervals are When Speeds & Mentioned Time intervals are mentioned
3
Average Speed ( ̅ ) = ….
…. ….
Average Speed ( ̅ ) =
+
Average Speed ( ̅ ) = … …
If s1 = s2 = s3........ =sN= then Average Speed ( ̅ ) =
…
If t1 = t2 = t3...........tn= then Average Speed ( ̅ ) =
…
i.e. Harmonic Mean of Velocities i.e. Arithmetic Mean of Velocities These are Basic independent cases. In practice you may have combination of two or more. Average velocity: It is defined as the ratio of displacement to time taken by the body Average Speed ( ̅ ) =
;
⃗=
⃗
Instantaneous velocity: The term “instantaneous” is used to explain an event happening in a very smallinterval of time. Usually instantaneous time is denoted as “dt”. The numerical value of ‘dt” is nota definite constant, but we can say that it less than one and close to zero i.e. about 0.01s or 0.001s. (Let us not assign a fixed value for it. Its depends on the context of explanation). Let us consider a situation of a particle moving along a straight line from A to B. At a time t = 0 (usually called initial time) the particle is at A and at a time t = 10s. Let AB = 100m. As the particle crosses the point C, can we guess time taken to cross this point. The answer is simple. “Impossible”. If we consider two nearest points, one point D just before reaching C and another point E just after crossing C. The
4
time interval from D to E is taken as instantaneous time i.e. “dt”. The displacement/ distance between D and E are denoted as / i.e. is called infinitesimal (very small) displacement ( ) DE is called infinitesimal distance (ds) Instantaneous speed: It is the infinitesimal distance travelled per infinitesimal time. Vinst = Instantaneous velocity: It is the ratio of infinitesimal displacement to infinitesimal time. in st =
Note: Always it is assumed that the particle is along a straight line path in this infinitesimal time. Uniform speed: Speed of the particle is said to be uniform, if it covers equal distances in equal intervals of time taken, however small the intervals are. i.e. VAB = VAc = VCB = VDE Non-uniform Speed: Speed of the particle is said to be non-uniform, if it covers unequal distances in equal intervals of time, however small the intervals are. Non-uniform velocity: Velocity of the particle is said to be nonuniform, if it travels unequal displacements in equal intervals of time, however small the intervals are. Uniform speed is possible for any path of particle i.e. straight line, curved paths. Uniform velocity is possible only for a straight line path, because velocity is a vector quantity i.e. for velocity to be constant, the direction of velocity should be same. 5
CLASS EXERCISE : 1] A circular park has a radius of 1 km. A man standing from the center of park, walking towards East and going along the circumference of the park reaches the North gate. What is his displacement? 2]
a)√2 km b) 1.57 km c) 1 km d) 2.57 km If a cyclist takes one minute to complete half revolution on a circular path 120m radius, what is the average velocity? a) 1 m/s
3]
4]
5]
b) 2 m/s c) 3 m/s d) 4 m/s A car covers the 1st half of the distance between two places at a speed of 40 km/hr and the second half at 60 km/hr. The average speed of the car is a) 100 km/hr b) 48 km/hr c) 50 km/hr d) 25 km/hr An aeroplane moves 400m towards north, 300m towards west and then 1200m vertically upwards. Then its displacement from the initial position is a) 1300m b) 1400m c) 1500m d) 1600m An ant starts from one corner of a cube of side length 3m and reaches the diagonally opposite corner. The displacement is
6
SESSION-3, 4 & 5 AIM Definition of acceleration, explanation of equations of motion. Definition of acceleration, explanation of equations of motion. Acceleration: The rate of change of velocity is acceleration. The direction of acceleration is the direction of change in velocity There is misconception that direction of acceleration is either along the direction of motion or opposite to direction of motion. It need not be parallel to direction of motion. = ; 1 and 2 need not be in the same direction 2– 1need not be either in the direction of 1 or 2. For a particle moving along a straight line, direction of acceleration is either in the direction of motion (in case of increasing velocity) or in the opposite to direction of motion (in case of decreasing velocity) Uniform acceleration: The body is said possess uniform acceleration if it executes equal change in velocity in equal intervals of time, however small the intervals are. There are two possible paths for this type of motion (i) straight line (ii) parabolic Non-uniform acceleration: The body is said to possess nonuniform acceleration, if it executes unequal change in velocity in equal intervals of time, however small the intervals are.
7
Average Acceleration: - If Velocity of an object changes from to during an interval of t=t1 to t=tz, the Average Acceleration is given as ⃑ Average=< ⃑>=
=
⃑ Instantaneous = ⃑ = lim
→
=
You know that the definitions might be boring but simple but applications are always interesting. Therefore, let's come to application For Motion in One Dimension you have already learnt Equations of Motion in One Dimension v=u+at s = ut + at2 vz = uz + 2as We can not only establish same set of equation without graphs but we can derive any result if we use Calculus Technique. Calculus Techniques:- The Equations of Motion in One Dimension are consistent to constant acceleration only. Therefore to start with consider constant acceleration. i.e. ⃑ = constant t Therefore by definition For 2nd Equation we can use ⃑ = ∴ d ⃑ = ⃑dt⟹ ∫ ⃑ = ∫ ⃑. ⟹[ ⃑] = ⃑ [ ] ⟹ ⃑ − ⃑ = ⃑ = ⃑ ( − 0) ⟹ ⃑ = ⃑ + Therefore you got your 1st Equation ⃑ = ⃑ + ⃑t or v = u + at because for Motion in One Dimension only “+ve” & “-ve” sign are more than sufficient to rep represent direction. "+ve" means away from Origin i.e. towards Right or Upwards and "-ve" means towards the Origin i.e. towards left or Downwards. ⃑ = ⃑ + ⃑t Sufficient for any type of motion with Constant Acceleration.
8
For 2nd Equation use ⃑ = ∴ d ⃑ = ⃑
⃑[
⟹ ∫ ⃑ ⟹ ∫⃑
⃑
⃑ = ∫ ( ⃑ + ⃑ ).
⟹ [ ⃑] ⃑⃑ = ⃑. +
⃑. ⟹ ⃑ − ⃑ = ⃑. + ⃑.
Where ⃑ − ⃑ = ⃑ (Displacement)
Therefore you got your 1st Equation ⃑= ⃑ + ⃑ = v = u + at because for Motion in One Dimension only "+ve" &" -ve "sign are more than sufficient to represent direction. ”+ve" means away from Origin i.e. towards Right or Upwards and "-ve" means towards the Origin i.e. towards left or Downwards. NOTE: ⃑ = ⃑ + ⃑ is sufficient for any type of motion with Constant Acceleration. For 2nd Equation use ⃑ = ∴ d ⃑= ⃑.
⃑
⟹ ∫⃑
⃑
⃑ = ∫ ⃑
⟹
⃑ = ∫ (⃑+ ⃑ )
⟹
[ ⃑] ⃑⃑ = ⃑. + ⃑. ⟹ ⃑ − ⃑ = ⃑. + ⃑.
Where ⃑ − ⃑ = ⃑ (Displacement)
Therefore you got your 2nd Equation. ⃑ = ⃑. + ⃑. or s = u.t. + a.t2 because for Motion in One Dimension only "+ve" &”-ve" sign are more than sufficient to represent direction. "+ve" means away from Origin i.e. towards Right or Upwards and "- ve" means towards the Origin i.e. towards left or Downwards. Note: ⃑ = ⃑. + ⃑. Constant Accelection.
is sufficient for any type of motion with
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For 3rd Equation method is little bit different. As we know by definition ⃑ = Therefore ⃑ = ⃑
=∫ ⃑ ⃑ = ⃑
⃑ ∫⃑
⃑ ⃑
but we require Velocity and Displacement form. ×
⃑
⟹ ⃑ = ⃑.
⃑⟹
⃑
⃑ ⃑
⃑
⃑. ⃑ ⟹ ⃑.d ⃑
= ⃑. [ ⃑] ⃑⃑
⟹ ⃑ ⃑ =2. ⃑( ⃑ − ⃑ ) Where ⃑ − ⃑ = ⃑ (Displacement) Therefore you got your 3rd Equation. ⃑ = + 2. ⃑. ⃑or ⃑ = + 2. . because for Motion in One Dimension only "+ve" & "-ve" sign are more than sufficient to represent direction. "+ve" means away from Origin i.e. towards Right or Upwards and " - ve" means towards the Origin i.e. towards left or Downwards. NOTE: ⃑ = + 2. ⃑. ⃑ is sufficient for any type of motion with Constant Acceleration. Instantaneous Acceleration: Instantaneous Acceleration is the Acceleration of the object at a particular instant. Here we get to use Calculus once more. Sn = u + (2n - 1); Sn is displacement of the particle in the n-th second The term n-th second is to be understood with clarity. This instant with duration of 1 second. The instant with duration of ‘t’ seconds can have duration less than or greater than or equal to one second. But the n-th second always has the interval “one second”. The last equation should not be used to calculate displacement if the interval of time not equal to 1 second.
10
Example: (i) A car travels for six seconds. The last second of travel is 6th second. (Interval of last second is one second) (ii) A vertical travel for 6.1 seconds. The last second of travel has an interval of one second. i.e. from 5.1 s to 6.1 s (iii) A cyclist travels for 0.95 seconds. There is no last second, because the travel time is less than one second. To calculate displacement from t1 to t2 seconds, use s = ut + at2 i.e. s1 = ut1 + a and s2 = ut2+ a ⟹(s2-s1) = u (t2 -t1) + a ( -
)
To calculate displacement from 5.1s to 6.1s, also use s = ut + at2 along with Sn = u + (2n -1) To calculate displacement from 5s to 6s. Use Sn = u + (2n -1) with n= 6 For a body moving with uniform acceleration, average velocity is given by and by definition, average velocity =
⟹ =
11
EXERCISE (A) 1]
2]
3]
4]
5]
Moving with uniform acceleration, a body covers 150m during 10sec so that it covers 24 m during the tenth second. Find the initial velocity and acceleration of the body a) 2 ms–1; 5 ms–2 b) 5ms–1; 2 ms–2 c) 3 ms–1; 4 ms–2 d) 4ms–1;3ms–2 A body moving with uniform acceleration covers 100m in the first 10 seconds and 150m in the next 10 seconds. The initial velocity of the body is a) 15 ms–1 b) 7.5 ms–1 c) 5 ms–1 d) 2.5 ms–1 A particle moving with a constant acceleration describes in the last second of its motion th of the whole distance. If it starts from rest, how long is the particle in motion a) 5s b) 10s c) 15s d) none A body starts with initial velocity u and moves with uniform acceleration f. If when the velocity has increased to 5u, the acceleration is reversed in direction, the magnitude remaining constant; it turns to the starting point with velocity of a) – u b) –6u c) –7u d) –9u A body moving with uniform acceleration travels a distance Sn = (0.4n + 9.8) m in nth sec. Find the initial velocity of the body in ms–1 a) 0.4 b) 10
c) 5
d) 4
12
6]
7]
8]
The displacement of particle is zero at t = 0 and it is x at t = t. It starts moving in the positive x direction with a velocity which varies as V = K, where K is constant. The relation between V and t is a) Kt/2 b) Kt3/2 c) K2t/2 d) 2Kt2 A particle starts from rest with constant acceleration for 20sec. If it travels a distance y1 in the first 10 sec and a distance y2 in the next 10 sec then a) y2 =2y1 b) y2 =3y1 c) y2 =4y1 d) y2 =5y1 The ends of a moving train with a constant acceleration pass a certain point with velocities u and v. What is the velocity V with which the mid–point of the train passes through the same point? a) v = b) v = √ + c) v =
d) V =
The velocity of a body at an instant is 20 ms–1. After 5s the velocity is 30 ms–1. How many seconds earlier from the instant, it might have started? Assume acceleration is uniform a) 8sec b) 10sec c) 6sec d) 7sec 10] The engine of a train moving with uniform acceleration passes an electric pole with a velocity u and the last compartment with a velocity V, the length of train that has passed the pole when the velocity is , if the total length of the train is l 9]
a)
( (
) )
b)
c)
d)
(
)
(
)
13
SESSION - 6 AIM Motion with Variable Acceleration Motion with Variable Acceleration (i) If acceleration is a function of time then a = f (t) then v = u +∫ ( ) and s = ut +∫(∫ ( ) ) (ii) If acceleration is a function of distance a = f(x) then v2=u2+2∫ ( ) (iii) If acceleration is a function of velocity a = f (v) then t = ∫ and x = x0 + ∫
( )
( )
ILLUSTRATION: Problem1. An electron starting from rest has a velocity that increases linearly with the time that is v = kt where k = 2 m/sec2. The distance travelled in the first 3 seconds will be a) 9 m b) 16 m c) 27 m d) 36 m Sol: (a) x = ∫
=∫ 2
=2
= 9m
14
Problem2.The acceleration of a particle is increasing linearly with time t as bt. The particle starts from the origin with an initial velocity The distance travelled by the particle in time t will be a) V0t + bt2 b) V0t + bt3 c) V0t + bt3 d) V0t + bt2 Sol: (c) ∫
=∫
=∫ ( )
⟹ v2 – v1=
=
S=∫
=
+∫
+
=
+
+
Problem3. The motion of a particle is described by the equation u = at. The distance travelled by the particle in the first 4 seconds a) 4 b)12 c)6 d) 8 Sol: d) u = at⟹ =at S= ∫
=a
=8a
Problem4. A body A moves with a uniform acceleration and zero initial velocity. Another body B starts from the same point moves in the same direction with a constant velocity v. The two bodies meet after a time t. The value of t is a)
b)
c)
d)
Sol: (a) Let they meet after time’t’. Distance covered by body A Distance covered by body B= vt and = ∴ =
15
Problem5. A student is standing at a distance of 50metres from the bus. As soon as the bus starts its motion with an acceleration of 1ms-2, the student starts running towards the bus with a uniform velocity Assuming the motion to be along a straight road, the minimum value of , so that the students is able to catch the bus is a) 5 ms-1 b)8 ms-1 c) 10 ms-1 d)12 ms-1 Sol: (c) Let student will catch the bus after sec. So it will cover distance ut. Similarly distance travelled by the bus will be for the given condition ut = 50 + (As a = 1 m/s2)
=50 +
=
+
Problem6. A particle is moving eastwards with velocity of 5 m/s. In 10 sec the velocity changes to 5 m/s northwards. The average acceleration in this time is a) Zero b) toward north-west c) m/s2toward north-east √
d) m/s2 toward north-west Sol: b)Δ ⃑ = ⃑ – ⃑ Δ =
+
−2
90 =
5 + 5 = 5√2
v1
2 5m / s 90o 1 5 m / s
16
Δ = 5√2 Average acceleration =
=
√
=
√
/ toward north-
west (As clear from the figure). Problem7. A body starts from the origin and moves along the xaxis such that velocity at any instant is given by where t is in second and velocity is in m/s. What is the acceleration of the particle, when it is 2m from the origin? a) 28 m/s2 b) 22 m/s2 c) 12 m/s2 d) 10 m/s2 Sol: (b) Given that = 4 − 2 x=∫ at x = t4-t2+c at t =0, x=0 ⟹C=0 When particle is 2m away from the origin 2= t4–t2 ⟹t4–t2–2=0 ⟹ (t2–2) (t2+1) = C ⟹ t=√2 sec a=
(4
=
− 2 ) = 12
−2⟹
= 12
− 2for
t = √2 sec ⟹a = 12×(√2)2–2⟹ =22 m/s2 Problem 8. A body of mass 10 kg is moving with a constant velocity of 10 m/s. When a constant force acts for 4 sec on it, it moves with a velocity 2 m/sec in the opposite direction. The acceleration produced in it is a) 3 m/s2 b) –3 m/s2 c) 0.3 m/s2 d) –0.3 m/s2 Sol: (b) Let particle moves towards east and by the application of constant force it moves towards west ⃑ = +10 / dx and ⃑ = −2 / Acceleration =
= ⟹
=
⃑
⃑
(−2) − (10) 12 =− = −3 / 4 4
17
Problem9. A car, moving with a speed of 50 km/hr, can be stopped by brakes after at least 6m. If the same car is moving at a speed of 100 km/hr, the minimum stopping distance is a) 6m b) 12m c) 18m d) 24m Sol (d): v2 = u2 - 2as ⟹ s = ⟹s u2 (As a = constant) =
⟹ s2 = 4s1 = 4 x 12 = 24m
Problem10. The velocity of a bullet is reduced from 200m/s to 100m/s while travelling through a wooden block of thickness 10cm. The retardation, assuming it to be uniform, will be a) 10 x 104 m/s2 b) 12 x 104 m/s2 c) 13.5 x 104 m/s2 d) 15 x 104 m/s2 Sol: (d) u = 200 m/s, v = 100 m/s, s = 0.1 m; ( ) ( ) a= = = 15 x 104 m/s2 × .
Problem 11.A body A starts from rest with an acceleration. After 2 seconds, another body B starts from rest with an acceleration. If they travel equal distances in the 5th second, after the start of A, then the ratio is equal to a) 5 : 9 b) 5 : 7 c) 9 : 5 d)9 : 7 Sol: (a) By using Sn = u+ (2n -1), Distance travelled by body A in 5th second= 0 +
(2 x 5 -1)
Distance travelled by body B in 3rd second is = 0+ According to problem: 0+ (2×5–1) = 0+ 5a2 ⟹
(2×3–1)
(2×3–1)⟹ 9 a1 =
=
18
Problem12. The average velocity of a body moving with uniform acceleration travelling a distance of 3.06 m is 0.34 ms-1. If the change in velocity of the body is 0.18ms-1 during this time, its uniform acceleration is a) 0.01 ms-2 b) 0.02 ms-2 c) 0.03 ms-2 d) 0.04 ms-2 . Sol: (b) Time = = =9sec . and Acceleration =
=
.
=0.02 m/s2.
Problem13. A particle travels 10m in first 5 sec and 10m in next 3 sec. Assuming constant acceleration what is the distance travelled in next 2 sec a) 8.3 m b) 9.3 m c) 10.3 m d) None of above Sol: a) Let initial (t = 0) velocity of particle = u for first 5sec of motion s5 = 10 metre, so by using s= ut + at2 10 = 5u + a (5)2⟹ 2u + 5a = 4 for first 8 sec of motion s8 = 20 metre 20 = 8u + a (8)2⟹ 2u + 8a = 5
......... (i) ........ (ii)
By solving (i) and (ii) u = m/s a = m/s2 Now distance travelled by particle in total 10 sec. s10 = u×10+ a(10)2 by substituting the value of u and a we will get s10 = 28.3 m So the distance in last 2 sec = s10 - s8 = 28.3 - 20 = 8.3 m
19
Problem14. A body travels for 15 sec starting from rest with constant acceleration. If it travels distances S1, S2 and S3 in the first five seconds, second five seconds and next five seconds respectively the relation between S1, S2 and S3 is a) S1 = S2 = S3 b) 5S1 = 3S2 = S3 c) S1 = S2 = d) S1 = S2 = S3 Sol: (c) Since the body starts from rest. Therefore u = 0 S1 = a(5)2 = S1 + S2 = a (10)2 =
⟹ S2 =
-S1 = 75
S1 + S2 + S3 = a (15)2 = ⟹S3 = -S2 - S1 = Thus Clearly S1 = S2 = S3 Problem15. If a body having initial velocity zero is moving with uniform acceleration the distance travelled by it in fifth second will be a) 36 metres b) 40 metres c) 100 metres d) Zero Sol: (a) sn= u + a (2n - 1) = 0 + (8) [2 x 5 - 1] = 36 metres Problem16. The engine of car produces acceleration 4m/sec2 in the car, if this car pulls another car of same mass, what will be the acceleration produced a) 8 m/s2 b) 2 m/s2 c) 4 m/s2 d) m/s2 Sol: (b) F = ma a if F = constant. Since the force is same and the effective mass of system becomes double
=
=
a2=
=2m/s2
20
Problem17. A body starts from rest. What is the ratio of the distance travelled by the body during the 4th and 3rd second? a) 7/5 b) 5/7 c)7/3 d)3/7 Sol: (a) As Sn (2 − 1), = CLASS EXERCISE : 1] If the relation between distance x and time t is of the from t = x2+ x here and being appropriate constants, then the retardation of the particle is a) 2 v3 b) 2 v3 c) 2 v3 d)2 2v3 2] A particle moves along a straight line according to the law = + 2 + . . The acceleration of the particle varies as a) S–3 b) S2/3 c) S3 d) S5/2 3] A point moves rectilinearly with deceleration whose modulus depends on the velocity v of the particle as = √ , where k is a positive constant. At the initial moment the velocity of the point is equal to v0. What distance will it traverse before it stops? What time will it take to cover that distance? 4] If the displacement of the particle varies with time as x1/2=t+7, then a) velocity of the particle is inversely proportional to t b) velocity of the particle is proportional to t c) velocity of the particle is inversely proportional t1/2. d) the particle moves with a constant acceleration. 5] A point moves in a straight line under the retardation kv2. If the initial velocity is u, the distance covered in t second is a) kut b) log (k ut) c) log(1+ kut)
d) k log (k ut)
21
6]
7]
A particle of mass 10–2 kg is moving along the positive x-axis under the influence of a force F(x) = – (k/2x2) where k = 10–2 Nm2. At time t = 0 it is at x = 1 m and its velocity v = 0. (i) Find its velocity when it reaches x = 0.5 m A particle moving in a straight line has an acceleration of (3t - 4) ms-2 at timet seconds. The particle is initially 1m from O, a fixed point on the line, with a velocity of 2 ms-1. Find the times when the velocity is zero. Find also the displacement of the particle from O whent = 3.
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SESSION 7, 8 & 9 AIM Graphical Representation of motion Motion of Body under Gravity (Free Fall). The force of attraction of earth on bodies is called force of gravity. Acceleration produced in the body by the force of gravity, is called acceleration due to gravity. It is represented by the symbol g. In the absence of air resistance, it is found that all bodies (irrespective of the size, weight or composition) fall with the same acceleration near the surface of the earth. This motion of a body falling towards the earth from a small altitude (h << R) is called free fall. An ideal one-dimensional motion under gravity in which air resistance and the small changes in acceleration with height are neglected. (1) If a body dropped from some height (initial velocity zero) (i) Equation of motion: Taking initial position as origin and direction of motion (i.e., downward direction) as a positive, here we have u=0 2h
t
g
h
v
h
v g
2 gh
v
2
2g
v
23
u=0 a = +g
[As body starts from rest] [As acceleration is in the direction of motion] v=gt … (i) h= .... (ii) =2gh .... (iii) hn= (2n+1) .... (iv) (ii) Graph of distance velocity and acceleration with respect to time: s
v
a g tan = g
t
t
t
(iii) As h = (1/2)gt2, i.e., h t2, distance covered in time t, 2t, 3t, etc., will be in the ratio of 12 : 22 : 32, i.e., square of integers. (iv) The distance covered in the nth sec, hn = (2 − 1)So distance covered in I, II, III sec, etc., will be in the ratio of 1: 3: 5, i.e., odd integers only. (2) If a body is projected vertically downward with some initial velocity Equation of motion: h = ut + gt2 = u2+2gh hn= u+ (2n –1) (3) If a body is projected vertically upward (i) Equation of motion: Taking initial position as origin and direction of motion (i.e., vertically up) as positive a =–g [As acceleration is downwards while motion upwards] 24
So, if the body is projected with velocity u and after time t it reaches up to height h then v=0
2h
t
h
g
u g
u u
h
2 gh
u
2
2g
=
−
h = ut– gt2
= u2–2gh
hn= u – (2n –1)
(ii) For maximum height v = 0 So from above equation u = gt, h = gt2 and =2 ℎ (iii) Graph of distance, velocity and acceleration with respect to time (for maximum height) : s
(u2/2g)
a
v +
O (u/g)
(u/g)
(2u/g)
+
t
O
–
g
–v
–a
t
t
It is clear that both quantities do not depend upon the mass of the body or we can say that in absence of air resistance, all bodies fall on the surface of the earth with the same rate. (4) In case of motion under gravity for a given body, mass, acceleration, and mechanical energy remain constant while speed, velocity, momentum, kinetic energy and potential energy change. 25
(5) The motion is independent of the mass of the body, as in any equation of motion, mass is not involved. That is why a heavy and light body when released from the same height, reach the ground simultaneously and with same velocity i.e., t = (2ℎ/ )and v = 2 ℎ (6) In case of motion under gravity time taken to go up is equal to the time taken to fall down through the same distance. Time of descent (t1) = time of ascent (t2) = u/g Total time of flight T = t1 + t2 = (7) In case of motion under gravity, the speed with which a body is projected up is equal to the speed with which it comes back to the point of projection. As well as the magnitude of velocity at any point on the path is same whether the body is moving in upwards or downward direction. (8) A ball is dropped from a building of height h and it reaches after t seconds on earth. From the same building if two ball are thrown (one upwards and other downwards) with the same velocity u and they reach the earth surface after t1 and t2 seconds respectively then
u
t1
u
t2
u=0
t
t=√ 26
(9) A body is thrown vertically upwards. If air resistance is to be taken into account, then the time of ascent is less than the time of descent. t2> t1 Let u is the initial velocity of body then time of ascent = and h =
(
)
where g is acceleration due to gravity and a is
retardation by air resistance and for upward motion both will work vertically downward. For downward motion a and g will work in opposite direction because a always work in direction opposite to motion and g always work vertically downward. So h = (g-a)
⟹
(
)
= ( − )
⟹ t2=
(
)(
)
Comparing t1 and t2 we can say that t2> t1 since (g + a) > (g - a) (10) A particle is dropped vertically from rest from a height. The time taken by it to fall through successive distance of 1m each will then be in the ratio of the difference in the square roots of the integer’s i.e. u=0 1m
t1 1
1m
t2 2 1
1m
t3 3 2
1m
t4 4 3
√1,(2–√1),(√1–√1)…….(√4 − √3),……. 27
CLASS EXERCISE : 1]
The variation of velocity with time of a particle moving along a straight line is illustrated in the following figure. The distance traveled by the particle in 4sec. is m/s
30 A
20
B C D
10 1
3]
E 4
3
a) 60m b) 45m c) 55m d) 50m In the given v-t graph, the distance travelled by the body in 5 second will be
a) 20 m b) 40 m c) 80 m d) 100 m For the displacement- time graph shown in fig. the ratio of the magnitudes of the speeds during the first two second and the next four second is displacement
2]
2
a) 1: 1
b) 2 : 1
20m
0s
2s Time
6s
c) 1 : 2
d) 3 : 2
28
The velocity- time graph of a particle moving along a straight line is shown in fig. The displacement of the body in 5 second is 2
1.5 1 V (m /s -1 )
4]
1.5
a) 2.5m
b) 1m
0 1 1
2
3
4 5 t(ins)
2
c) 2m
d) 3m
29
SESSION 10, 11 & 12 AIM Motion of a body under gravity Projectile -Motion in Two Dimensions Motion in a plane is called as motion in two dimensions e.g., projectile motion, circular motion etc. For the analysis of such motion our reference will be made of an origin and two co-ordinate axes x and y. Position of particle is known by knowing its coordinate (x, y). Velocity of particle will be resultant of velocities in x and y direction Vx and Vy. Similarly acceleration will be in the two directions. For analysis of such motion we analyses the motion along two axes independently i.e., while dealing motion in xdirection we need not to think what is going on in y-direction and vice versa. We have to study about projectile motion, circular motion and relative velocity under the head of motion in two dimensions. Projectile Motion A projectile is a particle, which is given an initial velocity, and then moves under the action of its weight alone. If the initial velocity is vertical, the particle moves in a straight line and such motion we had already discussed in ‘motion in one dimension as motion under gravity’. Here we are going to discuss the motion of particle which is projected obliquely near the earth’s surface. While discussing such motion we shall suppose the motion to be within such a moderate distance from the earth’s surface, that acceleration due to gravity may be considered to remain sensibly constant. We shall 30
also neglect the resistance of air and consider the motion to be in vacuum. Important Terms used in projectile motion When a particle is projected into air, the angle that the direction of projection makes with horizontal plane through the point of projection is called the angle of projection; the path, which the particle describes, is called the trajectory, the distance between the point of projection and the point where the path meets any plane draws through the point of projection is its range; the time that elapses in air is called as time of flight and the maximum distance above the plane during its motion is called as maximum height attained by the projectile. Analytical treatment of projectile motion Consider a particle projected with a velocity u of an angle with the horizontal from earth’s surface. If the earth did not attract a particle to itself, the particle would describe a straight line; on account of attraction of earth, however, the particle describes a curved path. This curve will be proved later to be always a parabola. Let us take origin at the point of projection and x-axis and y-axis along the surface of earth and perpendicular to it respectively as shown in figure. y
P
u
O
O x
By the principle of physical independence of forces, the weight of the body only has effect on the motion in vertical direction. It, 31
therefore, has no effect on the velocity of the body in the horizontal direction, and horizontal velocity therefore remains unaltered. Motion in x- direction: Motion in x - direction is motion with uniform velocity. At t = 0, x0 = 0 and ux = u cos Position after time t, x = x0 + uxt ⟹ x = (u cos ) t Velocity of any time t, vx = ux ⟹vx= u cos Motion in y-direction: Motion in y-direction is motion with uniformly acceleration When, t = 0, y0 = 0, uy = u sin and ay = -g ∴ After time‘t’, vy = uy + ay t ⟹vy = u sin - gt y = y0 + uyt + ayt2 ⟹y = uyt + ayt2 y = (u sin ) t Also, = + 2ayy ⟹ = u2 sin2 - 2 gy Time of Flight (T): Time of flight is the time during which particle moves from O to O i.e., when t = T, y = 0 ∴ From equation (iv)
32
O = u sin T- gT2
T=
Range of projectile (R): Range is horizontal distance travelled in time T. i.e., R = x (in time T) ∴ From equation (ii) R = ucos .T = ucos R= Maximum height reached (H): At the time particle reaches its maximum height velocity of particle becomes parallel to horizontal direction i.e., vy = 0 when y = H ∴ From equation (v) 0 = u2 sin2 − 2 gH; H = Equation of trajectory: The path traced by a particle in motion is called trajectory and it can be known by knowing the relation between x and y. From equation (1) and (2) eliminating time t we get y = x tan sec2 This is trajectory of path and is equation of parabola. So we can say the path of particle is parabolic. Velocity and direction of motion after a given time: After time‘t’ vx = ucos and vy = usin -gt Hence resultant velocity v =
+ 33
= +( − ) If direction of motion makes an angle tan = = ⟹ =
with horizontal
Velocity and direction of motion at a given height: At a height ‘h’, Vx = ucos And Vy =
−2 ℎ
Resultant velocity v =
+
;v =
−2 ℎ
Note that this is the velocity that a particle would have at height h if it is projected vertically from ground with u. SOME IMPORTATN POINTS REGARDING PROJECTILE MOTION OVER A HORIZONTAL PLANE i) For a given velocity of projection, the range of horizontal plane will be maximum when angle of projection is 450. We have range of projectile. R = Therefore if we keep on increasing range will increase and then decrease. Its value will be maximum when sin2 is maximum i.e., = 450 Also, maximum range Rmax = ii) For a given range and given initial speed of projection, there are two possible angle of projection which are complementary angle i.e., if one is other will be (900- ).
34
ClASS EXERCISE (A) : 1]
2]
3]
In the last second of free fall, a body covered 3/4 of its total path. Then the height from which the body is released will be a) 4.9m b) 9.8m c) 19.6m d) 39.2m A freely falling body travels a distance X in the nth second. In the next second if it travels a distance, Y. Then a) X+Y=g b) X – Y=g c) Y – X =g d) X = Y/g A body is released from height h above the ground which takes ‘t’ seconds to reach the ground. The position of the body after t/2 seconds is a) above the ground b)
above the ground
c) above the ground d) Depends upon the size of the body 4]
5]
6]
The average velocity of freely falling body is 21 ms-1. Then it is released from a height equal to a) 5m b) 90m c) 30m d) 60m A window is 0.5m high. A stone is released from a height 0.4m above the top of the window. The time taken by the stone to cross the window is a) 4 s b) 3/7 s c) 1/7 s d) 2/7 s A stone is dropped into a well and sound of the splash is heard 3 s later. If the depth of well is 44.1m find the velocity of sound in air (g = 9.8ms-2) a) 252.6ms-1 b) 352.8ms-1 c) 300ms-1
d) 350ms-1 35
7]
A body slides down a smooth inclined plane of inclination 300 with horizontal and length 19.6m starting from rest at the top. Find the speed and time to reach the bottom. (g = 9.8ms-2) a) 2.5s, 15ms-1 b) 3.83s, 15.4ms-1 c) 2.83s, 13.87ms-1 d) None
8]
A stone falls from the top of a tower in 8s. How much time will it take to cover the first quarter of distance starting from top? a) 2s b) 3s c) 4s d) 8s
CLASS EXERCISE (B): 1]
2]
3]
Two bodies one held 10m vertically above the other are released simultaneously. After falling freely for 3 seconds under gravity, their relative separation is a) 10m b) 5m c) 1m d) none After falling through the first h metres, a freely falling body acquires velocity ‘V’. After falling through the next h metres, velocity acquired by it would be a) 2 V b) √2 c) V/√2 d) 4 V A body is released from height ‘h’ above the ground. Exactly at the midway if g vanishes suddenly, its total time of fall is a)
4]
b) 2
c)
d)
A parachutist after bailing out (drops out) falls a distance of 19.6m after which the parachute opens and he decelerates downwards at 1.1ms-2. He reaches the ground with a speed of 2ms-1. Find total time taken and the height from the ground where parachutist bails out. a) 20s, 200m b) 40s, 300m c) 18s, 192.4md) 16s, 172.8m 36
5]
6]
7]
A ball is dropped from a height. Another ball is dropped from the same height after 2s. Their separation after 2 more second is a) g b) 2g c) 4g d) 6g A stone is dropped into water from a bridge 44.1m above the water. Another stone is thrown vertically downwards 1second later. Both strike the water simultaneously. Then initial speed of the second stone is a) 24.5ms-1 b) 49 ms-1 c) 9.8 ms-1 d) 12.25ms-1 A stone is dropped from a multistored building. If it crosses 2 floors in the first second of its free fall, the numbers of floors it can cross in the 3rd second of its fall is a) 5 b) 7 c) 10 d) 9
Body projected vertically upward CLASS EXERCISE (C): 1] A body is projected upwards with a velocity 98 m/s. Find: a) The maximum height reached b) The time taken to reach the maximum height c) Its velocity at a height 196 m from the point of projection d) Velocity with which it will cross down the point of projection e) The time taken to reach back the point of projection 2] A boy throws a stone vertically up and catches it after time ‘t’ seconds. Then maximum height reached by the stone is a)
b)
c)
d)
37
3]
4]
A body is projected vertically up with a velocity 19.6ms-1. Find the i) displacement after 4s ii) displacement in 2nd second. iii) Velocity after 3 seconds iv) ratio of displacements in 1st and 2nd second. iii) Velocity after 3 seconds iv) ratio of displacements in 1st and 2nd seconds respectively. A body projected vertically up reaches a maximum height h in time t. The time taken to reach half of the maximum height is a) t/2 b) t/4 c) d) t 1 − √
5]
6]
7] 8] 9]
√
A stone projected up with a velocity U reaches two points A and B at a distance ‘h’ with velocities U/2 and U/3. The maximum height reached by the stone is a) b) c) d) A body is projected vertically upwards. If t1 and t2 be the times at which it is at a height h above the point of projection while ascending and descending respectivley then a) h = t1t2 b) h = 2gt1t2 c) h= d) h = In above problem the velocity of projection is a) g(t1+t2) b) g(t1+t2) c) 2g(t1 + t2) d) 4g(t1 + t2) In above problem the maximum height reached by the body is a) g b) g c) g( + )2 d) g(t1+t2)2 In above problem the velocity of the body at height is a) gt1 b) gt2 c) d) √ −
38
CLASS EXERCISE (D): 1]
Assume that there is tower of sufficient height H. A particle is projected from bottom of tower with velocity u1 and another particle is dropped from top of tower with velocity u2. The time when the particles meet is T then a) T = ( b) T = ( ) ) c) T = (
2]
3]
4]
5]
)
d) T = (
)
If a particle occupies x seconds less and acquires a velocity y ms----1 more at one place than at another in falling through the same distance. If g1 and g2 are accelerations due to gravity at those two places, then x: y is equal to a) g1 / g2 b) g2 / g1 c) d) 1/ A rocket is fired vertically from the ground with a resultant vertical acceleration of 10 ms---2. The fuel is finished in one minute and the rocket continues to move up. If g = 10 ms---2, maximum height reached by the rocket is a) 36 Km b) 18 Km c) 9 Km d) none A body is projected vertically up from ground. In the last 2 seconds of its ascent distance travelled by it will be numerically a) b) g c) 2g d) 4g A body is projected vertically up from ground. In the last of its total journey distance travelled by it is h. Then its initial velocity of projection is numerically a) (h + g) b) (h _ g) c) (h + g / 2) c) (h - g / 2)
39
6]
Water drops fall from a tap on the floor 5m below at regular intervals of time. The first drop striking the floor when the fifth drop begins to fall. The height at which the third drop will be, from ground, at that instant when first strikes the ground, will be (g=10 ms-2). a) 1.25 m b) 2.15 m c) 2.75 m d) 3.75 m
Vertical projection from a height CLASS EXERCISE (E): 1] A balloon is ascending up with a velocity of 4 ms--1. An object is dropped from it when it is at a height of 100 m above the ground. The distance between the object and balloon after 2 seconds is (g = 10 ms--2) a) 10 m b) 20 m c) 30 m d) 40 m 2] A balloon rises from rest with a constant acceleration. A stone is released from it when it has risen to a height h. The time taken by the stone to reach the ground is a) 3]
b)
c)2
d) 4
From the top of a tower a stone is projected vertically up with a velocity 20 ms---1. After t seconds another3] from the top of a tower a stone is projected vertically up with a velocity 20 ms---1. After t seconds another stone is projected vertically down with a velocity 20 ms---1 so that both the stones reach the ground simultaneously. The t = (g = 10 ms--2) a) 2 b) 4 c) 5 d) 8
40
4]
A body projected vertically up with velocity u from the top of a tower reaches the foot of the tower with velocity 2u. Then height of that tower is a)
5]
6] 7]
8]
9]
b)
c)
d)
A rocket fired vertically upwards with constant acceleration has its engine exhausted in 10 second. What is the maximum height reached by the rocket if its velocity at the end of 10th second 600 ms-1? [g = 10 ms---2] a) 600 x 5 m b) 600 x 25 m c) 600 x 35 m d) 600 x 40 m A particle is projected vertically upwards. Prove that it will be at of their greatest heights at times which are in the ratio 1: 3 A stone is released from a hydrogen balloon, going upwards with velocity 12 m/s, when it is at height of 65m from the ground. Time the stone will take to reach the ground is a) 5 sec b) 6 sec c) 7 sec d) 8 sec From the top of a tower a stone is projected vertically upward. When it reaches a distance h below that point, its velocity is double that of its velocity when it was at a height ‘h’ above the top of the tower. Then greatest height attained by the stone above the top of the tower is a) 2h b) h/3 c) 5h/3 d) 4h A balloon rises from rest on the ground with a constant acceleration . A stone is dropped when the balloon has rises to a height of Hm. Find the time taken by the stone to reach the ground
41
SESSION – 13, 14 & 15 Aim: Projectile -Motion in Two Dimensions PROJECTILE -MOTION IN TWO DIMENSIONS Motion in a plane is called as motion in two dimensions e.g., projectile motion, circular motion etc. For the analysis of such motion our reference will be made of an origin and two co-ordinate axes x and y. Position of particle is known by knowing its coordinate (x, y). Velocity of particle will be resultant of velocities in x and y direction Vx and Vy. Similarly acceleration will be in the two directions. For analysis of such motion we analyses the motion along two axes independently i.e., while dealing motion in xdirection we need not to think what is going on in y-direction and vice versa. We have to study about projectile motion, circular motion and relative velocity under the head of motion in two dimensions. Projectile Motion A projectile is a particle, which is given an initial velocity, and then moves under the action of its weight alone. If the initial velocity is vertical, the particle moves in a straight line and such motion we had already discussed in ‘motion in one dimension as motion under gravity’. Here we are going to discuss the motion of particle which is projected obliquely near the earth’s surface. While discussing such motion we shall suppose the motion to be within such a moderate distance from the earth’s surface, that acceleration due to gravity may be considered to remain sensibly constant. We shall
42
also neglect the resistance of air and consider the motion to be in vacuum. Important Terms used in projectile motion When a particle is projected into air, the angle that the direction of projection makes with horizontal plane through the point of projection is called the angle of projection; the path, which the particle describes, is called the trajectory, the distance between the point of projection and the point where the path meets any plane draws through the point of projection is its range; the time that elapses in air is called as time of flight and the maximum distance above the plane during its motion is called as maximum height attained by the projectile. Analytical treatment of projectile motion Consider a particle projected with a velocity u of an angle with the horizontal from earth’s surface. If the earth did not attract a particle to itself, the particle would describe a straight line; on account of attraction of earth, however, the particle describes a curved path. This curve will be proved later to be always a parabola. Let us take origin at the point of projection and x-axis and y-axis along the surface of earth and perpendicular to it respectively as shown in figure. y
P
u
O
O x
By the principle of physical independence of forces, the weight of the body only has effect on the motion in vertical direction. It, 43
therefore, has no effect on the velocity of the body in the horizontal direction, and horizontal velocity therefore remains unaltered.
Motion in x- direction: Motion in x - direction is motion with uniform velocity. At t = 0, x0 = 0 and ux = u cos Position after time t, x = x0 + uxt ⟹ x = (u cos ) t Velocity of any time t, vx = ux ⟹vx= u cos Motion in y-direction: Motion in y-direction is motion with uniformly acceleration When, t = 0, y0 = 0, uy = u sin and ay = -g ∴ After time‘t’, vy = uy + ay t ⟹vy = u sin - gt y = y0 + uyt + ayt2 ⟹y = uyt + ayt2 y = (u sin ) t Also, = + 2ayy ⟹ = u2 sin2 - 2 gy Time of Flight : Time of flight is the time during which particle moves from O to O i.e., when t = T, y = 0 ∴ From equation (iv) O = u sin T- gT2 T= Range of projectile (R): 44
Range is horizontal distance travelled in time T. i.e., R = x (in time T) ∴ From equation (ii) R = ucos .T = ucos R= Maximum height reached (H): At the time particle reaches its maximum height velocity of particle becomes parallel to horizontal direction i.e., vy = 0 when y = H ∴ From equation (v) 0 = u2 sin2 − 2 gH; H = Equation of trajectory: The path traced by a particle in motion is called trajectory and it can be known by knowing the relation between x and y. From equation (1) and (2) eliminating time t we get y = x tan sec2 This is trajectory of path and is equation of parabola. So we can say the path of particle is parabolic. Velocity and direction of motion after a given time: After time‘t’ vx = ucos and vy = usin -gt Hence resultant velocity v =
+
= +( − ) If direction of motion makes an angle tan = = ⟹ =
with horizontal
45
Velocity and direction of motion at a given height: At a height ‘h’, Vx = ucos And Vy =
−2 ℎ
Resultant velocity v =
+
;v =
−2 ℎ
Note that this is the velocity that a particle would have at height h if it is projected vertically from ground with u. SOME IMPORTANT POINTS REGARDING PROJECTILE MOTION OVER A HORIZONTAL PLANE i]
For a given velocity of projection, the range of horizontal plane will be maximum when angle of projection is 45°. We have range of projectile. R = Therefore if we keep on increasing range will increase and then decrease. Its value will be maximum when sin2 is maximum i.e., = 450 Also, maximum range Rmax =
ii]
For a given range and given initial speed of projection, there are two possible angle of projection which are complementary angle i.e., if one is other will be (900- ).Class Exercise solved
Problem 1: The trajectory of a projectile is represented by y =√ 3 − /2. The angle of projection is a) 30o b) 45o c) 60o d) None of these Sol: (c) By comparing the coefficient of x in given equation with standard equation y = x tan −
;
tan = √3 ∴ = 60 46
Problem 2: The path followed by a body projected along y-axis is given as by y = √3 – (1/2)x2 if g = 10 m/s, then the initial velocity of projectile will be - (x and y are in m) a) 3√10 m/s b) 2√10 m/s c) 10√3 m/s d) 10√2 m/s Sol: (b) By comparing the coefficient of x2 in given equation with standard equation y = x tan − Substituting = 600 we get u =2√10m/s
.
=
Problem3: The equation of projectile is y = 16x– The horizontal range is a) 16 m b) 8 m c) 3.2 m d) 12.8 m Sol: (d) Standard equation of projectile motion y = x tan 1 − Given equation: y = 16x –
or y = 16x 1 −
/
By comparing above equations R = =12.8 m. Problem4: A body of mass 2 kg has an initial velocity of 3 m/s along OE and it is subjected to a force of 4 Newton's in OF direction perpendicular to OE. The distance of the body from O after 4 seconds will be a) 12 m b) 28 m c) 20 m d) 48 m Sol: (c) Body moves horizontally with constant initial velocity 3 m/s upto 4 seconds ∴ = = 3 × 4 = 12m and in perpendicular direction it moves under the effect of constant force with zero initial velocity upto 4 seconds. ∴ = + = 4 =16m So its distance from O is given by d = + = (12) + (16) ∴ = 20
47
Problem 5: A body starts from the origin with an acceleration of 6 m/s2 along the x-axis and 8 m/s2 along the y-axis. Its distance from the origin after 4 seconds will be a) 56 m b) 64 m c) 80 m d) 128 m Sol: (c) Displacement along X- axis x = + = ×6× (4) = 48 Displacement along Y- axis: y = + = × 8 × (4) = 48 Total distance from the origin = + = (48) + (64) = 80 Problem 6: In a projectile motion, velocity at maximum height is a) b) c) d) None of these Sol: (b) In a projectile motion at maximum height body possess only horizontal component of velocity i.e. u cos. Problem7: A body is thrown at angle 30o to the horizontal with the velocity of 30 m/s. After 1 sec, its velocity will be (in m/s) ( = 10 / ) a) 10√7 b) 700√10 c) 100√7 d) √40 Sol: (a) From the formula of instantaneous velocity = + −2 = (30) + (10) × 1 − 2 × 30 × 10 × 1 × 30 = 10√7 / Problem 8: A projectile is fired at 30o to the horizontal. The vertical component of its velocity is 80 ms-1. Its time of flight is T. What will be the velocity of the projectile at t = T/2 a) 80 ms-1 b) 80/√3ms-1 c)(80/√3) ms-1 d)40 ms-1
48
Sol: (b) At half of the time of flight, the position of the projectile will be at the highest point of the parabola and at that position particle possess horizontal component of velocity only. Given uvertical =usin = 80 ⟹ = = 160 / ∴ horizontal= ucos = 160cos 30 = 80√3 / Problem 9. A particle is projected from point O with velocity u in a direction making an angle with the horizontal. At any instant its position is at point P at right angles to the initial direction of projection. Its velocity at point P is 90o
P v
u
O
a) u tan b) u cot c) u cosec d) u sec Sol: (b) Horizontal velocity at point ‘O’ = ucos
90o
v sin
90o –
u
u sin
P
v
O
u cos
Horizontal velocity at point ‘P’ = v sin In projectile motion horizontal component of velocity remains constant throughout the motion ∴ v sin = ⟹ =
49
Problem 10: A particle P is projected with velocity u1 at an angle of 30o with the horizontal. Another particle Q is thrown vertically upwards with velocity u2 from a point vertically below the highest point of path of P. The necessary condition for the two particles to collide at the highest point is u1
u2
30o Q
P
a) = b) =2 c) = d) =4 Sol: (b) Both particle collide at the highest point it means the vertical distance travelled by both the particle will be equal, i.e. the vertical component of velocity of both particle will be equal 30 = ⟹ = ∴ = 2 Problem 11.Two seconds after projection a projectile is travelling in a direction inclined at 30o to the horizontal after one more sec, it is travelling horizontally, and the magnitude and direction of its velocity are a) 2√20 / , 60 b) 20√3 / , 60 c) 6√40 / , 30 d) 40√6 / , 30 Sol: (b) Let in 2 sec body reaches upto point A and after one more sec upto point B. Total time of ascent v A
B
o
30
u
O
u cos
50
For a body is given 3 sec i.e. t =
=3
∴ = 10 × 3 = 30 ….. (i) Horizontal component of velocity remains always constant ucos = 30 ......... (ii) For vertical upward motion between point O and A 30 = − × 2[ = − ] 30 = 30 − 20[ = 30] ∴ = 20 / Substituting this value in equation (ii) ucos = 20 30 = 10√3….. (iii) From equation (i) and (iii) and u =20√3 = 60 Problem12: A body is projected up a smooth inclined plane (length =20√2m) with velocity u from the point M as shown in the figure. The angle of inclination is 45o and the top is connected to a well of diameter 40 m. If the body just manages to cross the well, what is the value of .
45o M 40 m
a) 40ms–1 b) 40√2 ms–1 c) 20 ms–1 d) 20√2 ms–1 Sol: (d) At point N angle of projection of the body will be 45°. Let velocity of projection at this point is v. If the body just manages to cross the well then
51
v N R 45o u M
= 40 [as
40 m
= 45 ]
= 400⟹ v = 20 m/s But we have to calculate the velocity (u) of the body at point M. For motion along the inclined plane (from M to N) Final velocity (v) = 20 m/s, acceleration (a) = - g sin = - g sin 45o, distance of inclined plane = 20√2(20)
−2
√
. 20√2[Using v2 = u2 + 2as]
= 20 +
√
400 ⟹ = . Problem 13: A projectile is fired with velocity u making angle with the horizontal. What is the change in velocity when it is at the highest point? a) ucos b) u c) u sin d) (u cos - u) Sol: (c) Since horizontal component of velocity remain always constant therefore only vertical component of velocity changes. Initially vertical component usin .Finally it becomes zero. So change in velocity = u sin Problem14: A body of mass 0.5 kg is projected under gravity with a speed of 98 m/s at an angle of 30o with the horizontal. The change in momentum (in magnitude) of the body is a) 24.5 N-s b) 49.0 N-s c) 98.0 N-s d) 50.0 N-s
52
Sol: (b) Change in momentum between complete projectile motion = 2musin =2×0.5×98×sin300=49 N-s. Problem15: A particle of mass 100 g is fired with a velocity 20 m sec-1 making an angle of 30o with the horizontal. When it rises to the highest point of its path then the change in its momentum is a) √3 kg m sec–1 b) 1/2 kg m sec-1 c) √2 kg m sec–1 d)1 kg m sec-1 Sol: (d) Horizontal momentum remains always constant So change in vertical momentum ( Δ ⃗ ) = Final vertical momentum - Initial vertical momentum = 0– mu sin |ΔP| = 0.1 × 20 × 30 = 1 / Problem 16: Two equal masses (m) are projected at the same angle ( ) from two points separated by their range with equal velocities (v). The momentum at the point of their collision is a) Zero b) 2 mv cos c) - 2 mv cos d) None Sol: (a) Both masses will collide at the highest point of their trajectory with equal and opposite momentum. So net momentum of the system will be zero. mv cos
mv cos
v
v
53
Problem17: A particle of mass m is projected with velocity making an angle of 45o with the horizontal. The magnitude of the angular momentum of the particle about the point of projection when the particle is at its maximum height is (where g = acceleration due to gravity) a) Zero b) mv3/ (4√2 ) c) mv3/(√2 ) d) mv2/2g Sol: (b) = L = = [As = 45o] √
Problem18: A body is thrown with a velocity of 9.8 m/s making an angle of 30o with the horizontal. It will hit the ground after a time a) 1.5 s b) 1 s c) 3 s d) 2 s Sol: (b) T =
=
× . × .
= 1sec
Problem 19: Two particles are separated at a horizontal distance as shown in figure. They are projected at the same time as shown in figure with different initial speed. The time after which the horizontal distance between the particles become zero is a) u/2x b) x/ u c) 2u / x d) u /x Sol: (b) Let x1 and x2 are the horizontal distances travelled by particle A and B respectively in time t. x1 = . 30 × .......... (i) and x2= 60 × ......... (ii) √
x1+x2=
√
.
30 × +
60 × =
∴ = /
Problem 20: A particle is projected from a point O with a velocity in a direction making an angle upward with the horizontal. After some time at point P it is moving at right angle with its initial direction of projection. The time of flight from O to P is a) b) c) d) 54
Sol: (b) When body projected with initial velocity by making angle with the horizontal. Then after time t, (at point P) its direction is perpendicular to.
o
u sin
90
P
u
O
v cos
(90 – ) v
u cos
Magnitude of velocity at point P is given by = (from sample problem no. 9).For vertical motion: Initial velocity (at point O) = usin Final velocity (at point P) = –v cos = − Time of flight (from point O to P) = t Applying first equation of motion = − − = usin − [ = = + cos ] = Problem 21: A ball is projected upwards from the top of tower with a velocity 50 ms-1 making angle 30o with the horizontal. The height of the tower is 70 m. After how many seconds from the instant of throwing will the ball reach the ground? a) 2.33 sec b) 5.33 sec c) 6.33 sec d) 9.33 sec Sol: (c) Formula for calculation of time to reach the body on the ground from the tower of height 'h' (If it is thrown vertically up with velocity u) is given by t =
1+ 1+
So we can
resolve the given velocity in vertical direction and can apply the above formula. Initial vertical component of velocity
55
U sin = 50 sin30 = 25 m/s. t=
.
1+ 1+
× . × ( )
=6.33 sec
Problem 22: If for a given angle of projection, the horizontal range is doubled, the time of flight becomes a) 4 times b) 2 times c) √2 times d) 1/√2times Sol: (c) =
and T =
∴ and T (If and g are constant). In the given condition to make range double, velocity must be increased upto √2 times that of previous value. So automatically time of flight will becomes √2 times. Problem 23. A particle is thrown with velocity u at an angle from the horizontal. Another particle is thrown with the same velocity at an angle from the vertical. The ratio of times of flight of two particles will be a) tan 2 : 1 b) cot 2 : 1 c) tan : 1 d) cot : 1 Sol: (c) For first particles angle of projection from the horizontal is So T1= For second particle angle of projection from the vertical is . It mean from the horizontal is (90- ) ( ) ∴ T1= = So ratio of time of flight
=
56
Problem 24: The friction of the air causes vertical retardation equal to one tenth of the acceleration due to gravity (Take g = 10 ms-2). The time of flight will be decreased by a) 0% b) 1% c) 9% d) 11% Sol: (c) T =
∴
=
=
=
Fractional decrease in time of flight =
=
Percentage decrease = 9% Problem25: A boy playing on the roof of a 10m high building throws a ball with a speed of 10 m/s at an angle of 30o with the horizontal. How far from the throwing point will the ball be at the height of 10 m from the ground (g = 10 m/s2, sin 30o = √ sin 30o = ) a) 8.66 m b) 5.20 m c) 4.33 m d) 2.60 m Sol: (a) Simply we have to calculate the range of projectile u 30o
10 m
R=
=
(
)
( ×
10 m
)
R=5 √3
57
SESSION – 16 & 17 Aim: Projection on Inclined Plane. Horizontal projection PROJECTION ON INCLINED PLANE. A particle is projected from a point A on an inclined plane, which is inclined at an angle to the horizon with a velocity at an elevation. The direction of projection lines in the vertical plane through AB, the line of the greatest slope of the plane. Let the particle strike the plane at B so that AB is the range on the inclined plane.
u
B C
A The initial velocity of projection u can be resolved into a component u cos ( - ) along the plane and a component u sin ( - ) perpendicular to the plane. The acceleration due to gravity g which acts vertically down can be resolved into components g sin up the plane and g cos perpendicular to the plane. By the principle of physical independence of forces the motion along the plane may be considered independent of the motion perpendicular to the plane. Let T be the time, which the particle takes to go from A to B. Then in this time the distance traversed by the projectile perpendicular to the plane is zero. ∴ 0 = u sin ( - ) T - g cos T2 T=
(
)
58
During this time the horizontal velocity of the projectile (u cos ) remains constant. Hence the horizontal distance described is given (
by AC = u cos T =
)
∴ (
Range on the inclined plane =
=
=
(
)
)
Maximum range on the inclined plane (
R=
)
=
[sin (2 - ) - sin ]
For given values of u and , R is maximum when sin (2 − i.e., (2 − ) = 90° = (450 + /2)
)=1
If Rm represents the maximum range on the inclined plane. Rm =
(1- sin );
Rm =
(
)
For a given velocity of projection, it can be shown that there are two directions of projection which are equally inclined to the direction of maximum range. Now R =
[sin (2 - ) - sin ]
For given values of u, and R, sin (2 − ) is constant. There are two values of (2 − ) each less than 1800 that can satisfy the above equation. Let (2+ − ) and (2 − ) be the two values. Then _ 0 2 - = 180 - (2 - ) ; - /2 = 900 - ( - /2) - (450 + /2) = (450 + /2) Since (45° + /2) is the angle of projection giving the maximum range, it follows that the direction giving maximum range bisects 59
the angle between the two angles of projection that can give a particular range. The initial component of velocity perpendicular to PQ is u sin ( − ) and the acceleration in this direction is (-g cos ) .If T is the time the particle takes to go from P to Q then in time T the displacement described in a direction perpendicular to PQ is zero. 0 = u sin ( − ). T- gcos .T2 If the direction of motion at the instant when the particle hits the plane be perpendicular to the plane, then the velocity at that instant parallel to the plane must be zero. ∴ u cos ( − ) - g sin T = 0 ( ) ( ) =T= ∴ cot = 2 tan ( −
)
Horizontal projection: Consider a body projected horizontally with a velocity ℎ. O
at a height
Vel=u th Pa of dy bo
h
A
B
Ground
60
Components Along x-axis Initial Velocity ux = u cos 0 = u Acceleration ax = 0 Displacement x Velocity at time Vx = ux = u cos 0 = u displacement after x = ux t + axt2 t seconds x = ut Velocity at time V= u+2axx
Components Along y - axis uy = u sin 0 = O ay = + g y Vy = uy + ayt y = uyt +ayt2 y = gt2 V= u+ 2ay
after a displacement
V = 2gy
Vx = ux
Resultant velocity = V = =
+
+
+
= + = +( +2 ) = +2 Time of descent (td): As the body reaches ground (Point B) from point of projection (Point O), the time taken is tA.The vertical displacement is OA y = uyt + ayt2 ⟹ ℎ = g
⟹td =
Range (R): AB is range of body x = ux t+ axt2⟹R = ut = u
61
CLASS EXERCISE (A): 1]
If time taken by the projectile to reach Q is T, then PQ is equal to
v
90 P
O
a) Tv Sin
b) Tv Cos
c) Tv Sec
d) Tv Tan
2]
A heavy particle is projected from a point at the foot of a fixed plane, inclined at an angle 45o to the horizontal, in the vertical plane containing the line of greatest slope through the point. If (>450) is the inclination to the horizontal of the initial direction of projection, for what value of tan will the particle strike the plane? i) horizontally ii) at right angle
3]
Two bodies are projected from the same point with equal speeds in such directions that they both strike the same point on a plane whose inclination is . If be the angle of projection of the first body with the horizontal find the ratio of their times of flight
62
CLASS EXERCISE (B): 1]
An aeroplane flying horizontally with speed 90 km/hr releases a bomb at a height of 78.4m from the ground, when will the bomb strike the ground? [g = 10 m/s2] a) 8 sec b) 6 sec c) 4 sec d) 10 sec 2] A body thrown horizontally with velocity 10 m/s from certain height of 10m List – I A) Horizontal displacement covered by it when it reaches the ground B) Horizontal displacement covered by it when its velocity components are equal C) Vertical component of velocity when it reaches the ground D) Horizontal component of velocity when its velocity components are equal List – II P) 5 Q) 10 R) 14.14 S) 20 a) A - Q, B - S; C - R, D- P b) A- R, B-S; C - P, D-Q c) A - R, B- Q; C - R, D- Q d) A -Q, B - S; C - R, D - P 3] A ball is dropped from the top of a tower and another ball is thrown horizontally at the same time. Then a) first ball reaches the ground earlier b) second ball reaches the ground earlier c) both reach the ground simultaneously d) we cannot decide
63
SESSION – 18 & 19 AIM: RELATIVE MOTION RELATIVE MOTION Motion is a combined property of the object under study and the observer. Motion is always relative, there is no such term like absolute motion or absolute rest. Motion is always defined with respect to an observer or reference frame. Reference frame: Reference frame is an axis system from which motion is observed. A clock is attached to measure time. Reference frame can be stationary or moving. There are two types of reference frame: i)
Inertial reference frame: A frame of reference in which Newton’s first law is valid is called as inertial reference frame. ii) Non-inertial reference frame: A frame of reference in which Newton’s first law is not valid is called as non-inertial reference frame. Note: Earth is by definition a non-inertial reference frame because of its centripetal acceleration towards sun. But, for small practical applications earth is assumed stationary hence, it behaves as an inertial reference frame.
64
RELATIVE VELOCITY Definition: Relative velocity of a particle (object) A with respect to B is defined as the velocity with which A appears to move is B if considered to be at rest. In other words, it is the velocity with which A appears to move as seen by the B considering itself to be at rest.
RELATIVE MOTION ALONG STRAIGHT LINE –
vA = , vB = = − =
−
=
−
⟹ = − = 0( ℎ ) Note: velocity of an object w.r.t. itself is always zero. Velocity of Approach It is the rate at which a separation between two moving particles decreases. If separation decreases velocity of approach is positive, Velocity of approach = 22 m/s Velocity of approach = 15 m/s
65
If separation increases, velocity of approach is negative. It is mainly called velocity of separation. It is the rate with which separation between two moving object increases. Velocity of separation = 2 m/s Velocity of separation = 15 m/s Illustration: Two balls A and B are moving in the same direction with equal velocities, find out their relative velocity.
Velocity of A with respect to B ⃗
=0
Illustration: A and B are thrown vertically upward with velocity, 5 m/s and 10 m/s respectively (g = 10 m/s2. Find separation between them after one second Sol. SA = ut – gt2 = 5t – × 10 × t2 = 5 × 1 – 5 × 12 =5–5 =0 SB = ut – gt2. = 10 × 1 – × 10 × 12 = 10 – 5 =5 ∴ SB – SA = separation = 5m.
66
Alter: By relative ⃗ = ⃗ − ⃗ = (–10) – (–10) = 0 Also ⃗ = ⃗ − ⃗ = 10 – 5 = 5 m/s ∴ ⃗ (in 1 sec) = ⃗ × t = 5 × 1 = 5 m ∴ Distance between A and B after 1 sec = 5 m.
Illustration: A ball is thrown downwards with a speed of 20 m/s from top of a building 150 m high and simultaneously another ball is thrown vertically upwards with a speed of 30 m/s from the foot of the building. Find the time when both the balls will meet. (g = 10 m/s2)
Sol. (I) S1 = 20 t + 5 t2 + S2 = 30 t – 5 t2 ____________________ 150 = 50 t ⟹ t = 3 s. (II) Relative acceleration of both is zero since both have acceleration in downward Direction ⃗ = ⃗ − ⃗ =g – g = 0 67
⃗ == 30 – (–20) = 50 sBA = vBA × t t=
=
=3s
Ex: Two cars C1 and C2 moving in the same direction on a straight road with velocities 12 m/s and 10 m/s respectively. When the separation between the two was 200 m C2 started accelerating to avoid collision. What is the minimum acceleration of car C2 so that they don’t collide.
Sol. By relative ⃗ = ⃗ − ⃗ = 0 – a = (–a) ⃗ = ⃗ − ⃗ = 12 – 10 = 2 m/s. So by relativity we want the car to stop. v2 – u2 = 2as. Þ 0 – 22 = – 2 × a × 200 Þ a = m/s2 = 0.1 m/s2 = 1 cm/s2. Minimum acceleration needed by car C2 = 1 cm/s2 RELATIVE MOTION IN LIFT: Illustration: A lift is moving up with acceleration a. A person inside the lift throws the ball upwards with a velocity u relative to hand. (a) What is the time of flight of the ball? (b) What is the maximum height reached by the ball in the lift? 68
Sol. (a)
⃗ = ⃗ − ⃗ = (g + a) downwards
⃗ = ⃗ t + t2 T=(
0= uT – (g + a)T2
)
(b) v2 – u2 = 2 as H=
(
0 – u2 = –2(g + a) H
)
Lift moving with acceleration a upwards 1) Initial velocity = u 2) Velocity at maximum height = u cos q 3) T = 4)
Maximum height (H) =
5)
Range =
(
(
)
)
69
RELATIVE MOTION IN TWO DIMENSIONS ⃗ = position of A with respect to O ⃗ = position of B with respect to O ⃗ = position of A with respect to B. ⃗ = ⃗ − ⃗
∴
( ⃗
( ⃗
)
)
=
( ⃗ )
( ⃗ )
( ⃗ )
–
.
⟹⃗ = ⃗ − ⃗
( ⃗ )
= – ⟹ ⃗ = ⃗ − ⃗ Note: These formulae are not applicable for light. Illustration: Object A and B has velocities 10 m/s. A is moving along East while B is moving towards North from the same point as shown. Find velocity of A relative to B ( ⃗ )
Sol. ⃗ = ⃗ − ⃗ ∴ ⃗ = 10√2
70
Note:
⃗ − ⃗ =
+
−2
Illustration: Two particles A and B are projected in air. A is thrown horizontally, B is thrown vertically up. What is the separation between them after 1 sec?
Sol. ⃗ = ⃗ − ⃗ = 0 ∴ ⃗ = √10 10 = 10√2
∴
=
t = (10√2) t = 10√2 m
Consider the situation, shown in figure
71
Ex:
1] 2] 3] Sol:
Find out motion of tree, bird and old man as seen by boy. Find out motion of tree, bird, boy as seen by old man Find out motion of tree, boy and old man as seen by bird. (1) With respect to boy: vtree = 16 m/s (←) vbird = 12 m/s (↑) vold man = 18 m/s (←) (2) With respect to old man: vBoy = 18 m/s (→) vTree = 2 m/s (→) vBird = 18 m/s (→) and 12 m/s (↑) (3) With respect to Bird: vTree = 12 m/s (↓) and 16 m/s (←) vold man = 18 m/s (←) and 12 m/s (↓). vBoy = 12 m/s (↓).
72
MOTION OF A TRAIN MOVING ON EQUATOR: If a train is moving at equator on the earth’s surface with a velocity vTE relative to earth’s surface and a point on the surface of earth with velocity vE relative to its centre, then ⃗ = ⃗ − ⃗ or ⃗ = ⃗ + ⃗ So, if the train moves from west to east and if the train moves from east to west (the direction of motion of earth on its axis) (i.e. opposite to the motion of earth)
⃗ = ⃗
+ ⃗
⃗ = ⃗
− ⃗
Relative Motion on a moving training: If a boy is running with speed on a train moving with velocity relative to ground, the speed of the boy relative to ground will be given by: ⃗ = ⃗ − ⃗ or ⃗ = ⃗ + ⃗ so, if the boy is running in the direction of train vB = u + v and if the boy is running on the train in a direction opposite to the motion of train vB = u – v 73
RELATIVE MOTION IN RIVER FLOW: If a man can swim relative to water with velocity ⃗ and water is following relative to ground with velocity ⃗ , velocity of man relative to ground ⃗ will be given by: ⃗ = ⃗ − ⃗ or ⃗ = ⃗ + ⃗ So, if the swimming is in the direction of flow of water, vm = vmR + vR
and if the swimming is opposite to the flow of water, vm = vmR – vR
Illustration: A swimmer capable of swimming with velocity v relative to water jumps in a flowing river having velocity u. The man swims a distance d downstream and returns back to the original position. Find out the time taken in complete motion. Sol. t = tdown + tup =
+
=
CROSSING RIVER: A boat or man in a river always moves in the direction of resultant velocity of velocity of boat (or man) and velocity of river flow.
74
1]
Shortest Time: The person swims perpendicular to the river flow crossing a river: consider a river having flow velocity ⃗ and swimmer jump into the river from a point A, from one bank of the river, in a direction perpendicular to the direction of river current. Due to the flow velocity of river the swimmer is drifted along the river by a distance BC and the net velocity of the swimmer will be ⃗ along the direction AC. If we find the components of velocity of swimmer along and perpendicular to the flow these are. y B
drift = x
d
C
vr vmR
vm
x
A
Velocity along the river, vx = vR. Velocity perpendicular to the river, vf = vmR The net speed is given by vm = at an angle of tan =
+
(downstream with the direction of
flow). Velocity of vy is used only in crossing the river, ∴time taken to cross the river is t= = .
75
Velocity vx is only used to drift the motion of the swimmer in the river, drift is along the river flow, x = (vx) or
x=
Illustration: A river 400 m wide is flowing at a rate of 2.0 m/s. A boat is sailing at a velocity of 10 m/s with respect to the water, in a direction perpendicular to the river. (a) Find the time taken by the boat to reach the opposite bank. (b) How far from the point directly opposite to the starting point does the boat reach the opposite bank. (c) In what direction does the boat actually move. Sol:
(a) time taken to cross the river t= = = 40 s Ans. / (b) drift (x) = (vx) = (2m/s) (40s) = 80 m Ans. (c) Actual direction of boat, = tan–1 = tan–1 5, (downstream) with the river flow.
76
2] SHORTEST PATH: When the person crosses the river perpendicularly (along the shortest path). It should swim upstream making an angle with AB such that the resultant velocity, of man must be perpendicular to the flow of river along AB. If we find the components of velocity of swimmer along and perpendicular to the flow, these are, velocity along the river, vx = 0 and velocity perpendicular to river vy = − The net speed is given by vm = with the river direction.
−
at an angle of 90º
Velocity vy is used only to cross the river, therefore time to cross the river, t= = . and velocity vx is zero, therefore, in this case the drift (x) should be zero. and velocity vx is zero, therefore, in this case the drift (x) should be zero. x=0 or vx = vR – vmR sin = 0 or vR = vmR sin 77
or
= sin–1
Hence, to cross the river perpendicular (along the shortest path) the man should swim at an angle of + upstream from the direction of river flow. Further, since sin < 1, Swimmer can cross the river perpendicularly only when vmR> vR ie., Practically it is not possible to reach at B if the river velocity (vR) is too high. Illustration: A man can swim at the rate of 5 km/h in still water. A river 1 km wide flows at the rate of 3 km/h. The man wishes to swim across the river directly opposite to the starting point. (a) Along what direction must the man swim? (b) What should be his resultant velocity? (c) How much time he would take to cross? Sol: The velocity of man with respect to river vmR= 5 km/hr, this is greater than the river flow velocity, therefore, he can cross the river directly (along the shortest path). The anlge of swim must be = + sin–1 = 90º + sin-1 = 90° + sin–1 (upstream)
= 90° + 37° = 127°, with the river flow
Ans. (b) Resultant velocity will be vm = = √5 − 3 = 4 km/hr
−
78
along the direction perpendicular to the river flow. (c) time taken to cross the t= = / = h = 15 min Illustration: A man wishes to cross a river flowing with velocity u jumps at an anlge with the river flow. Find out the net velocity of the man with respect to ground if he can swim with speed v. Also find How far from the point directly opposite to the starting point does the boat reach the opposite bank? In what direction does the boat actually move? If the width of the river is d. x
B
C' C
d u
v
x
A
Sol: Velocity of man = vM = √
+
+2
tanϕ = (v sin ) t = d
⟹ t=
x = (u + v cos ) t = (u + v cos )
Illustration: Shortest Drift (v
79
Sol: In this problem, one thing should be are fully meted that the velocity of boat is less than the river flow velocity. In such a case, boat cannot reach the point directly opposite to its starting point. i.e. drift can never be zero. Thus, to minimize the drift, boat starts at an angle from the normal direction up stream as shown.
Now, again if we find the components of velocity of boat along and perpendicular to the flow, these are, velocity along the river, vx = u – v sin . and velocity perpendicular to the river, vy = v cos . time taken to cross the river is t = = . In this time, drift x = (vx)t = (u – v sin ) or
x=
sec
– d tan
The drift x is minimum, when = 0, or (sec . tan ) – d sec2 = 0 or sin = 1 or sin = = (as v = )
80
so, for minimum drift, the boat must move at an angle = sin–1 from normal direction or an angle + sin–1 from stream direction. CLASS EXERCISE: Ex.1 An object A is moving with 5 m/s and B is moving with 20 m/s in the same direction. (Positive x-axis) (i) Find velocity of B with respect to A. Sol. vB = 20 ̂ m/s Þ vA = 5 ̂ m/sÞ vB – vA = 15 ̂ m/s (ii) Find velocity of A with respect to B Sol. vB = 20 ̂ m/s, vA = 15 ̂ m/s Þ vAB = vA – vB = – 15 ̂ m/s Note: vBA = – vAB Ex.2 Two objects A and B are moving towards each other with velocities 10 m/s and 12 m/s respectively a shown.
(i) Find out velocity of A with respect to B. Sol. vAB = vA – vB = (10) – (–12)= 22 m/s towards right. (ii) Find out velocity of B with respect to A vBA = vB – vA = (–12) – (10)= –22 m/s towards left.
81
SESSION – 20 AIM To study Rain Problems To find the Shortest Distance To Solve problems related to relative velocity RAIN PROBLEMS: If rain is falling vertically with a velocity and on observer is moving horizontally with velocity, the velocity of rain relative to observer will be: ⃗ = ⃗ − ⃗ or ⃗ = + and direction = tan–1 with the vertical as shown in figure.
Illustration: Rain is falling vertically and a man is moving with velocity 6 m/s. Find the angle with which umbrella should be hold by man to avoid getting wet.
Sol: ⃗
⃗ =6 = –10 Velocity of rain with respect to man = ⃗ = –10 ̂ – 6 ̂ tan = = tan–1 Where
is angle with vertical 82
Illustration: A man moving with 5m/s observes rain falling vertically at the rate of 10 m/s. Find the speed and direction of the rain with respect to ground.
VRM = 10 m/s, VM = 5 m/s ⃗ = ⃗ – ⃗ ⃗ =⃗ – ⃗ ∴ ⃗ = 5√5 ∴ tan = , = tan–1 Illustration: A man standing, observes rain falling with velocity of 20 m/s at an angle of 30º with the vertical. 1] Find out velocity of man so that rain appears to fall vertically. 2] Find out velocity of man so that rain again appears to fall at 30º with the vertical. Sol: (1) ⃗ = –v ̂ (let) ⃗ = –10 ̂– 10√3 ̂ ⃗ = – (10 – v) ̂ – 10√3 ̂ ⟹ – (10 – v) = 0 (for vertical fall, horizontal component must be zero) or v = 10 m/s
83
Ans.
(2) ⃗ = –10 ̂– 10√3 ̂ ⃗ = –vx ̂ ⃗ = – (10 – vx) ̂ – 10√3 ̂ Angle with the vertical = 30º ⟹ tan 30º = √
⟹ vx = 20 m/s
WIND AIRPLANE: This is very similar to boat river flow problems the only difference is that boat is replaced by also plane and river is replaced by wind. Thus, velocity of aeroplane with respect to wind ⃗ = ⃗ − ⃗ or ⃗ = ⃗ + ⃗ Where, ⃗ = absolute velocity of aeroplane and, ⃗ = velocity of wind.
84
Illustration: An aeroplane flies along a straight path A to B and returns back again. The distance between A and B is and the aeroplane maintains the constant speed v. There is a steady wind with a speed u at an angle with line AB. Determine the expression for the total time of the trip.
Sol: A to B: Velocity of plane along AB = v cos – ucos , and for no-drift from line AB: v sin = usin ⟹ sin = time taken from A to B : tAB = B to A: velocity of plane along BA = v cos + u cos and for no drift from line AB : v sin = usin ⟹ sin = time taken from B to A: tBA = total time taken = tAB + tBA =
=
+
=
.
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Condition to collide or to reach at the same point When the relative velocity of one particle w.r.t. to other particle is directed towards each other then they will collide. (If there is a zero relative acceleration). Illustration: Three particles A, B and C are situated at the vertices of an equilateral triangle ABC of side a at t = 0. Each of the particles moves with constant speed v. A always has its velocity along AB, B along BC and C along CA. At what time will the particle meet each other? Sol: The motion of the particles is roughly sketched in figure. By symmetry they will meet at the centroid O of the triangle. At any instant the particles will from an equilateral triangle ABC with the same.
Centroid O. All the particles will meet at the centre. Concentrate on the motion of any one particle, say B. At any instant its velocity makes angle 30º with BO. The component of this velocity along BO is v cos 30º. This component is the rate of decrease of the distance BO. Initially. / BO = = = displacement of each particle. √
Therefore, the time taken for BO to become zero =
/√
=
√
×√
=
.
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Alternative: Velocity of B is v along BC. The velocity of C is along CA. Its component along BC is v cos 60º = v/2. Thus, the separation BC decreases at the rate of approach velocity.
∴ ∴ approach velocity = v + = Since, the rate of approach is constant, the time taken in reducing the separation BC from a to zero is t = = Illustration: Two cars A and B are moving west to east and south to north respectively along crossroads. A moves with a speed of 72 kmh–1 and is 500 m away from point of intersection of cross roads and B moves with a speed of 54 kmh–1 and is 400 m away from point of intersection of cross roads. Find the shortest distance between them? Sol:
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Method – I (Using the concept of relative velocity) In this method we watch the velocity of A w.r.t. B. To do this we plot the resultant velocity Vr. Since the accelerations of both the bodies is zero, so the relative acceleration between them is also zero. Hence the relative velocity will remain constant. So the path of A with respect to B will be straight line and along the direction of relative velocity of A with respect to B. The shortest distance between A & B is when A is at point F (i.e. when we drop a perpendicular from B on the line of motion of A with respect to B). From figure tan = = = ........................(i) This q is the angle made by the resultant velocity vector with the xaxis. Also we know that from figure OE = = .............................. (ii) From equation (i) & (ii) we get x = 375 m ∴ EB = OB – OE = 400 – 375 = 25 m But the shortest distance is BF. From magnified figure we see that BF = EB cos = 25 × ∴ BF = 20 m
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Method II (Using the concept of maxima – minima)
A & B be are the initial positions and A’,B’ be the final positions after time t. B is moving with a speed of 15 m/sec so it will travel a distance of BB’ = 15t during time t. A is moving with a speed of 20 m/sec so it will travel a distance of AA’ = 20t during time t. So OA’ =500 – 20 t OB’ = 400 – 15 t ∴ A’B’2 = OA’2 + OB’2 = (500 – 20t)2 + (400 – 15t)2 ........ (i) For A’B’ to be minimum A’B’2 should also be minimum
∴
∴
(
)
(
)
(
)
= =0 = 2(400 – 15t) (–15) + 2(500 – 20t) (–20) = 0 = – 1200 + 45t = 2000 – 80 t 125 t = 3200 89
∴
t=
s.
Hence A and B will be closest after (
s.
)
Now comes out to be positive hence it is a minima. On substituting the value of t in equation (i) we get ∴
A’B’2 = 400 − 15 × = 16 + (12) = 20 m ∴
+ 500 − 20 × Minimum distance A’B’ = 20 m.
Method III (Using the concept of relative velocity of approach) After time t let us plot the components of velocity of A & in the direction along AB. When the distance between the two is minimum; the relative velocity of approach is zero. ∴ VA cos f + VB sin f = 0 (where af is the angle made by the line A’B’ with the x-axis) 20 cos f = – 15 sin f ∴ tan f = − = − Here do not confuse this angle with the angle in method (I) because that is the angle made by the resultant with x-axis. Here f is the angle made with x-axis when velocity of approach in zero, ∴ =− ∴
t=
So, OB’ = 16 m and OA’ = – 12m A’B’ = 16 + (−12) = 20 m
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CLASS EXERCISE: Qus.Two ships are 10 km apart on a line running south to north. The one farther north is steaming west at 20 km h–1. The other is steaming north at 20 km h–1. What is their distance of closest approach? How long do they take to reach it? Ans. 5√2 km/h; 1/4 h = 15 min consider the situation shown in figure for the two particle A and B.
Qus.(1)Will the two particles will collide (2) Find out shortest distance between two particles Ans. (1) The particles will not collide (2) 4√5m.
Note: Muzzle Velocity is the velocity of bullet with respect to the gun i.e. it is Relative Velocity. EXERCISE 1 1. A stone is thrown upwards with a velocity 50 ms–1. Another stone is simultaneously thrown downwards from the same location with a velocity 50 ms–1. When the first stone is at the highest point, the relative velocity of the second stone w.r.t. the first stone is: (A) Zero (B) 50 ms–1 (C) 100 ms–1 (D) 150 ms–1
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2.
3.
4.
5.
A thief is running away on a straight road in a jeep moving with a speed of 9 m s–1. A police man chases him on a motor cycle moving at a speed of 10 m s–1. If the instantaneous separation of the jeep from the motorcycle is 100m, how long will it take for the police man to catch the thief? (A) 1s (B) 19s (C) 90s (D) 100s Two cars are moving in the same direction with a speed of 30 km h–1. They are separated from each other by 5 km. Third car moving in the opposite direction meets the two cars after an interval of 4 minutes. What is the speed of the third car? (A) 30 km h–1 (B) 35 km h–1 (C) 40 km h–1 (D) 45 km h–1 Shown in the figure is the displacement time graph for two children going home from the school. Which of the following statements about their relative motion is true? Their relative velocity: (A) first increases and then decreases (B) first decreases and then increases (C) is zero (D) is non-zero constant. A person standing on the escalator takes time t1 to reach the top of a tower when the escalator is moving. He takes time t2 to reach the top of the tower when the escalator is standing. How long will he take if he walks up a moving escalator? (A) t2 – t1 (B) t1 + t2 (C) t1 t2 /(t1 – t2) (D) t1 t2/(t1 + t2)
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6.
7.
8.
9.
Shown in the figure are the velocity time graphs of the two particles P1 and P2. Which of the following statements about their relative motion is true? Their relative velocity: (A) is zero (B) is non-zero but constant (C) continuously decreases (D) continuously increases Two particles P1 and P2 are moving with velocities v1 and v2 respectively. Which of the statement about their relative velocity vr 12 is true? (A) vr 12> (v1 + v2) (B) vr 12 cannot be greater than v1 – v2 (C) vr 12 cannot be greater than v1 + v2 (D) vr 12< (v1 + v2) Two identical trains take 3 sec to pass one another when going in the opposite direction but only 2.5 sec if the speed of one is increased by 50 %. The time one would take to pass the other when going in the same direction at theiroriginal speed is: (A) 10 sec (B) 12 sec (C) 15 sec (D) 18 sec Two billiard balls are rolling on a flat table. One has velocity components vx = 1m/s, vy =√3 m/s and the other has components vx = 2m/s and vy = 2 m/s. If both the balls start moving from the same point, the angle between their path is (A) 60° (B) 45° (C) 22.5° (D) 15°
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10. A battalion of soldiers is ordered to swim across a river 500 ft wide. At what minimum rate should they swim perpendicular to river flow in order to avoid being washed away by the waterfall 300 ft downstream? The speed of current is 3 m.p.h.: (A) 6 m.p.h. (B) 5 m.p.h. (C) 4 m.p.h. (D) 2 m.p.h. 11. A boat, which has a speed of 5 km/hr in still water, crosses a river of width 1 km along the shortest possible path in 15 minutes. The velocity of the river water in km/hr is (A) 1 (B) 3 (C) 4 (D) √41 12. A bucket is placed in the open where the rain is falling vertically. If a wind begins to blow at double the velocity of the rain, how will be rate of filling of the bucket change? (A) Remain unchanged (B) Doubled (C) Halved (D)Become four times 13. A car with a vertical wind shield moves along in a rain storm at the speed of 40 km/hr. The rain drops fall vertically with a terminal speed of 20 m/s. The angle with the vertical at which the rain drop strike the wind shield is (A) tan–1 (5/9) (B) tan–1(9/5) (C) tan–1 (3/2) (D) tan–1(3) SUBJECTIVE PROBLEMS: 14. Men are running along a road at 15 km/h behind one another at equal intervals of 20 m. Cyclist are riding in the same direction at 25 km/h at equal intervals of 30 m. At what speed an observer travel along the road in opposite direction so that whenever he meets a runner he also meets a cyclist? 94
15. Two perpendicular rail tracks have two trains A & B respectively. Train A moves north with a speed of 54 km h–1 and train B moves west with a speed of 72 km h–1. Assume that both trains starts from same point. Calculate the (a) rate of separation of the two trains (b) relative velocity of ground with respect to B (c) relative velocity of A with respect to B. 16. A man is swimming in a lake in a direction of 30° East of North with a speed of 5 km/hr and a cyclist is going on a road along the lake shore towards East at a speed of 10 km/hr. In what direction and with what speed would the man appear to swim to the cyclist. 17. A motor boat has 2 throttle position on its engine. The high speed position propels the boat at 10 km hr -1 in still water and the low position gives half the higher speed. The boat travels from its dock downstream on a river with the throttle at low position and returns to its dock with throttle at high position. The return trip took 15% longer time than it did for the downstream trip. Find the velocity of the water current in the river. 18. (I) A man can swim with a speed of 4 km h–1 in still water. How long does he take to cross a river 1 km wide if the river flows steadily at 3 km h–1 and he makes his strokes normal to the river current? How far down the river does he go when he reaches the other bank? (II) If he keeps himself always at an angle of 120°C with the river flow while swimming. (a) Find the time he takes to cross the river. (b) At what point on the opposite bank will he arrive?
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19. A river is flowing from west to east at a speed of 5 m/min. A man on the south bank of the river, capable of swimming at 10 m/min in still water, wants to swim across the river in shortest distance. In what direction should he swim? 20. An airplane is flying with velocity 50√2km/hour in north-east direction. Wind is blowing at 25 km/hr from north to south. What is the resultant displacement of airplane in 2 hours? 21. When a train has a speed of 10 m s–1 eastward, raindrops that are falling vertically with respect to the earth make traces that are inclined 30º to the vertical on the windows of the train. (a) What is the horizontal component of a drop's velocity with respect to the earth? With respect to the train? (b) What is the velocity of the raindrop with respect to the earth? With respect to the train? 22. To a man walking at 7 km/h due west, the wind appears to blow from the north-west, but when he walks at 3 km/h due west, the wind appears to blow from the north. What is the actual direction of the wind and what is its velocity? 23. When a motorist is driving with velocity 6 ̂ + ̂, the wind appears to come from the direction ̂. When he doubles his velocity the wind appears to come from the direction ̂ + ̂. Then the true velocity of the wind expressed in the form of ̂ + is ______. 24. 'n' numbers of particles are located at the vertices of a regular polygon of 'n' sides having the edge length 'a'. They all start moving simultaneously with equal constant speed 'v' heading towards each other all the time. How long will the particles take to collide?
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25. Two ships are 10 km apart on a line running south to north. The one further north is streaming west at 40 km/hr. The other is streaming north at 40 km/hr. What is their distance of closest approach and how long do they take to reach it? 26. A ship is sailing towards north at a speed of√2m/s. The current is taking it towards East at the rate of 1 m/s and a sailor is climbing a vertical pole on the ship at the rate of 1 m/s. Find the velocity of the sailor in space. 27. A motorboat is observed to travel 10 km hr–1 relative to the earth in the direction 37º north of east. If the velocity of the boat due to the wind only is 2 km hr–1 westward and that due to the current only is 4 km hr–1 southward, what is the magnitude and direction of the velocity of the boat due to its own power? 28. A person P sitting on a wooden block (which does not move relative to water) in a flowing river sees two swimmers A and B. A and B both have constant speed vm relative to water. P observes that A starts from one point of the river bank and appears to move perpendicular to the river flow. P also observes that B starts from some point on the other bank at the same time and moves downstream. The width of the river is 'd' and it flows with velocity vr. If A and B both reach a point at the same time, than find the initial separation between A and B. 29. A motorboat going down stream overcome a float at a point M.60 minutes later it turned back and after some time passed the float at a distance of 6 km from the point M. Find the velocity of the stream assuming a constant velocity for the motorboat in still water.
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30. 2 swimmers start from point A on one bank of a river to reach point B on the other bank, lying directly opposite to point A. One of them crosses the river along the straight line AB, while the other swims at right angles to the stream and then walks the distance which he has been carried away by the stream to get to point B. What was the velocity (assumed uniform) of his walking if both the swimmers reached point B simultaneously? Velocity of each swimmer in still water is 2.5 km hr-1 and the stream velocity is 2 km hr-1.
31. An airplane pilot sets a compass course due west and maintains an air speed of 240 km. hr–1. After flying for hr, he finds himself over a town that is 150 km west and 40 km south of his starting point. (a) Find the wind velocity, in magnitude and direction. (b) If the wind velocity were 120 km. hr–1 due south, in what direction should the pilot set his course in order to travel due west? Take the same air speed of 240 km. hr–1. 32. Two straight AOB and COD meet each other right angles. A person walking at a speed of 5 km/hr along AOB is at the crossing O at noon. Another person walking at the same speed along COD reaches the crossing O at 1:30 PM. Find at what time the distance between them is least and what is its value? ***** 98
KEY SESSION – 1 & 2 CLASS EXERCISE 1. c 2.d 3. b
4.a
5. Ans: 3√3m
SESSION – 3, 4 & 5 CLASS EXERCISE (A) 1. b 2. b 3.a 8.d 9.b 10.a
4.c
5.b
6.c
7.b
SESSION – 6 CLASS EXERCISE 1. a 2. 3.Ans: i) 5.
6. Ans:
(V0) 3/2
ii)
4.d
7. Ans: O, (8/3) s; -7/2
SESSION – 7, 8 & 9 CLASS EXERCISE 1. c 2.d 3.b
4.a SESSION – 10, 11 & 12
CLASS EXERCISE (A) 1.c 2.c 3.b 8.c CLASS EXERCISE (B) 1.a 2.b 3.d
4.b
5.
6.b
7.c
4.c
5.d
6.d
7.c 99
CLASS EXERCISE (C) 2.a 3.Ans: i) zero 5.d
ii) 4.9m iii) -9.8ms-1 iv) 3 :1
CLASS EXERCISE (D) 1.a 2.d 3.c
4.c
5.c
6.d
CLASS EXERCISE (E) 1.b 2. c
3.b
4.d
5.a
7.a
8.b
4.c
6.
9.Ans: 2
10.Ans: ( 3s after B is released and 1.6 m above the ground)
SESSION – 13, 14 & 15 CLASS EXERCISE (A) 1.c 2. 3.a 8.c 9.b 10.b CLASS EXERCISE (B) 1.b,c 2.d 3.b 10.b 11.b CLASS EXERCISE (C) 1.d 2.c 3.b 8.c 9.c 10.b
4.a 11.c
5.c 12.c
6.d
7.d
5.b
6.d
7.b
9.a
4.c 11.d
5.a 12.c
6.b 13.b
7.c
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SESSION – 16 CLASS EXERCISE 1.d 2.Ans: i) 900 ( ) 3.Ans:
ii) Tan 450 = 2 Tan = 3
SESSION – 18, 19 & 20 OBJECTIVE PROBLEMS 1. C 2. D 3. D 8. C 9. D 10. B
4. D 11. B
5.D 12. A
6. D 13.A
7. C
SUBJECTIVE PROBLEMS 14. 5 km/h 15. (a) 25 m/s or 90 km/hr (b) 20 m/s or 72 km/hr due east (c) 25 m/s or 90 km/hr at 37°N of E 16. 30º N of W at 5√3 km/hr. 17.3 km/hr. 18. (I) 0.75 km (II) (a) h (b) km. √
√
19. At an angle 30° west of north. 20. 50√5 km 21. (a) 0, 10 m/s West (b) 10√3 m/s, 20 m/s 22. Coming from 5 km/hr, 53° N of E 23. (4 ̂ + 8 ̂) 25.
√
24.
, hr
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26. 2m/s in a direction making an angle of 60º with E, 45º with N and 60º with the vertical 27. 10√2km/hr, 45º N of E 28. √2d 29.3 km/hr 30. 3 km/hr towards B 31. (a) 100 km/hr, 37° W of S (b) 30° N of W 32. 12: 45 PM EXERCISE # 2 1. a > 10 m/s2 2.3.6 minute 2. (b) 2 m/s
3. (a)
= 45º ;
*****
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