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To describe motion, you can use terms like displacement, speed, velocity, and acceleration. Displacement is a distance in a particular direction. Speed is the distance traveled in a certain amount of time. If you combine speed with a direction, you get velocity. Acceleration measures how quickly velocity changes.

The Problems You’ll Work On

In this chapter you’ll move through the following topics:

✓ Finding the displacement in one and two dimensions

✓ Using velocity to determine the displacement

✓ Taking the average of the instantaneous speed

✓ Determining the change in velocity using acceleration

✓ Relating displacement, velocity, acceleration, and time

What to Watch Out For

You’ll speed through these questions if you keep the following in mind:

✓ Using the distance traveled, not the displacement, to determine average speed

✓ Remembering that velocity is the change in displacement in a certain amount of time

✓ Remembering that acceleration is the change in velocity in a certain amount of time

Determining Displacement Using Positions in One Dimension

26–29

26. You leave your apartment and walk 2 blocks north, only to realize that you forgot your keys. You turn around and walk back 2 blocks south to get them. What is your total displacement?

27. A car is driven north for 5 miles, then south for 3 miles, and then north again for 2 miles. What is its displacement?

28. A pair of figure skaters skates together for 10 meters. Then the man launches his partner through the air. She lands 15 meters ahead of where the pair began skating. What is her displacement with respect to the launching point when she lands?

29. An elevator is at the ground floor. It goes once to the first floor, twice to the second floor, three times to the third floor, and then four times to the fourth floor. What is the displacement of the elevator when Ms. Smith gets on at the third floor?

Getting Displacement in Two Dimensions with Axes

30–35

30. You move a marker on a board from one point (2 centimeters, 4 centimeters) to another point (5 centimeters, 8 centimeters). What is the magnitude of the displacement of the marker?

31. To get to your friend’s house, you walk 4 blocks north and 1 block east. What is the direction of your displacement with respect to the direction east?

32. A basketball player shoots the ball, releasing it at 8 feet above the floor and 5 feet from the basket. The ball goes straight through the basket, which is 10 feet above the floor. What is the magnitude of the displacement of the ball from the point at which it is released and the point at which it passes through the hoop?

33. A chess board is 8 squares by 8 squares. You move your bishop from one square (3, 1) to another square (7, 5). What is the magnitude (in squares) and angle of the displacement?

34. A child is scooting around on his toy truck. He scoots 5 meters down the hall, then turns 90 degrees to the right and scoots 3 meters, then turns again 90 degrees to the right and scoots 2 more meters. What is his displacement over this trip?

35. You want to shoot a laser beam from the edge of a stage to a disco ball hanging from the ceiling. The stage is 1 meter above the floor, the disco ball is 1 meter below the ceiling, and the height of the ceiling is 4 meters. The horizontal distance from the edge of the stage where the laser is mounted to the point directly under the disco ball is 5 meters. At what angle above the horizontal should you aim the laser?

Traveling with Average Speed and at Instantaneous Speed

36–40

36. You run from your house to the grocery store in 1.0 minute, and the store is 300 meters from your house. What is your average speed in meters per second for the trip?

37. In a traffic jam, you drive at 10 miles per hour for 10 minutes, at 20 miles per hour for 1 minute, at 15 miles per hour for 5 minutes, at 30 miles per hour for 2 minutes, and at 5 miles per hour for 15 minutes. What is your maximum instantaneous speed?

38. A ball is dropped from the top floor of your five-floor apartment building. You’re on the bottom floor and see the ball go past your small window. If you measure the ball’s speed at this point, is it the average speed, the instantaneous speed, both the average and the instantaneous speed, or neither the average nor the instantaneous speed?

39. The average speed of a car being driven in London is 11 miles per hour. If you have to drive 15 miles from your home to work in London, how long do you expect it to take?

40. The average speed of runner A is 10 percent greater than that of runner B. If runner B is given a 10-meter head start in a 100-meter dash, which runner will finish first?

Distinguishing between Average Speed and Average Velocity

41–45

41. You travel north for 80 miles and then east for 30 miles. What is the magnitude of your average velocity if the entire trip takes 4 hours?

42. You travel north for 80 miles and then east for 30 miles. What is your average speed if the entire trip takes 4 hours?

43. You travel 35 miles north and 20 miles east. If the trip takes 30 minutes, what is the magnitude (in miles per hour) and direction of your average velocity?

44. A postman walks 10 blocks north, then 3 blocks east, and then south for an unknown number of blocks. The time for his trip is 1.0 hour, and each block is 100 meters long. If his average speed is 1.0 meter per second, what is the magnitude of his average velocity in meters per second?

45. You travel 40 miles north, then 30 miles east, then 20 miles north, and then 10 miles south. If your trip takes 2 hours, what is your average speed?

Speeding Up and Down with Acceleration

46–50

46. It takes you 2.0 seconds to accelerate from a standstill to a running speed of 7.0 meters per second. What is the magnitude of your acceleration?

47. The acceleration due to gravity at the surface of Earth is about 9.8 meters per second per second. If you drop a small heavy ball from the fourth floor of a building, how fast is the ball moving after 0.5 seconds?

48. Your infant daughter has a maximum crawling velocity of 0.3 meters per second. If she accelerates at 2 meters per second per second, how long does it take her to reach her maximum velocity when she starts from rest?

49. A plane’s takeoff speed is 300 kilometers per hour. If it accelerates at 2.9 meters per second per second, how long is it on the runway after starting its takeoff roll?

50. You ride your bicycle at 10 meters per second and accelerate at −2.3 meters per second per second for 10 seconds. What is your final velocity?

Finding Displacement with Acceleration and Time

51–54

51. Starting from rest, you accelerate at 2 meters per second per second for 2 seconds to get up to full speed on your bicycle. How far do you travel during this time?

52. A car accelerates northward at 4.0 meters per second per second over a distance of 30 meters. If it starts at rest, for how long does it accelerate?

53. Starting from rest, a motorcycle rider covers 200 meters in 10 seconds. What was his acceleration?

54. A tennis player serves a ball at 100 miles per hour. If the ball accelerates over a period of 0.05 seconds from essentially a standstill, how far (in meters) does the ball travel during its acceleration? One mile consists of 1,609 meters.

Finding Displacement with Acceleration and Velocities

55–58

55. You’re driving at 20 meters per second northbound and brake to slow to 10 meters per second. During that time, you cover 50 meters. What was your acceleration?

56. You ski along at 3.0 meters per second, and your friend whizzes by at a greater speed. You have to accelerate at 2.0 meters per second per second for 20 meters to attain the same speed. At what speed was your friend skiing?

57. A baseball pitcher throws a fastball at 90 miles per hour. He accelerates the ball over a distance of 2.0 meters. What is the acceleration of the ball?

58. In a spaceship, you accelerate from 200 meters per second to 500 meters per second at 10 meters per second per second. How many kilometers do you travel during this acceleration?

Finding Acceleration with Displacement and Time

59–62

59. A speed skater accelerates from a standstill to full speed over a distance of 12 meters. If she takes 2.6 seconds to do this, what is the magnitude of her acceleration?

60. Your car can accelerate at 3.4 meters per second per second. You are stopped at a red light and have 20 meters to accelerate onto the freeway when the light turns green. How long will it take you to accelerate over this distance?

61. You’re driving at 18 meters per second when you apply the brake for 4 seconds. If the magnitude of your acceleration is 2.8 meters per second per second, how far did you travel in this time?

62. A ferry boat is traveling east at 1.3 meters per second when the captain notices a boat in its path. The captain engages the reverse motors so that the ferry accelerates to the west at 0.2 meters per second per second. After 20 seconds of this acceleration, what is the boat’s position with respect to its initial position?

Finding Acceleration with Velocities and Displacement

63–66

63. A cheetah can accelerate from 0 miles per hour to 60 miles per hour in 20 meters. What is the magnitude of its acceleration?

64. A speedboat can accelerate from an initial velocity of 3.0 meters per second to a final velocity that is 3 times greater over a distance of 42 meters. What is the magnitude of its acceleration?

65. You drop a feather from your balcony, which is 4.5 meters above the ground. After falling 0.20 meters, it moves at the speed of 0.30 meters per second. What is the magnitude of its acceleration?

66. A boat moving north at 2.3 meters per second undergoes constant acceleration until its speed is 1.2 meters per second northward. With respect to its initial position, its final position is 200 meters northward. What is its acceleration?

Finding Velocities with Acceleration and Displacement

67–70

67. A boat accelerates at 0.34 meters per second per second northward over a distance of 100 meters. If its initial velocity is 2.0 meters per second northward, what is its final velocity?

68. A train brakes to a stop over a distance of 3,000 meters. If its acceleration is 0.1 meter per second per second, what is its initial speed?

69. To pass another race car, a driver doubles his speed by accelerating at 4.5 meters per second per second for 50 meters. What are his initial and final speeds?

70. From a position 120 meters above a pigeon, a falcon dives at 9.1 meters per second per second, starting from rest. After diving 25 meters, the falcon stops accelerating. What is the falcon’s speed when it strikes the pigeon?