Chapter 3 Motion in a Plane Fundamentals of Physics: A Senior Course SHORT ANSWER
1. For each of the followin following, g, perform perform the vector operatio operation n indicated indicated to find either either the sum sum or the difference vector. (a) Determine
(b) Determine
(c) Determine
(d) Determine
ANS: (a)
(b)
(c)
(d)
REF: K/U MSC: P 2. The vector vector
OBJ: Appendix A
is illust illustrat rated. ed.
KEY: FOP 3.4, p.97
Draw the following vectors, to the same scale. (a) (b) (c) (d) ANS: (a)
(b)
(c)
(d)
REF: K/U MSC: P
OBJ: Appendix A
KEY: FOP 3.5, p.98
3. A car travels at 20 m/s m/s [N35°E] for for 3.0 s. Draw its velocity velocity vector, vector, and then, using scalar multiplication, multiplication, draw the vector representing its displacement during this time interval. ANS: 1 cm = 5 m/s
Draw the following vectors, to the same scale. (a) (b) (c) (d) ANS: (a)
(b)
(c)
(d)
REF: K/U MSC: P
OBJ: Appendix A
KEY: FOP 3.5, p.98
3. A car travels at 20 m/s m/s [N35°E] for for 3.0 s. Draw its velocity velocity vector, vector, and then, using scalar multiplication, multiplication, draw the vector representing its displacement during this time interval. ANS: 1 cm = 5 m/s
REF: K/U MSC: P
OBJ: 1.1
LOC: FMV.01
KEY: FOP 3.5, p.98
4. An airplane airplane is climbing climbing at an angle angle of 15º to the horizon, horizon, with the the sun directly directly overhead. overhead. Its shadow is observed to be moving across the ground at 200 km/h. (a) What is the actual air speed of the plane? (b) How long does it take to increase the airplane’s altitude altitude by 1000 m? ANS:
(a)
(b)
REF:
K/U, MC
OBJ: 1.1
LOC: FM1.02
KEY: FOP 3.6, p.103
MSC: P 5. A quarterback is running across the field, parallel to the line of scrimmage, at a constant speed of 2.5 m/s, when he spots an open, stationary receiver straight downfield from him (i.e., in a line parallel to the sidelines). (a) If he can throw the football at a speed of 8.0 m/s, relative to himself, at what angle, relative to the sidelines, must he throw it in order to hit the receiver? (b) How far downfield was the receiver, if the pass took 3.0 s to reach him? ANS:
(a)
(b)
REF: K/U, MC MSC: P
OBJ: 1.5
6. From the following, find
.
LOC: FM1.02
KEY: FOP 3.7, p.109
ANS:
REF: K/U MSC: P
OBJ: 1.2
7. Using the following, find
ANS:
.
LOC: FM1.02
KEY: FOP 3.9, p.111
REF: K/U MSC: P
OBJ: 1.2
LOC: FM1.02
KEY: FOP 3.9, p.112
8. A car turns a circular curve with a speed of 20 m/s. If the radius of the curve is 100 m, what is the centripetal acceleration of the car? ANS:
REF: K/U MSC: SP
OBJ: 3.1
LOC: FMV.01
KEY: FOP 3.9, p.117
9. A ball on a string swings in a horizontal circle of radius 2.0 m. If its centripetal acceleration is 15 m/s2, what is the speed of the ball? ANS:
REF: K/U MSC: SP
OBJ: 3.1
LOC: FMV.01
KEY: FOP 3.9, p.117
10. What is the centripetal acceleration of a stone being whirled in a circle, at the end of a 1.5 m string, on a smooth sheet of ice, with a frequency of 1.25 Hz? ANS:
REF: K/U MSC: SP
OBJ: 3.1
LOC: FMV.01
KEY: FOP 3.9, p.118
11. The planet Mercury moves in an approximately circular path around the sun at an average distance of 5.8 × 1010 m, accelerating centripetally at 0.04 m/s 2. What is its period of revolution around the sun? ANS:
REF: K/U, MC MSC: SP
OBJ: 3.1
LOC: FMV.01
KEY: FOP 3.9, p.117
12. What is the centripetal acceleration of a locomotive that travels around a circular curve of radius 250 m at a constant speed of 70 km/h? ANS:
REF: K/U MSC: P
OBJ: 3.1
LOC: FMV.01
KEY: FOP 3.9, p.118
13. What is the centripetal acceleration of a small girl standing at the outer edge of a carousel 4.0 m in diameter, which makes one complete rotation in 6.0 s?
ANS:
REF: K/U MSC: P
OBJ: 3.1
LOC: FMV.01
KEY: FOP 3.9, p.118
14. Patrons on the midway ride called the Rotor stand with their backs against the wall of a revolving cylinder, cylinder, while the floor drops away from beneath them. To keep from falling, they require a centripetal acceleration of about 25 m/s 2. If the rotor has a diameter of 5.0 m, with what minimum frequency does it revolve? (The vertical force required to support their weight is supplied by friction with the wall.) ANS:
REF: K/U, MC MSC: P
OBJ: 3.1
LOC: FM3.01
KEY: FOP 3.9, p.118
15. An airplane flies in a horizontal circle of radius radius 500 m. If its centripetal centripetal acceleration acceleration is 20 20 m/s 2, how long does it take to complete the circle? ANS:
REF: K/U MSC: P
OBJ: 3.1
LOC: FMV.01
KEY: FOP 3.9, p.118
16. A helico helicopter pter rises rises with with a uniform uniform speed of 30 m/s at an angle angle of 50º 50º to the horizont horizontal. al. (a) What are the vertical and horizontal components of its velocity? (b) How long will it take to reach an altitude of 1.00 km? (c) What horizontal distance will it have covered by that time? ANS:
(a)
(b)
(c) REF: K/U MSC: P
OBJ: 1.1
LOC: FM1.02
KEY: FOP 3.11, p.123
17. A car, moving initially at 32 km/h [N], turns turns a corner and and continues at at 32 km/h [W]. The turn takes 3.0 s to complete. Find (a) the change in velocity (b) the average acceleration during the turn ANS: (a)
(b) REF: K/U MSC: P
OBJ: 1.2
LOC: FM1.02
KEY: FOP 3.11, p.125
18. A car, travelling travelling at 25 m/s around a circular curve, has a centripetal centripetal acceleration acceleration of 8.3 m/s 2. What is the radius of the curve? ANS:
REF: K/U MSC: P
OBJ: 3.1
LOC: FMV.01
KEY: FOP 3.11, p.126
19. What is the centripetal acceleration of a point on the rim of a bicycle wheel of radius 0.25 m, if the bicycle’s speed is 5.0 m/s? (HINT: Take the acceleration relative to the bicycle frame.) ANS:
REF: K/U MSC: P
OBJ: 3.1
LOC: FMV.01
KEY: FOP 3.11, p.126
20. The moon, an Earth satellite with a period of about 27.3 d and a nearly circular orbit, has a centripetal acceleration of 2.7 × 10 –3 m/s2. What is the average distance from the Earth to the moon? ANS:
REF:
K/U, MC
OBJ: 3.1
LOC: FMV.01
KEY: FOP 3.11, p.126
MSC: P 21. What is the centripetal acceleration due to the daily rotation of an object at the Earth’s equator if the equatorial radius is 6.4 × 106 m? ANS:
REF: K/U MSC: P
OBJ: 3.1
LOC: FMV.01
KEY: FOP 3.11, p.126
22. A biophysicist is able to separate very small subcellular particles, using an analytic ultracentrifuge. The physicist must determine the amount of acceleration provided by the centrifuge at various speeds and radii. (1 g of acceleration is equal to the acceleration due to gravity (i.e., about 9.8 m/s 2.) Calculate the number of g ’s of acceleration at 8.4 cm from the centre of the centrifuge when it is spinning at 60 000 r/min. ANS:
REF: K/U, MC MSC: P
OBJ: 3.1
LOC: FM3.01
KEY: FOP 3.11, p.126
23. This diagram shows a racing car’s position at three equally spaced points in time. It also shows the car’s instantaneous velocity vectors at points 1 and 2. If the acceleration of the car is uniform, determine, using an accurate vector diagram, the instantaneous velocity of the car at point 3. Scale: 1 cm = 5.0 m/s.
ANS:
REF: K/U MSC: P
OBJ: 1.2
LOC: FM1.02
KEY: FOP 3.11, p.122
24. A boat sails 8.0 km [S10ºE] through still water. What are the components of its displacement in each of the following directions? (a) [S] (b) [E] (c) [S20ºE] (d) [E10°N] ANS:
(a)
(b)
(c)
(d) REF: K/U MSC: P
OBJ: 1.1
LOC: FM1.02
KEY: FOP 3.11, p.123
25. A newspaper boy throws papers sideways onto the porches of his customers while riding his bicycle along the sidewalk. The sidewalk is 15 m in front of the porches. The boy throws the papers at the horizontal speed of 6.0 m/s relative to himself, and rides the bicycle at a speed of 4.0 m/s relative to the sidewalk. (a) With what horizontal speed do the papers actually travel relative to the ground? (b) How far in advance of a porch should the boy throw a paper, so that it lands on target? (c) If he waits until he is directly opposite a porch, at what horizontal angle with respect to the sidewalk will he have to throw the paper, to hit the porch? ANS:
(a)
(b)
(c )
REF: K/U, MC MSC: P
OBJ: 1.5
LOC: FM1.02
KEY: FOP 3.11, p.124
26. A train has a speed of 20 km/h. Raindrops falling against its side windows makes traces inclined at 30º to the vertical. We ignore air turbulence, and there is no wind. (a) What is the horizontal component of a raindrop’s velocity with respect to Earth? With respect to the train? (b) What is the velocity of the raindrop with respect to Earth? With respect to the train? ANS:
(a) For the raindrop, wrt Earth, vh = 0 km/h wrt the train, vh = –20 km/h
(b) REF: K/U MSC: P
OBJ: 1.5
LOC: FM1.02
KEY: FOP 3.11, p.125
PROBLEM
1. A hockey puck hits the boards with a velocity of 10 m/s at an angle of 20º to the boards. It is deflected with a velocity of 8.0 m/s at 24º to the boards. If the time of impact is 0.03 s, what is the average acceleration of the puck?
ANS:
Using cosine law:
Using the sine law:
REF: K/U MSC: P
OBJ: 1.2
LOC: FM1.02
KEY: FOP 3.8, p.113
2. A train moving at a constant speed of 100 km/h travels east for 40 min, then 30º east of north for 20 min, and finally west for 30 min. What is the train’s average velocity for the trip? ANS:
REF: K/U MSC: P
OBJ: 1.1
LOC: FM1.02
KEY: FOP 3.11, p.122
3. A man walks 600 m [E47°N], then 500 m [N38°W], then 300 m [W29°S], and finally 400 m [S13°E]. Find his resultant displacement. ANS: Using components in the x-y plane:
REF: K/U MSC: P
OBJ: 1.1
LOC: FM1.02
KEY: FOP 3.11, p.122
4. A snowmobile is being driven across a frozen lake, and in the diagram represent its velocity vectors at two instants 3.0 s apart. The scale is 1 cm = 5.0 m/s. Using these vectors create an accurate vector diagram to determine . What is the average acceleration of the snowmobile?
ANS:
Scale: 1 cm = 5.0 m/s
REF: K/U MSC: P
OBJ: 1.2
LOC: FM1.02
KEY: FOP 3.11, p.122
5. This diagram shows a racing car’s position at three equally spaced points in time. It also shows the car’s instantaneous velocity vectors at points 1 and 2. If the acceleration of the car is uniform, determine, using an accurate vector diagram, the instantaneous velocity of the car at point 3. Scale: 1 cm = 5.0 m/s
ANS:
REF: K/U MSC: P
OBJ: 1.2
LOC: FM1.02
KEY: FOP 3.11, p.122
6. The current in a river moves at 2.0 m/s [S]. How fast and in what direction must a swimmer move through the water in order to have a resultant velocity relative to the river bank of (a) 3.6 m/s[S] (b) 3.6 m/s[N] (c) 3.6 m/s[E] ANS:
(a)
(b)
(c) Using a vector diagram:
REF: K/U MSC: P
OBJ: 1.5
LOC: FM1.02
KEY: FOP 3.11, p.123
7. A ball is thrown from the top of a building with a speed of 20 m/s and at a downward angle of 30° to the horizontal, as shown. What are the horizontal and vertical components of the ball’s initial velocity?
ANS:
REF: K/U MSC: P
OBJ: 1.4
LOC: FM1.03
KEY: FOP 3.11, p.123
8. A boat travelling at 3.0 m/s through the water keeps its bow pointing north across a stream that flows west at 5.0 m/s. What is the resultant velocity of the boat with respect to the shore? ANS:
REF: K/U MSC: P
OBJ: 1.5
LOC: FM1.02
KEY: FOP 3.11, p.123
9. A dog walks at 1.6 m/s on the deck of a boat that is travelling north at 7.6 m/s with respect to the water. (a) What is the velocity of the dog with respect to the water if it walks towards the bow? (b) What is the velocity of the dog with respect to the water if it walks towards the stern? (c) What is the velocity of the dog with respect to the water if it walks towards the east rail, at right angles to the boat’s keel? ANS: (a)
(b)
(c)
REF: K/U MSC: P
OBJ: 1.5
LOC: FM1.02
KEY: FOP 3.11, p.123
10. An airplane maintains a heading due west at an air speed of 900 km/h. It is flying through a hurricane with winds of 300 km/h, from the northeast. (a) What is the plane’s ground speed? (b) In which direction is the plane m oving relative to the ground? (c) How long would it take the plane to fly from one city to another 500 km away, along the path in (b)? ANS: (a)
Using the cosine law,
(b) Using the sine law,
(c) REF: K/U, MC MSC: P
OBJ: 1.5
LOC: FM1.02
KEY: FOP 3.11, p.123
11. Two boathouses are located on a river, 1.0 km apart on the same shore. Two men make round trips from one boathouse to the other, and back. One man paddles a canoe at a velocity of 4.0 km/h relative to the water, and the other walks along the shore at a constant velocity of 4.0 km/h. The current in the river is 2.0 km/h in the starting direction of the canoeist. (a) How much sooner than the walker does the canoeist reach the second boathouse? (b) How long does it take each to make the round trip? ANS: (a) Walker:
Canoeist:
The canoeist arrives 5 min before the walker. (b) Now for the canoeist’s return trip
So the total time for the canoeist is
The walker takes 30 min to make the trip. The canoeist takes 40 min to make the trip. REF: K/U, I MSC: P
OBJ: 1.5
LOC: FM1.02
KEY: FOP 3.11, p.124
12. A 70 m wide river flows at 0.80 m/s. A girl swims across it at 1.4 m/s relative to the water. (a) What is the least time she requires to cross the river? (b) How far downstream will she be when she lands on the opposite shore? (c) At what angle to the shore would she have to aim, in order to arrive at a point directly opposite the starting point? (d) How long would the trip in part (c) take? ANS:
(a)
(b) (c)
The angle with respect to shore is 90°
(d)
35° = 55°.
REF: K/U, I MSC: P
OBJ: 1.5
LOC: FM1.02
KEY: FOP 3.11, p.124
13. An ocean liner is steaming at 18 km/h due south. A passenger strolling on the deck walks toward the rear of the ship at 3.0 m/s. After walking for 12 s, he turns right and walks at the same speed towards the rail, 15 m from his turning point. (a) What is his velocity, relative to the water, while walking towards the rear? While walking towards the rail? (b) Draw a scale vector diagram, or make a sketch and use the mathematical approach, to determine his resultant displacement relative to the water at the end of his walk. ANS:
(a)
(b)
REF: K/U, I MSC: P
OBJ: 1.5
LOC: FM1.02
KEY: FOP 3.11, p.124
14. A pilot maintains a heading due west with an air speed of 240 km/h. After flying for 30 min, he finds himself over a town that he knows is 150 km west and 40 km south of his starting point. (a) What is the wind velocity, in magnitude and direction? (b) What heading should he now maintain, with the same air speed, to follow a course due west from the town? ANS: Find
Using the cosine law,
(a) Using the sine law,
The wind velocity is 100 km/h [W53°S], or [S37°W]. (b)
Using the sine law,
The heading is [W19°N]. REF: K/U, MC MSC: P
OBJ: 1.5
LOC: FM1.02
KEY: FOP 3.11, p.124
15. The navigator of an airplane plans a flight from one airport to another 1200 km away, in one direction 30° east of north. The weather office informs him of a prevailing wind from the west, of 80 km/h. The pilot wants to maintain an air speed of 300 km/h. (a) What heading should the navigator give the pilot? (b) How long will the flight take? (c) How much time did the wind save? ANS:
(a) Using the sine law,
The heading is [N17°E].
(b)
The flight will take 3.6 h. (c) Without wind,
The time saved by the wind is 0.4 h, or 24 min. REF: K/U, MC MSC: P
OBJ: 1.5
LOC: FM1.02
KEY: FOP 3.11, p.125
16. An airplane flying at a constant speed of 1000 km/h executes a slow, level turn that changes its direction from west to east. If the turn takes 80 s, calculate the plane’s average acceleration. ANS:
REF: K/U MSC: P
OBJ: 1.2
LOC: FM1.02
KEY: FOP 3.11, p.125
17. A traditional watch has a second hand 1.5 cm long, from centre to tip.
(a) What is the speed of the tip of the second hand? (b) What is the velocity of the tip at 15 s? at 45 s? at 60 s? (c) What is its change in velocity between 30 s and 45 s? (d) What is its average acceleration during the same interval? ANS:
(a) (b) at t = 15 s, = 0.16 cm/s [D] at t = 45 s, = 0.16 cm/s [U] at t = 60 s, = 0.16 cm/s [R]
(c)
REF: K/U MSC: P
OBJ: 1.1
LOC: FM1.02
KEY: FOP 3.11, p.125
18. A puck sliding across the ice at 20 m/s [E] is struck by a stick and moves at 30 m/s, at an angle of 120° to its original path. Find its change in velocity. ANS:
REF: K/U MSC: P
OBJ: 1.2
LOC: FM1.02
KEY: FOP 3.11, p.125
19. A batter strikes a baseball moving horizontally towards him at 15 m/s. The ball leaves the bat horizontally at 24 m/s, 40° to the left of a line from the plate to the pitcher. The ball is in contact with the bat for 0.01 s. Determine the following. (a) the change in velocity of the ball (b) its average acceleration while being hit by the bat ANS:
(a)
(b) REF: K/U MSC: P
OBJ: 1.2
LOC: FM1.02
KEY: FOP 3.11, p.125
20. A racing car starts into a circular portion of a Grand Prix course at 200 km/h [E], travelling in a clockwise direction. By the time it is headed due south, its speed has increased to 240 km/h. (a) If this took 12.0 s, find the average acceleration during the turn. (b) Estimate the radius of the curve. ANS:
(a)
(b) To estimate the radius, consider it a circular path, so the Assume vavg = 220 km/h.
is the centripetal acceleration.
REF: KEY:
K/U OBJ: 1.2, 3.1 FOP 3.11, p.125
LOC: FM1.02, FMV.01 MSC: P
21. A person walks up a stalled escalator in a department store in 90 s. When standing still on the same escalator in motion, he is carried up in 60 s. (a) How much time would it take him to walk up the moving escalator? (b) Could he walk down the escalator while it was moving up? If so, how long would that take? ANS: The length of the escalator is and the person’s walking speed is v. All speeds and distances are relative to the building and up is positive.
(a) The speed of the escalator is given by
When walking on the moving escalator the velocity is:
(b) When waking “down” the moving escalator, This is positive, meaning he will be moving up.
REF: K/U, I MSC: P
OBJ: 1.5
LOC: FM1.02
KEY: FOP 3.11, p.126
22. A slightly disoriented homing pigeon flies the following course at a constant speed of 15 m/s: (i) 800 m, 37° east of north, (ii) 300 m due west, and (iii) 400 m, 37° south of east A crow flies in a straight line (as the crow flies) between the starting and finishing points. At what speed must the crow fly if the birds leave and arrive together? ANS: Using components in the x and y directions, for the homing pigeon
Time for homing pigeon
REF: K/U MSC: P
OBJ: 1.1
LOC: FM1.02
KEY: FOP 3.11, p.126
23. A power boat travels down the St. Lawrence River from Montreal to Quebec City, at full throttle. The trip takes 3.0 h. The boat then heads back to Montreal, again at full throttle. This time, the trip takes 15 h. With no gas left, the boat now drifts with a steady current back to Quebec City. How long does the third trip take? ANS: Let the speed of the boat in still water be x. Let the speed of the current be y. From Montreal to Quebec:
From Quebec to Montreal:
From Montreal to Quebec:
From (1) and (2)
Then, in (3)
REF: K/U MSC: P
OBJ: 1.5
LOC: FM1.02
KEY: FOP 3.11, p.126
24. Snoopy is flying his plane, the Sopwith Camel, in search of the Red Baron. He flies with a constant speed of 120 km/h relative to the air, and makes instantaneous turns, when necessary. He follows a perfectly square path on the ground, using north-south and east-west roads as a guide for each of the 60 km sides. On a day when there is a steady 60 km/h wind blowing diagonally across the square (say from the northeast) how long does the trip take? ANS: The pilot flies “off course” in such a way that, when the wind’s velocity is added to his, he is on the intended course. For the northbound leg,
Virtually the same situation occurs on the westbound leg, and For the southbound leg,
Again, the same situation occurs on the westbound leg, so that
REF: K/U MSC: P
OBJ: 1.5
LOC: FM1.02
KEY: FOP 3.11, p.127
25. A sunbather, drifting downstream on a raft, dives off the raft just as it passes under a bridge and swims against the current for 15 min. She then turns and swims downstream, making the same total effort and overtaking the raft when it is 1.0 km downstream from the bridge. What is the speed of the current in the river? ANS: Let the swimmer’s speed in still water be x. Let the speed of the current in the river be y. The trip consists of 3 intervals:
For the sunbather swimming upstream
and swimming downstream
and for the boat drifting downstream
From (2) and (1)
From (3)
Then,
The speed of the current in the river is 2.0 km/h Note: As an interesting alternative, go into the frame of reference of the water and raft. She swims away from the raft for h, so must swim back to it in drifts 1 km. Therefore the speed of the current is 2 km/h. REF: K/U, I MSC: P
OBJ: 1.5
LOC: FM1.02
h. During her
h swim, the raft
KEY: FOP 3.11, p.127
26. Two boys are at point X on one side of a river, 40 m wide and having a current of 1.0 m/s, flowing as shown. Simultaneously, they dive into the water in an attempt to reach point Y, directly opposite X. Both swim at 2.0 m/s relative to the water, but one directs himself so that his net motion corresponds to XY, while the other keeps his body perpendicular to the current and consequently lands at point Z. After landing, he runs along the shore to point Y at a speed of 6.0 m/s. Which boy arrives at Y first, and by how much time does he beat the other?
ANS: Boy 1:
Boy 2:
Time to cross river
Time to run along the shore
Therefore, the time for boy 2 is 23.3 s. Boy 1 arrives first by 0.2 s. REF: K/U, I MSC: P
OBJ: 1.5
LOC: FM1.02
KEY: FOP 3.11, p.127
27. A police cruiser chasing a speeding motorist travelled 60 km [S], then 35 km [NE], and finally 50 km [W]. (a) Calculate the total displacement of the cruiser. (b) If the chase took 1.3 h, what was the cruiser’s (i) average speed and (ii) average velocity, for the trip. ANS:
(a) Final displacement is AD. AB = 60 km, BC = 35 km, CD = 50 km
If BC = 35 km,
(b)
.
(i)
(ii) REF: K/U MSC: P
OBJ: 1.1
LOC: FM1.02
KEY: FOP 3.3, p.93
28. An express bus travels directly from A-town to B-ville. A local bus also links these two towns, but it goes west 30 km from A-town to C-city, then 30 km south to D-ville, and finally 12 km west to E-town and 30 km northeast to B-ville. (a) What is the shortest distance from A-town to B-ville? (b) In what direction does the express bus travel? (c) If the express bus takes 0.45 h to go from A-town to B-ville, and the local bus takes 3.0 h, calculate the average speed and the average velocity for each bus. ANS:
(a)
(b)
(c) For the express bus,
For the local bus,
REF: K/U MSC: P
OBJ: 1.1
LOC: FM1.02
KEY: FOP 3.3, p.93
29. A hiker walks 10.0 km [NE], 5.0 km [W], and then 2.0 km [S] in 2.5 h. (a) What is the hiker’s displacement? (b) In what direction must the hiker set out, in order to return by the most direct route to the starting point? (c) If the hiker walks at a constant speed for the entire trip and returns by the most direct route, how long will the total walk take? ANS: (a)
(b) [S22°W] (i.e., along DA in the opposite direction) (c)
REF: K/U MSC: P
OBJ: 1.1
LOC: FM1.02
KEY: FOP 3.3, p.93
30. A plane’s velocity changes from 200 km/h [N] to 300 km/h [N30°W]. Find the change in velocity,
. (Hint:
)
ANS:
REF: K/U MSC: P
OBJ: 1.2
LOC: FM1.02
KEY: FOP 3.4, p.97
31. A car travelling at 15 m/s [N] executes a gradual turn, so that it then moves at 18 m/s [E]. What is the car’s change in velocity? ANS:
REF: K/U MSC: P
OBJ: 1.2
LOC: FM1.02
KEY: FOP 3.4, p.97
32. A plane is flying at 100 m/s [E]. The pilot changes its velocity by 30 m/s [E30°N]. What is the plane’s final velocity? ANS:
REF: K/U MSC: P
OBJ: 1.2
LOC: FM1.02
KEY: FOP 3.4, p.97
33. The easterly and northerly components of a car’s velocity are 24 m/s and 30 m/s, respectively. In what direction and with what speed is the car moving? In other words, what is the car ’s velocity? ANS: Mathematical Solution:
Graphical Solution:
Therefore, the car is moving in a direction 51° to the north of due east, with a speed of 38 m/s ( = 38 m/s [E51°N]). REF: K/U MSC: SP
OBJ: 1.1
LOC: FM1.02
KEY: FOP 3.6, p.102
34. The displacement of an airplane from its starting point is 100 km [E30°N]. Determine the components of its displacement in the easterly and northerly directions. ANS: Mathematical Solution:
Graphical Solution:
Therefore, the components of a displacement of 100 km [E30°N] are 86.6 km [E] and 50.0 km [N]. REF: K/U MSC: SP
OBJ: 1.1
LOC: FM1.02
KEY: FOP 3.6, p.101
35. A boat sails in a straight line 20 km [N30°E]. What are the components of its displacement to the north and east? ANS:
REF: K/U MSC: P
OBJ: 1.1
LOC: FM1.02
KEY: FOP 3.6, p.102
36. A cannon fires a cannonball with a speed of 100 m/s at an angle of 20° above the horizontal. What are the horizontal and vertical components of the initial velocity of the cannonball? ANS:
REF: K/U MSC: P
OBJ: 1.1
LOC: FM1.02
KEY: FOP 3.6, p.102
37. A football player is running at a constant speed in a straight line up the field at an angle of 15° to the sidelines. The coach notices that it takes the player 4.0 s to get from the 25 m line to the goal line. How fast is the player running? ANS:
REF: K/U MSC: P
OBJ: 1.1
LOC: FM1.02
KEY: FOP 3.6, p.103
38. A girl swims at 3.0 m/s across a swimming pool at an angle of 30° to the side of the pool, as shown. What are the components of her swimming velocity in each of the following directions? (a) across the pool (b) along the pool’s edge
ANS:
REF:
K/U
OBJ: 1.1
LOC: FM1.02
KEY: FOP 3.6, p.102
MSC: P 39. The pilot of a light plane heads due north at an air speed of 400 km/h. A wind is blowing from the west at 60 km/h. (a) What is the plane’s velocity with respect to the ground? (b) How far off course would the plane be after 2.5 h, if the pilot had hoped to travel due north but had forgotten to check the wind velocity? ANS:
(a)
(b) REF: K/U, MC MSC: P
OBJ: 1.5
LOC: FM1.02
KEY: FOP 3.7, p.109
40. A canoeist paddles “north” across a river at 3.0 m/s. (The canoe is always kept pointed at right angles to the river.) The river is flowing east at 4.0 m/s and is 100 m wide. (a) What is the velocity of the canoe relative to the river bank? (b) Calculate the time required to cross the river. (c) How far downstream is the landing point from the starting point? ANS:
(a)
(b) The canoe crosses the river at
(c) The canoe “moves downstream” at
REF: K/U MSC: P
OBJ: 1.5
LOC: FM1.02
KEY: FOP 3.7, p.109
41. A pilot wishes to make a flight of 300 km [NE] in 45 min. On checking with the meteorological office, she finds that there will be a wind of 80 km/h from the north for the entire flight. What heading and airspeed must she use for the flight? ANS:
Using cosine law,
Using the sine law,
REF: K/U, MC MSC: P
OBJ: 1.5
LOC: FM1.02
KEY: FOP 3.7, p.109
42. A helicopter travelling horizontally at 150 km/h [E] executes a gradual turn, and eventually is moving at 120 km/h [S]. If the turn takes 50 s to complete, what is the average acceleration of the helicopter? ANS:
REF: K/U MSC: P
OBJ: 1.2
LOC: FM1.02
KEY: FOP 3.8, p.113
43. A clock has a second hand that is 12 cm long. Find each of the following. (a) the average speed of the tip of the second hand (b) its instantaneous velocity as it passes the 6 and the 9 on the clock face (c) its average velocity in moving from the 3 to the 12 on the clock face Note: the circumference of a circle is . ANS: (a) The tip of the second hand makes one complete revolution in 60 s.
(b) Since the speed is constant, | | = v = 1.3 cm/s Therefore,
= 1.3 cm/s [left]
= 1.3 cm/s [up] (c) The displacement of the tip of the second hand in moving from the 3 to the 12 may be found from the following diagram: Mathematical Solution:
