AP Physics Velocity and Linear Acceleration Unit 1 Problems:

Unit 1 Problems: Linear Velocity and Acceleration

This entire set of problems is due the day of the test. I will not accept these for a late grade.

1. *A horse canters away from its trainer in a straight line moving 150m in 14s. It then turns abruptly and gallops halfway back in 4.5s.

a. What is its average speed? (12.16 m/s)

b. What is its average velocity for the entire trip? (4.05 m/s)

2. *At 1 p.m., a car, traveling at a constant velocity of 94 Km/h toward the west, is 17 Km to the west of your school. Where will it be at 3:30 p.m.? (252 km West of the school)

3. *Suppose the car in the previous problem started 17 Km east of your school at the same time, moving in the same direction with the same velocity.

a. Where would it be at 3:30 p.m.? (218 km W of school)

b. When would it be at your school? (.18 hr later or 10.85 min later or ~1:10 pm) II. Linear Accelerated Motion

4. *A bus is moving at 25 m/s. The driver steps on the brakes, and the bus stops in 3.0 s. a. What is the average acceleration of the bus while braking? (-8.33 m/s/s)

b. Suppose the bus took twice as long to stop. How would the acceleration compare to the acceleration you found above? (1/2 as much; -4.17 m/s/s)

5. *A car accelerates at a constant rate from 15 m/s to 25 m/s while it travels 125 m. How long does this motion take? (6.25 s)

6. *Starting from rest, a racecar moves 110 m in the first 5.0 s of uniform acceleration. What is the car’s acceleration? (8.8 m/s/s)

7. *An airplane accelerates from a velocity of 21 m/s at the constant rate of 3.0 m/s2 over + 535 m. a. What is its final velocity? (60.42 m/s)

b. The pilot stops the same plane in 484 m using a constant acceleration of – 8.0 m/s2. How fast was the plane moving before braking began? (88 m/s)

8. An Indy-500 racecar’s velocity increases from 4.0 m/s to 36 m/s over a 4.0-s period. What is its average acceleration? (8 m/s/s)

9. A car is coasting backwards down a hill at 3.0 m/s when the driver gets the engine started. After 2.5 s, the car is moving uphill at a speed of 4.5 m/s. What is the car’s average acceleration? (3 m/s/s)

10. A golf ball rolls up a hill toward a Putt-Putt hole.

a. If it starts with a velocity of +2 m/s and accelerates at a constant rate of –0.5 m/s2, what is its velocity after 2 s? (1 m/s)

b. If the acceleration occurs for 6 s, what is its final velocity? (-1 m/s) c. In words, describe the motion of the ball.

11. If a car accelerates from rest at a constant 5.5 m/s2, how long will be required to reach 28 m/s? (5.09 s)

12. A bike rider accelerates constantly to a velocity of 7.5 m/s during 4.5 s. The bike’s displacement is 19 m. What was the initial velocity of the bike? (0.94 m/s)

13. A driver brings a car traveling at 22 m/s to a full stop in 2.0 s. Assume its acceleration in constant. a. What is the car’s acceleration? (-11 m/s/s)

b. How far does it travel before stopping? (22 m/s)

14. A car is initially sliding backwards down a hill at –2.5 m/s. The driver guns the car. By the time the car’s velocity is +3.5 m/s, it is 3.2 m from its starting point. Find the acceleration. (+0.94 m/s/s)

15. A spacecraft traveling at a velocity of +1210 m/s is uniformly accelerated at – 150 m/s2. If the acceleration lasts for 8.68 s, what is the final velocity of the craft? Explain your results in words. (-92 m/s)

16. * A race driver makes a pit stop to refuel. After refueling, he leaves the pit area with an acceleration of 6 m/s/s and after 4 s he enters the main speedway. At the same instant he enters the speedway, another race-car that is on the speedway and traveling with a constant speed of 70 m/s overtakes and passes the entering car. If the entering car maintains its acceleration, how much time is required for it to catch the other car? (15.33 s, hint: both d and t are the same for both cars)

III. Acceleration due to Gravity

17. *A brick falls freely from a high scaffold.

a. What is its velocity after 4.0 s? (-39.2 m/s)

b. How far does the brick fall during the first 4.0 s? (78.4 m) 18. *If you drop a golf ball, how far does it fall in ½ s? (1.23 m) 19. *A man falls 1.0 m to the floor.

a. How long does the fall take? (0.45 s)

20. *During a baseball game, a batter hits a high pop-up. If the ball remains in the air for 6.0 s, how high does it rise? (44.1m)

21. *A pitcher throws a baseball straight up with an initial speed of 27 m/s. a. How long does it take the ball to reach its highest point? (2.76 s) b. How high does the ball rise above its release point? (37.19 m)

22. Kyle is hovering in a helicopter when he drops a bag out the window. When the bag has fallen 2.0 s, a. What is the bag’s velocity?(-19.6 m/s)

b. How far has the bag fallen? (-19.6 m)

23. Kyle is flying the same helicopter and it is rising at 5.0 m/s when he releases the bag. After 2.0 s, a. What is the bag’s velocity? (-14.6 m/s) b. How far has the bag fallen? (-9.6 m) c. How far below the helicopter is the bag? (19.6 m below the helicopter)

24. Kyle’s helicopter now descends at 5.0 m/s as he releases the bag. After 2.0 s, a. What is the bag’s velocity? (-24. 6 m/s)

b. How far has the bag fallen? (-29.6 m)

c. How far below the helicopter is the bag? (19.6 m below the helicopter)

25. *Calculate how long it takes King Kong to fall down from the top of the Empire State Building (380m) and his velocity just before hitting the ground. (8.81 sec, -86.34 m/s)

26. *A ballplayer catches a ball 3.6s after throwing it vertically upward. a. What speed did he throw it with? (17.64 m/s)

b. What height did it reach? (15.88 m)

27. *A helicopter is ascending vertically with a speed of 5.5 m/s. At a height of 100m above the Earth, a package is dropped from a window. How much time does it take for the package to hit the ground? (5.11 sec and – 3.99 sec) 28. *A falling stone takes 0.3s to travel past a window that is 2.2 m tall. From what height above the top of the window

did the stone fall? (1.75 m)

29. *Suppose you adjust your garden hose nozzle for a hard stream of water. You point the nozzle vertically upward at a height of 1.5m above the ground. When you quickly move the nozzle away from the vertical, you hear the water striking the ground next to you for another 2s. What is the speed as it leaves the nozzle? (9.05 m/s)

30. *A person falls from a fourth story window 15m above a fireman’s safety net. The survivor stretches the net 1m before coming to rest.

a. What was the average deceleration experienced by the survivor as he was slowed by the net? (147.04 m/s/s) b. What could you do to the net to make the deceleration smaller/safer?

IV. Vectors and Motion

31. *After walking 11 km due north (900) from camp, a hiker then walks 21 km due east (00 ). a. What total distance was walked by the hiker? (32 Km)

b. Determine the total displacement from the starting point. (23.71 Km, 27.60)

32. While flying due east at 120 km/h, an airplane is also carried northward at 45 km/h by the wind blowing due north. What is the plane’s resultant velocity? (128.16 Km/h, 20.560₎

33. *An airplane flies toward 149o at 525 km/h. What is the component of the plane's velocity a. toward 90o? (270.39 Km/h)

b. toward 180o? (450.01 Km/h)

34. *A motorboat heads straight across a river at 16 m/s. The river flows downstream at 9.0 m/s.

a. What is the resultant velocity (speed and direction as taken from the shore) of the boat? (18.36 m/s, 60.640) b. If the river is 136 m wide, how long does it take the motorboat to reach the other side? (8.5 s)

c. How far downstream is the boat when it reaches the other side of the river? (76.5 m) 35. A motorboat travels at 8.5 m/s. It heads straight across a river 110 m wide.

a. If the water flows downstream at a rate of 3.8 m/s, what is the boat's resultant velocity, relative to the shore? (9.31 m/s, 65.910)

b. How long does it take the boat to reach the opposite shore? (12.94 s)

36. A boat heads directly across a river 41 m wide at 3.8 m/s. The current is flowing downstream at 2.2 m/s. a. What is the resultant velocity, relative to the shore, of the boat? (4.39 m/s, 59.930)

b. How much time does it take the boat to cross the river? (10.79 s)

c. How far downstream is the boat when it reaches the other side? (23.74 m)

37. Sue rows a boat directly across a river at 3.2 m/s. The river flows at 12 m/s and is 200m wide. a. What velocity, relative to the shore, does Sue’s boat go? (12. 42 m/s, 14.930)

b. How long does it take Sue to cross the river? (62.5 s) c. How far downstream is Sue's landing point? (750 m)

d. How long would it take Sue to cross the river if there were no current? (the same amount of time as the current is perpendicular to the boat’s velocity and, therefore, will not affect the boat’s velocity directed directly across the river)

VI. Projectile Motion (constant horizontal velocity with constant vertical acceleration) 38. *A stone is thrown horizontally at a speed of + 5.0 m/s off the top of a cliff 78.4m high.

a. How long does it take the stone to reach the bottom of the cliff? (4 s) b. How far from the base of the cliff does the stone strike the ground? (20 m)

c. What are the horizontal and vertical components of the velocity of the stone just before it hits the ground? (vyf = -39.2 m/s, vx = 5 m/s as there is no acceleration acting on the projectile in the horizontal dxn)

39. An auto, moving too fast on a horizontal stretch of mountain road, slides off the road, falling into deep snow 43.9 m below the road and 87.7 m beyond the edge of the road.

a. How long did the auto take to fall? (2.99 s)

b. How fast was it going when it left the road in m/s and km/h? ( horizontal 29.30 m/s and 105.48 Km/h) c. What was its acceleration 10 m below the edge of the road? (still 9.8 m/s/s, down)

40. A steel ball rolls with constant velocity across a tabletop 0.950 m high. It rolls off and hits the ground 0.352m horizontally from the edge of the table. How fast was the ball rolling on the table?(0.8 m/s)

41. A rude tourist throws a peach pit 7m/s horizontally out of the cage of an elevator.

a. If the elevator isn’t moving, how long will it take the pit to reach the ground, 17m below? (1.86 m) b. How far horizontally will the pit land? (13.04 m)

c. Repeat a and c if the elevator was moving up with a velocity of 2m/s. (2.08 s, 14.56 m)

42. *A player kicks a football from ground level with a velocity of 27m/s at an angle of 300 above the horizontal. Find a. The total time the ball is in the air. (2.76 s)

b. The distance the ball travels before it hits the ground. (64.41 m) c. Its maximum height. (9.33 m)

43. Repeat the previous problem for the same player now kicking the ball with the same speed, but at 600 to the horizontal. (4.77 s, 64.41 m, 27.88m)

44. Repeat the above problem (at the 60 degree angle) for someone on the moon where the acceleration due to gravity is 1.62m/s/s, down. (28.86 s, 389.67 m, 168.71 m)

45. In baseball, the pitcher’s mound is raised to compensate for the fact that the ball falls downward as it travels from the pitcher to the batter. If baseball were played on the moon, would the pitcher’s mound have to be higher than, lower than, or the same height as the pitcher’s mound on Earth? (lower…the ball wouldn’t fall as fast as it would on Earth)

46. *A projectile is shot from the edge of a cliff 140 m above the ground level with an initial speed of 100 m/s at an angle of 37.0 with the horizontal

a. Determine the time taken by the projectile to hit the ground. (14.28 s)

b. Determine the range of the projectile as measured from the base of the cliff. (1.14 E3 m)

c. At the instant just before the projectile hits the ground, find the total velocity it hits the ground with (112.87 m/s, 44.960)

Challenging HW problems

47. In a snow skiing long-jump competition, the skiers ski down a ramp that tilts up at the end with a 63-degree angle to the horizontal. With the launch angle, a skier attains a height of 13m above the edge of the ramp. What is the skier’s launch speed? (17.92 m/s)

48. A rocket is fired at a speed of 75 m/s from ground level at an angle of 60 degrees to the horizontal. The rocket is fired toward a 100m tall wall, which is located 270 m away. By how much does the rocket clear the wall? (113.62m above the wall)

49. When a cannon is aimed at an angle of 360 above the horizontal, a cannon ball lands 140 m down range. What was the muzzle velocity of the cannon? (37.98 m/s)

50. A diver springs upward from a diving board that is 3m above the surface of the water. At the instant she contacts the water her speed is 8.90 m/s and her body makes an angle of 75.00 to the surface of the water.

a. Determine her initial velocity (speed and dxn) that she left the board with. (4.52 m/s, 59.410) b. How much time did she spend in the air? (1.27 s)

V. Motion Graphs (on separate paper---print it off tonight, please)

All problems below you will do on your own (you can ask me for help but I will not be going over them in class). Conceptual

1. What quantity can be obtained from the slope of a position-time graph? 2. Does a car speedometer measure speed, velocity, or both?

3. Can you conclude a car is not accelerating if its speedometer indicates a steady 50Km/hour? 4. Is the acceleration of a projectile equal to zero when it reaches the top of its trajectory? 5. A person is riding a bike. Can the person have a

a. Constant speed and a changing velocity? Justify your answer. b. Constant velocity and a changing speed? Justify your answer.

6. A NASA team is present in Cape Canaveral, Florida, to oversee a space shuttle launch. The NASA team then hops on an airplane and travels to Edwards Air Force Base in California to supervise the shuttle landing (which is a week later as it has been orbiting the earth for awhile). Which group of people, the astronauts or the NASA team, had the greater displacement?

7. A walker and a runner leave your front door at the same time. They move in the same direction at different constant velocities. Draw the position-time graphs of each.

8. A physics book is moved once around the perimeter of a square table of dimensions 1.0 m by 2.0 m. a. If the book ends up at its initial position, what is its displacement?

b. What is the distance traveled?

9. If the average velocity of a particle is zero in some time interval, what can you say about its displacement for that interval?

10. What quantity is represented by the area under a velocity-time curve? 11. Look at the figure to the right

a. What kind of motion does this graph represent? d b. What does the slope of the line represent?

t 12. The figure to the lower right is a position-time graph of two people running.

a. Describe the position of runner A relative to runner B at the y-intercepts. P

b. Which runner is faster?

c. What occurs at point P and beyond? A d

B t

13. Look at the figure to the right.

a. What kind of motion does this graph represent? v b. What does the area under the curve of the graph represent?

t

14. Look at the figure to the right.

a. What kind of motion does this graph represent? v b. What does the area under the curve of the graph represent?

t

15. If a velocity-time curve is a straight line parallel to the t-axis, what can be deduced about the acceleration? 16. Refer to the figure to the right to find the distance the moving object travels between

a. t = 0 s and t = 5 s. 30 b. t = 5 s and t = 10 s. c. t = 10 s and t = 15 s. V (m/s) 20 d. t = 0 s and t = 25 s. 10 0 5 10 15 20 25 30 t (s) 17. Use the above figure (in the previous question) to find the acceleration of the moving object

a. During the first 5 s of travel. (6 m/s/s)

b. Between the fifth and the tenth second of travel. (0 m/s/s) c. Between the tenth and the fifteenth second of travel. (-2 m/s/s) d. Between the twentieth and twenty-fifth second of travel. (-4 m/s/s)

18. Explain why an aluminum ball and a steel ball of similar size and shape, dropped from the same height, reach the ground at the same time (assume no air friction).

19. Describe the changes in the velocity of a ball thrown straight up into the air. Then describe the changes in its

acceleration.

20. Rock A is dropped from a cliff; rock B is thrown downward from the same height at the same time. a. Which arrives first?

b. When they reach the bottom, which rock has a greater velocity? c. Which has a greater acceleration?

21. A rock is thrown vertically upward with a speed v0 from the edge of a cliff. A second rock is thrown vertically

downward with the same initial speed. Which rock has the greater speed when it reaches the ground?

C

22. Consider the ball's trajectory in the figure to the right. B D a. Where is the vertical velocity the greatest?

b. Where is the greatest horizontal velocity? A E c. Where is the vertical velocity least?

23. A rifle at some height above the ground fires a bullet parallel to the ground. At the same instant and at the same height, a second bullet is dropped from rest. In the absence of air resistance, which bullet hits the ground first? 24. A stone is thrown horizontally from the top of the cliff and eventually hits the ground below. A second stone is

dropped from rest from the same spot off the cliff. Describe the differences, if any, between the two regarding their a. Displacements (both vertical and horizontal)

b. Velocity upon impact (both vertical and horizontal) c. Time in flight

25. An airplane pilot flying at constant velocity and altitude drops a flare. Ignoring air resistance, where will the plane be relative to the flare when the flare hits the ground?

26. Sam throws a baseball horizontally from the top of a cliff. If Sam throws the baseball with higher speed a. how will the time it takes to hit the ground be affected?

b. how will the horizontal distance be affected?

27. *A zoologist standing on a cliff aims a tranquilizer gun directly at a monkey hanging from a distant tree branch. The barrel of the gun is horizontal. Just as the zoologist pulls the trigger, the monkey lets go of the branch and begins to fall. Will the dart hit the monkey?

Problems

28. You and a friend each drive 50 km. You travel at 90 km/h, your friend at 95 km/h. How long will your friend wait for you at the end of the trip?

29. Ann is driving down a street at 10 m/s. Suddenly a child runs into the street. If it takes Ann 0.75 s to react and apply the brakes, how many meters will she have moved before she begins to slow down?

30. A car traveling 90 Km/h strikes a tree. The front end of the car compresses and the driver comes to a rest after traveling 0.70 m. What is the average acceleration of the driver during the collision? How many g’s is this? (-446.43 m/s/s, ~45 g’s)

31. A kangaroo jumps to a vertical height of 2.8m. How long was it in the air before returning to Earth? (1.51 s) 32. A ballplayer catches a ball 3.6s after throwing it vertically upward (from the same height).

a. What speed did he throw it? (+17.64 m/s) b. What height did it reach? (15.88m)

33. A batter hits a pop-up straight up over home plate with an initial velocity of 20 m/s. The ball is caught by the catcher at the same height from which it was hit. What is the velocity of the ball when it lands in the catcher's mitt? (-20m/s) 34. A stone is thrown directly upward with a speed of 22m/s.

a. What is its velocity at a height of 15.0m? (+13.78 m/s AND –13.78 m/s) b. How long is required to reach this height? (0.84 s AND 3.65 s)

c. Why are there two answers for part b? (you SHOULD have two answers for part b!!) 35. A stone is thrown vertically upward with a speed of 12m/s from the edge of a cliff 75m high.

a. How much later does it reach the bottom of the cliff? (5.32 s) b. What is its velocity just before hitting the ground? (-40.17 m/s) c. What total distance did it travel? (89.70 m)

36. A baseball is traveling vertically upward in the air. When it is 25m off the ground, it has a velocity of 12.0m/s. If the ball was thrown from ground level

a. What was its initial speed? (25.18 m/s)

b. How much earlier was it thrown? (1.34 seconds earlier) c. What altitude does it reach? (32.35 m)

37. *A drag-racing car starting from rest speeds up for 402m with an acceleration of 17.0m/s/s. A parachute opens, slowing the car down with an acceleration of -6.10m/s/s. How fast is the car moving 350m after the parachute opens? (96.94 m/s)

38. *A cement block accidentally falls from rest from the ledge of a 53m tall building. When the block is 14m above the ground, a man who is 2.0m tall looks up and notices the block directly above him. How much time (at most) does he have to get out of the way? (0.40 s)

39. *A roof tile falls from rest from the top of a building. An observer in the building notices that it takes 0.20s for the tile to pass her window whose height is 1.6m. How far from the top of this window is the roof? (2.51m)

40. *A log is floating on swiftly moving water. A stone is dropped from rest from a 75m high bridge and lands on the log as it passes under the bridge. If the log moves with a constant speed of 5m/s, what is the horizontal distance between the log and the bridge when the stone is released? (19.55 m)

41. *A football player starting from rest at the line of scrimmage accelerates along a straight line for 3.0s. During a negligible amount of time, he changes his acceleration to a value of 1.1m/s/s. With this acceleration he continues in the same direction for another 2s until he reaches a speed of 6.4m/s. What was his acceleration during the initial 3s period? (1.4 m/s/s)

Vectors and Projectile Motion (draw projectile diagrams for ALL problems --with all your horizontal and vertical variables clearly labeled).

42. An airplane flies at 200 km/h. What is the resultant velocity of the airplane if a. it experiences a 50 km/h tail wind (pushing it from behind)?

b. it experiences a 50 km/h head wind (pushing it from the front)?

43. You walk 30m South and then 30m East. What is your resultant displacement from your start point? 44. *Dave can row a boat across a river at 4.0 m/s. The river flows at 6.0 m/s and is 360 m across.

a. In what direction, relative to the shore, does Dave's boat go? (33.690 to the shore) b. How long does it take Dave to cross the river? (90s)

c. How far downstream is Dave's landing point? (540m)

d. How long would it take Dave to cross the river if there were no current? (90 s)

45. *Diane can row a boat at 8.0 m/s. The river flows at 6.0 m/s. If the river is 240 m wide, what angle does she need to steer the boat to reach a point directly across the river from where she launched? (41.410 relative to the shore) 46. *A bullet is fired from a rifle that is held 1.6m above the ground in a horizontal position. The firing speed of the

bullet is 1100m/s. Find the horizontal distance traveled by the bullet. (627m)

47. *The cliff divers of Acapulco push off horizontally from rock platforms about 35 m above the water, but they must clear rocky outcrops at water levels that extend out into the water 5.0 m from the base of the cliff directly under their launch point. What minimum push-off speed is necessary to do this? (>1.87m/s)

48. *A hunter aims directly horizontally at a target (on the same level) 140 m away. If the bullet leaves the gun at a speed of 280 m/s, by how much (vertical) will it miss the target? (-1.225m)

49. * A dart player throws a dart horizontally at a speed of + 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? (3.17m)

50. *A projectile is fired with an initial speed of 75.2 m/s at an angle of 34.5 above the horizontal on a long flat firing range. Determine

a. The maximum height reached by the projectile. (92.56m) b. The total time in the air. (8.69s)

c. The total horizontal distance covered (the range). (538.56m)

d. The vertical and horizontal velocity of the projectile 1.50 s after firing. (x= 61.97m/s, y= 27.89m/s) 51. *A projectile is shot from the edge of a 200m tall cliff with an initial speed of 162 m/s at an angle of 42 with the

horizontal. Determine

a. The time taken by the projectile to hit the ground. (23.83s)

b. The range of the projectile as measured from the base of the cliff. (2.87 E3m)

c. At the instant just before the projectile hits the ground, find the horizontal and the vertical components of its velocity. (x= 120.39m/s, y= -125.18m/s)

52. *An Olympic long jumper is capable of jumping a horizontal distance of 8.0 m. Assuming his horizontal speed is 9.1 m/s as he leaves the ground, how long is he in the air and how high does he go? Assume that he lands standing upright, the same way he left the ground. (0.88s, 0.95m high)

53. * A fire hose ejects a stream of water at an angle of 35 degrees to the ground. The water leaves the nozzle with a speed of 25m/s. Assuming the water behaves like a projectile, how far from the building should the fire hose be located to hit the highest possible fire? (29.90m away)

54. *During a baseball game, a fly ball is hit to center field and is caught 142.64m from home plate. A moment after the ball was caught by the center-fielder, a runner on 3rd base takes off for home and the center fielder throws the caught ball to the catcher standing on home plate (the throw occurs simultaneously with the runner beginning to run). The

runner runs the 27.43m to home plate with an average velocity of 7.84m/s, while the baseball is thrown with a

velocity of 40.17m/s at 300 to the horizontal. Which reaches home first: The runner or the ball and by how much

time? (the runner by 0.6s)

55. *A baseball is hit into the air with an initial speed of 36.6 m/s at an angle of 50 degrees above the horizontal. At the same time the ball is hit, the center fielder starts running away from the batter and catches the ball 0.914m above the level at which it was hit (the ball was on its way down when it was caught). If the center fielder is initially 110m from home plate, what must be his average speed? (4.20m/s)

56. *A baseball is hit with a velocity of 30m/s at an angle of 530 with the horizontal. Immediately an outfielder runs 4m/s toward the infield and catches the ball at the same height it was hit. What was the original distance between the batter and the outfielder? (107.82m)