3 Rectilinear Motion3.1 Speed, velocity, acceleration 1. A bike travels at a constant speed of 4.0 m/s for 5.0 s. How far does it go?(20m)

2. Light from the Sun reaches Earth in 8.3 min. The speed of light is 3.00x108m/s. How far is Earth from the Sun?(1.5 x 1011 m)

3. A car is moving down a street at 55 km/h. A child suddenly runs into the street. If it takes the driver 0.75 s to react and apply the brakes, how many meters will the carhave moved before it begins to slow down?(11m)

4. Nora jogs several times a week and always keeps track of how much time she runs each time she goes out. One day she forgets to take her stopwatch with her and wondersif theres a way she can still have some idea of her time. As she passes a particular bank, she remembers that it is 4.3 km from her house. She knows from her previoustraining that she has a consistent pace of 4.0 m/s. How long has Nora been jogging when she reaches the bank?(18 min)

5. A golf ball rolls up a hill toward a miniature-golf hole. Assume that the direction toward the hole is positive.a. If the golf ball starts with a speed of 2.0 m/s and slows at a constant rate of 0.50 m/s2, what is its velocity after 2.0 s?(1.0 m/s)

b. What is the golf balls velocity if the constant acceleration continues for 6.0 s?(-1.0 m/s)

6. A bus that is traveling at 30.0 km/h speeds up at a constant rate of 3.5 m/s2. What velocity does it reach 6.8 s later?(120 km/h)

7. If a car accelerates from rest at a constant 5.5 m/s2, how long will it take for the car to reach a velocity of 28 m/s?(5.1 s)

8. A car slows from 22 m/s to 3.0 m/s at a constant rate of 2.1 m/s2. How many seconds are required before the car istraveling at 3.0 m/s?(9.0 s)

9. A woman driving at a speed of 23 m/s sees a deer on the road ahead and applies the brakes when she is 210 m from the deer. If the deer does not move and the car stops right before it hits the deer, what is the acceleration provided by the cars brakes?(-1,3 m/s2)

10. A plane travels a distance of 5.0x102m while being accelerated uniformly from rest at the rate of 5.0 m/s2. What final velocity does it attain?(70 m/s)

11. An airplane accelerated uniformly from rest at the rate of 5.0 m/s2for 14 s. What final velocity did it attain?(70 m/s)

12. An airplane starts from rest and accelerates at a constant 3.00 m/s2for 30.0 s before leaving the ground.a. How far did it move?(1.35 x 103 m)

b. How fast was the airplane going when it took off?(90 m/s)

13. A car is driven for 2.0 h at 40.0 km/h, then for another 2.0 h at 60.0 km/h in the same direction.a. What is the cars average velocity?(50 km/h)

b. What is the cars average velocity if it is driven 1.0x102km at each of the two speeds?(48 km/h)

14. Find the uniform acceleration that causes a cars velocity to change from 32 m/s to 96 m/s in an 8.0s period.(8 m/s2)

15. A car with a velocity of 22 m/s is accelerated uniformly at the rate of 1.6 m/s2for 6.8 s. What is its final velocity?(33 m/s)

16. Determine the final velocity of a proton that has an initial velocity of 2.35x105m/s and then is accelerated uniformly in an electric field at the rate of -1.10x1012m/s2for 1.50x10-7s.(7.0 x 104 m/s)

17. A supersonic jet flying at 145 m/s experiences uniform acceleration at the rate of 23.1 m/s2for 20.0 s. What is its final velocity?(607 m/s)

18. A car moves at 12 m/s and coasts up a hill with a uniform acceleration of -1.6 m/s2.a. What is its displacement after 6.0 s?(43 m)

19. A race car can be slowed with a constant acceleration of -11 m/s2. a. If the car is going 55 m/s, how many meters will it travel before it stops?(1.4 x102 m)

b. How many meters will it take to stop a car going twice as fast?(550 m)

Free Fall23. A construction worker accidentally drops a brick from a high scaffold.a. What is the velocity of the brick after 4.0 s?(-39 m/s)

b. How far does the brick fall during this time?(78 m)

24. A student drops a ball from a window 3.5 m above the sidewalk. How fast is it moving when it hits the sidewalk?(83 m/s)

25. A tennis ball is thrown straight up with an initial speed of 22.5 m/s. It is caught at the same distance above the ground.a. How high does the ball rise?(25.8 m)

b. How long does the ball remain in the air? Hint: The time it takes the ball to riseequals the time it takes to fall.(4.60 s)

26. You decide to flip a coin to determine whether to do your physics or English homework first. The coin is flipped straight up.a. If the coin reaches a high point of 0.25 m above where you released it, what was its initial speed?(2.2 m/s)

b. If you catch it at the same height as you released it, how much time did it spend in the air?(0.45 s)

27. Your sister drops your house keys down to you from the second floor window. If you catch them 4.3 m from where your sister dropped them, what is the velocity of the keys when you catch them?(9.2 m/s)

28. A student trying out for the football team kicks the football straight up in the air. The ball hits him on the way back down. If it took 3.0 s from the time when the student punted the ball until he gets hit by the ball, what was the footballs initial velocity?(15 m/s)

29. Suppose an astronaut drops a feather from 1.2 m above the surface of the Moon. If the acceleration due to gravity on the Moon is 1.62 m/s2downward, how long does it takethe feather to hit the Moons surface?(1.2 s)

30. A bag is dropped from a hovering helicopter. The bag has fallen for 2.0 s. What is the bags velocity? How far has the bag fallen?(-20 m/s, -20 m)

31. A weather balloon is floating at a constantheight above Earth when it releases a packof instruments.a. If the pack hits the ground with avelocity of -73.5 m/s, how far did thepack fall?(-276 m)

b. How long did it take for the pack to fall?(7.50 s)

32. A helicopter is rising at 5.0 m/s when a bag of its cargo is dropped. The bag falls for 2.0 s. a. What is the bags velocity?(-15 m/s)

b. How far has the bag fallen?(10 m)

c. How far below the helicopter is thebag?(10 m)

33. The diagram shows a mass attached by a piece of string to a glider which is free to glide along an air track.

A student finds that the glider takes 1.13 s to move a distance of 90 cm starting from rest. Calculate the speed of the glider after 1.13 s. (4) (A 1.59 ms-1)

Calculate its average acceleration during this time.(3) (A 1.41 ms-2)

3.2 Projectiles

1. A stone is thrown horizontally at a speed of 5.0 m/s from the top of a cliff that is 78.4 m high.a. How long does it take the stone to reach the bottom of the cliff?(4.00 s)

b. How far from the base of the cliff does the stone hit the ground?(20 m)

c. What are the horizontal and vertical components of the stones velocity just before it hits the ground?(39.2 m/s)

2. Lucy and her friend are working at an assembly plant making wooden toy giraffes. At the end of the line, the giraffes go horizontally off the edge of the conveyorbelt and fall into a box below. If the box is 0.6 m below the level of the conveyor belt and 0.4 m away from it, what must be the horizontal velocity of giraffes as they leavethe conveyor belt?(1 m/s)

3. You are visiting a friend from elementary school who now lives in a small town. One local amusement is the ice-cream parlor, where Stan, the short-order cook, slides hiscompleted ice-cream sundaes down the counter at a constant speed of 2.0 m/s to the servers. (The counter is kept very well polished for this purpose.) If the serverscatch the sundaes 7.0 cm from the edge of the counter, how far do they fall from the edge of the counter to the point at which the servers catch them?(0.0060 m)

4. A player kicks a football from ground level with an initial velocity of 27.0 m/s, 30.0 above the horizontal, as shown in Figure 6-4. Find each of the following. Assume that air resistance is negligible.

a. the balls hang time(2.76 s)

b. the balls maximum height(9.30 m)

c. the balls range(64.5 m)

5. The player in problem 4 then kicks the ball with the same speed, but at 60.0 from the horizontal. What is the balls hang time, range, and maximum height?(27.9 m)

6. A rock is thrown from a 50.0-m-high cliff with an initial velocity of 7.0 m/s at an angle of 53.0 above the horizontal. Find the velocity vector for when it hits theground below.(37 m/s. 83o from horizontal)

7. Projectile Motion A softball is tossed into the air at an angle of 50.0with the vertical at an initial velocity of 11.0 m/s. What is its maximum height?(2.55 m)

8. Projectile Motion A tennis ball is thrown out a window 28 m above the ground at an initial velocity of 15.0 m/s and 20.0 below the horizontal. How far does the ball move horizontally before it hits the ground?(27.1 m)

9. You accidentally throw your car keys horizontally at 8.0 m/s from a cliff 64-m high. How far from the base of the cliff should you look for the keys?(29 m)

10. The toy car in Figure 6-12 runs off the edge of a table that is 1.225-m high. The car lands 0.400 m from the base of the table.

a. How long did it take the car to fall?(0.500 s)

b. How fast was the car going on the table?(0.800 m/s)

11. A dart player throws a dart horizontally at 12.4 m/s. The dart hits the board 0.32 m below the height from which it was thrown. How far away is the player from the board?(3.2 m)

12. You took a running leap off a high-diving platform. You were running at 2.8 m/s and hit the water 2.6 s later. How high was the platform, and how far from the edge of the platform did you hit the water? Ignore air resistance.(7.3 m)

13. An arrow is shot at 30.0 above the horizontal. Its velocity is 49 m/s, and it hits the target.a. What is the maximum height the arrow will attain?(31 m)

b. The target is at the height from which the arrow was shot. How far away is it?(2.1 x 102 m)

14. An airplane traveling 1001 m above the ocean at 125 km/h is going to drop a box of supplies to shipwrecked victims below.a. How many seconds before the plane is directly overhead should the box be dropped?(14.3 s)

b. What is the horizontal distance between the plane and the victims when the box is dropped?(497 m)

15. Divers in Acapulco dive from a cliff that is 61 m high. If the rocks below the cliff extend outward for 23 m, what is the minimum horizontal velocity a diver musthave to clear the rocks?(6.5 m/s)

16.A body is projected horizontally from ground level with a speed of 24 m/s at an angleof 30 above the horizontal. Neglect air resistance and calculatea. the vertical resolved part of its velocityb. the time taken to reach its highest pointc. the greatest height reachedd. the horizontal range of the body.

17.The long jumper in the diagram is shown at the instant he leaves the ground, at three positions during his flight and at the instant he first touches the sand. His long jump measures 7.5 m and he is recorded as being in the air for 0.80 s. His centre of gravity falls 0.95 m between his take-off and landing.

a. Calculate his horizontal velocity at take off. (9.4 ms-1)b. Show that his vertical velocity at take-off is 2.7 m/s.c. Hence calculate the angle at which he projects himself at take-off. (16o)

18. The cliff divers of Acapulco, Mexico, take off horizontally from a rocky cliff face 26.5 mabove the surface of the water. In the course of their flight, of which all but the last 1.5 m of their horizontal motion is above rock, they travel 8.0 m forward.a. Make a sketch of the cliffs and water, and draw on the flight path of the diver.b. For how long are they in the air (ignore air resistance)? (2.3 s)c. For how long are they over the water during their flight? (0.44 s)d. What is their vertical velocity on entry? (23 ms-1)e. At what angle is their path to the vertical at entry? (8.6o)

19. A cricketer bowls a ball from a height of 2.3 m. The ball leaves the hand horizontally with a velocity u. After bouncing once, it passes just over the stumps at the top of its bounce. The stumps are 0.71 m high and are situated 20 m from where the bowler releases the ball.

a. Show that from the moment it is released, the ball takes about 0.7 s to fall 2.3 m. (2)

b. How long does it take the ball to rise 0.71 m after bouncing? (3) (A 0.38 s)

c. Use your answers to parts (a) and (b) to calculate the initial horizontal velocity u of the ball. You may assume that the horizontal velocity has remained constant. (2) (A 18.86 m s-1)

d. In reality the horizontal velocity would not be constant. State one reason why. (1)

3.3 Displacement-time and velocity-time graphs

1. The diagram shows a velocity-time graph for a ball bouncing vertically on a hard surface.

a. At what instant does the graph show the ball to be in contact with the ground for the third time? (2) (A 2.05 s and 2.10 s)

b. The downwards-sloping lines on the graph are straight. Why are they straight? (2)

c. Calculate the height from which the ball is dropped. (3) (A Between 1.2m and 1.3m)

d. Sketch a displacement-time curve on the axes below for the first second of the motion. (3)

e. What is the displacement of the ball when it finally comes to rest? (1) (A -1.25 m)

2. The graph shows the horizontal speed v of a long jumper from the start of his run to the time when he reaches the take-off board.

a. Use the graph to estimate his maximum acceleration. (3) (A 4 m s-2)

b. Use the graph to estimate the distance of the run-up.(2) (A 33 m)

3.The graph shows, in idealized form, a velocitytime graph for a typical short journey.

a. Calculate the acceleration at each stage of the journey and display your answers on accelerationtime graph.(A -5.0 ms-2)b. Sketch a displacementtime graph for this journey.

4.The graph on the next page shows the result of studying a sprint start.a. What, in m.p.h., was the maximum velocity reached? Take 1 m/s to equal 2.24 m.p.h. (A 21 m.p.h)b. Calculate the acceleration of the sprinter (i) as she leaves her blocks (ii) after 2.0 s. (i 8.5 ms-2, ii 1.5 ms-2)

5. The diagram below shows a trolley running down a slope.

a. Complete the diagram to show an experimental arrangement you could use to determine how the trolleys position varies with time. (2)

The data is used to produce a velocity-time graph for the trolley. Below is the graph for the motion from point A to point B. Time is taken to be zero as the trolley passes A, and the trolley passes B 0.70 s later.

b. The motion shown on the graph can be described by the equation = u + at. Use information from the graph to determine values for u and a. (3) (A 0.95 ms-1, 0.79 ms-2)

c. Determine the distance AB. (3) (A 0.86m)

d. On the axes below sketch a graph to show how the displacement x of the trolley from point A varies with time t. Add a scale to each axis. (3)

6. Two cars, A and B, are travelling along the outside lane of a motorway at a speed of 30.0 m s1. They are a distance d apart.

The driver of car A sees a slower vehicle move out in front of him, and brakes hard until his speed has fallen to 22.0 m s1. The driver of car B sees car A brake and, after a reaction time of 0.900 s, brakes with the same constant deceleration as A. The diagram below shows velocity-time graphs for car A (solid line) and car B (broken line).

a. Find the deceleration of the cars whilst they are braking. (3) (6.1-6.3 ms-2)

b. What does the area under a velocity-time graph represent? (1)

c. Determine the shaded area. (2) (A 6.9 -7.5)

d. State the minimum value of the initial separation d if the cars are not to collide. Explain how you arrived at your answer. (2)

e. Suppose that, instead of only slowing down to 22.0 ms1, the cars had to stop. Add lines to the grid above to show the velocitytime graphs in this case. (Assume that the cars come to rest with the same constant deceleration as before. (1)

f. Explain why a collision is now more likely.(2)

7. A ball is dropped from a high window onto a concrete floor. The velocitytime graph for part of its motion is shown.

a. Calculate the gradient from the origin to A. Comment on the significance of your answer. (3) (10 ms-2)

b.What happened to the ball at point A? (1)

c. Calculate the height of the window above the ground. (3) (A 45m)

8. The graph shows the variation of velocity with time for a body moving in a straight line.

Calculate (i) the total distance travelled, (300 m)

(ii) the average speed over the 20 seconds. (4) (15 ms-1)

9. An athlete runs a 100 m race. The idealised graph below shows how the athletes velocity v changes with time t for a 100 m sprint.

a. By considering the area under the graph, calculate the maximum velocity vmax of the athlete. (3) (A 10 ms-1)

b. Using the axes below, sketch a graph showing how the acceleration of this athlete changes with time during this race. Mark any significant values on the axes.(4)

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