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**James S. Walker, Physics 4th edition, **** Addison-Wesley 2009**.

**Chapter 2**

1. Referring to Figure 2-20, you walk from your home to the library, then to the park. (a) What is the distance traveled? (b) What is your displacement?

2. The two tennis players shown in Figure 2-21 walk to the net to congratulate one another. (a) Find the distance traveled and the displacement of player A. (b) Repeat for player B.

3. The golfer in Figure 2-22 sinks the ball in two putts, as shown. What are (a) the distance traveled by the ball, and (b) the displacement of the ball?

4. In figure 2-20, you walk from the park to your friend’s house, then back to your house. What is (a) distance traveled, and (b) displacement?

5. A jogger runs on the track shown in Figure 2-23. Neglecting the curvature of the corners, (a) what is the distance traveled and the displacement in running from point A to point B? (b) Find the distance and displacement for a complete circuit of the track.

6. A child rides a pony on a circular track whose radius is 4.5 m. (a) Find the distance traveled and the displacement after the child has gone halfway around the track. (b) Does the distance traveled increase, decrease, or stay the same when the child completes one circuit of the track? Explain. (c) Does the displacement increase, decrease, or stay the same when the child completes one circuit of the track? Explain. (d) Find the distance traveled and displacement after a complete circuit of the track.

7. You drive your car in a straight line at 15 m/s for 10 kilometers, then at 25 m/s for another 10 kilometers. (a) Is your average speed for the entire trip more than, less than, or equal to 20 m/s? (b) Choose the best explanation from among the following: (1) More time is spent at 15 m/s than at 25 m/s. (2) The average of 15 m/s and 25 m/s is 20 m/s. (3) Less time is spent at 15 m/s than at 25 m/s.

9. Joseph DeLoach of the United States set an Olympic record in 1988 for the 200-meter dash with a time of 19.75 seconds. What was his average speed? Give your answer in meters per second and miles per hour.

10. In 1992 Zhuang Yong of China set a women’s Olympic record in the 100-meter freestyle swim with a time of 54.64 seconds. What was her average speed in m/s and mi/h?

11. Kangaroos have been clocked at speeds of 65 km/h. (a) How far can a kangaroo hop in 3.2 minutes at this speed? (b) How long will it take a kangaroo to hop 0.25 km at this speed?

12. A severe storm on January 10, 1992, caused a cargo ship near the Aleutian Islands to spill 29,000 rubber ducks and other bath toys into the ocean. Ten months later hundreds of rubber ducks began to appear along the shoreline near Sitka, Alaska, roughly 1600 miles away. What was the approximate average speed in(a) m/s and (b) mi/h?

13. Radio waves travel at the speed of light, approximately 186000 miles per second. How long does it take for a radio message to travel from Earth to the Moon and back?

14. It was dark and stormy night, when suddenly you saw a flash of lightning. Three-and-a-half seconds later you heard the thunder. Given that the speed of sound in air is about 340 m/s, how far away was the lightning bolt?

15. The human nervous system can propagate nerve impulses at about 102 m/s. Estimate the time it takes for a nerve impulse generated when your finger touches a hot object to travel to your brain.

16. Estimate how fast your hair grows in miles per hour.

17. A finch rides on the back of a Galapagos tortoise, which walks at the stately pace of 0.060 m/s. After 1.2 minutes the finch tires of the tortoise’s slow pace, and takes flight in the same direction for another 1.2 minutes at 12 m/s. What was the average speed of the finch for this 2.4-minute interval?

18. You jog at 9.5 km/h for 8.0 km, then you jump into a car and drive an additional 16 km. With what average speed must you drive your car if your average speed for the entire 24 km is to be 22 km/h?

19. A dog runs back and forth between its two owners, who are walking toward one another. The dog starts running when the owners are 10.0 m apart. If the dog runs with a speed of 3.0 m/s, and the owners each walk with a speed of 1.3 m/s, how far has the dog traveled when the owners meet?

20. You drive in a straight line at 20.0 m/s for 10.0 minutes, then at 30.0 m/s for another 10.0 minutes. (a) Is your average speed 25.0 m/s, more than 25.0 m/s, or less than 25.0 m/s? Explain. (b) Verify your answer to part (a) by calculating the average speed.

21. In heavy rush-hour traffic you drive in a straight line at 12 m/s for 1.5 minutes, then you have to stop for 3.5 minutes, and finally you drive at 15 m/s for another 2.5 minutes. (a) Plot the position-versus-time graph for this motion. Your plot should extend from t=0 to t=7.5 minutes. (b) Use your plot from part (a) to calculate the average velocity between t=0 and t=7.5 minutes.

22. You drive in a straight line at 20.0 m/s for 10.0 miles, then at 30.0 m/s for another 10.0 miles. (a) Is your average speed 25.0 m/s, more than 25.0 m/s, less than 25.0 m/s? Explain. (b) Verify your response to part (a) by calculating the average speed.

23. An expectant father paces back and forth producing the position-versus-time graph shown in Figure 2-25. Without performing a calculation indicate whether the father’s velocity is positive, negative, or zero on the segments of the graph: (a) A, (b) B, (c) C, and (d) D. (b) Calculate the numerical value of the father’s velocity for the segments (e) A, (f) B, (g) C, and (h) D, and show that your results verify your answers to parts (a)-(d).

32. A 747 airliner reaches its takeoff speed of 173 mi/h in 35.2 s. What is the magnitude of its average acceleration?

33. At the starting gun, a runner accelerates at 1.9 m/s2 for 5.2 s. The runner’s acceleration is zero for the rest of the race. What is the speed of the runner (a) at t = 2.0 s, and (b) at the end of the race?

34. A jet makes a landing traveling due east with a speed of 115 m/s. If the jet comes to rest in 13.0 s, what are the magnitude and direction of its average acceleration?

35. A car is traveling due north at 18.1 m/s. Find the velocity of the car after 7.50 s if its acceleration is (a) 1.30 m/s2 due north, or (b) 1.15 m/s2 due south.

36. A motorcycle moves according to the velocity-versus-time graph shown in Figure 2-28. Find the average acceleration of the motorcycle during each of the following segments of the motion: (a) A, (b) B, and (c) C.

37. A person on horseback moves according to the velocity –versus-time graph shown in Figure 2-29. Find the displacement of the person for each of the following segments of the motion: (a) A, (b) B, and (c) C.

38. Running with an initial velocity of +11 m/s, a horse has an average acceleration of -1.81 m/s2. How long does it take for the horse to decrease its velocity to +6.5 m/s?

39. Assume that the brakes in your car create a constant deceleration of 4.2 m/s2 regardless of how fast you are driving. If you double your driving speed from 16 m/s to 32 m/s, (a) does the time required to come to a stop increase by a factor of two or a factor of four? Explain. Verify your answer to part (a) by calculating the stopping times for initial speeds of (b) 16 m/s and (c) 32 m/s.

40. In previous problem, (a) does the distance needed to stop increase by a factor of two or a factor of four? Explain. Verify your answer to part (a) by calculating the stopping distances for initial speeds of (b) 16 m/s and (c) 32 m/s.

41. As a train accelerates away from a station, it reaches a speed of 4.7 m/s in 5.0 s. If the train’s acceleration remains constant, what is its speed after an additional 6.0 s has elapsed?

42. A particle has an acceleration of +6.24 m/s2 for 0.300 s. At the end of this time the particle’s velocity is +9.31 m/s. What was the particle’s initial velocity?

43. Landing with a speed of 81.9 m/s, and traveling due south, a jet comes to rest in 949 m. Assuming the jet slows with constant acceleration, find the magnitude and direction of its acceleration.

44. When you see a traffic light turn red, you apply the brakes until you come to a stop. If your initial speed was 12 m/s, and you were heading due west, what was your average velocity during braking? Assume constant acceleration.

45. A ball is released at the point x = 2 m on an inclined plane with a nonzero initial velocity. After being released, the ball moves with constant acceleration. The acceleration and initial velocity of the ball are described by one of the following four cases: case 1, a > 0, v0 > 0; case 2, a > 0, v0 < 0; case 3, a < 0, v0 > 0; case 4, a < 0, v0 < 0. (a) In which of these cases will the ball definitely pass x = 0 at some later time? (b) In which of these cases is more information needed to determine whether the ball will cross x = 0? (c) In which of these cases will the ball come to rest momentarily at some time during its motion?

46. Suppose the car in Problem 44 comes to rest in 35 m. How much time does it take?

47. Starting from rest, a boat increases its speed to 4.12 m/s with constant acceleration. (a) What is the boat's average speed? (b) If it takes the boat 4.77 s to reach this speed, how far has it traveled?

48. A cheetah can accelerate from rest to 25.0 m/s in 6.22 s. Assuming constant acceleration, (a) how far has the cheetah run in this time? (b) After sprinting for just 3.11 s, is the cheetah’s speed 12.5 m/s, more than 12.5 m/s, or less than 12.5 m/s? Explain. (c) What is the cheetah's average speed for the first 3.11 s of its sprint? For the second 3.11 s of its sprint? (d) Calculate the distance covered by the cheetah in the first 3.11 s and the second 3.11 s.

49. A child slides down a hill on a toboggan with an acceleration with an acceleration of 1.8 m/s2. If she starts at rest, how far has she traveled in (a) 1.0 s, (b) 2.0 s, and (c) 3.0 s.

50. On a ride called the Detonator at Worlds Fun in Kansas City, passengers accelerate straight downward from rest to 45 mi/h in 2.2 seconds. What is the average acceleration of the passengers on this ride?

51. Air Bags. Air bags are designed to deploy in 10 ms. Estimate the acceleration of the front surface of the bag as it expands. Express your answer in terms of the acceleration of gravity g.

52. Jules Verne. In his novel From the Earth to the Moon (1866), Jules Verne describes a spaceship that is blasted out of a cannon, called the Columbiad, with a speed of 12,000 yards/s. The Columbiad is 900 ft long, but part of it is packed with powder, so the spaceship accelerates over a distance of only 700 ft. Estimate the acceleration experienced by the occupants of the spaceship during launch. Give your answer in m/s2.

53. Approximately 0.1% of the bacteria in an adult human's intestines are Escherichia coli. These bacteria have been observed to move with speeds up to 15 μm/s and maximum accelerations of 166 μm/s2. Suppose an E. coli bacterium in your intestines starts at rest and accelerates at 156 μm/s2. How much (a) time and (b) distance are required for the bacterium to reach a speed of 12 μm/s?

54. Two cars drive on a straight highway. At time t = 0, car 1 passes mile marker 0 traveling due east with a speed of 20.0 m/s. At the same time, car 2 is 1.0 km east of mile marker 0 traveling at 30.0 m/s due west. Car 1 is speeding up with an acceleration of magnitude 2.5 m/s2, and car 2 is slowing down with an acceleration of magnitude 3.2 m/s2. (a) Write x-versus-t equations of motion for both cars, taking east as the positive direction. (b) At what time do the cars pass next to one another?

55. A meteorite strikes. On October9, 1992, a 27-pound meteorite struck a car in Peekskill, NY, leaving a dent 22 cm deep in the trunk. If the meteorite struck the car with a speed of 130 m/s, what was the magnitude of its deceleration, assuming it to be constant?

56. A rocket blasts off and moves straight upward from the launch pad with constant acceleration. After 3.0 s the rocket is at a height of 77 m. (a) What are the magnitude and direction of the rocket’s acceleration? (b) What is its speed at this time?

57. You are driving through town at 12.0 m/s when suddenly a ball rolls out in front of you. You apply the brakes and begin decelerating at 3.5 m/s2. (a) How far do you travel before stopping? (b) When you have traveled only half the distance in part (a), is your speed 6.0 m/s, greater than 6.0 m/s, or less than 6.0 m/s? Support your answer with a calculation.

58. You are driving through town at 16.0 m/s when suddenly a car backs out of a driveway in front of you. You apply the brakes and begin decelerating at 3.2 m/s2. (a) How much time does it take to stop? (b) After braking half the time found in part (a), is your speed 8.0 m/s, greater than 8.0 m/s, or less than 8.0 m/s? Support your answer with a calculation. (c) If the car backing out was initially 55 m in front of you, what is the maximum reaction time you can have before hitting the brakes and still avoid hitting the car?

59. A tongue’s acceleration. When a chameleon captures an insect, its tongue can extend 16 cm in 0.10 s. (a) Find the magnitude of the tongue's acceleration, assuming it to be constant. (b) In the first 0.050 s, does the tongue extend 8.0 cm, more than 8.0 cm, or less than 8.0 cm? Support your conclusion with a calculation.

60. Coasting due west on your bicycle at 8.4 m/s, you encounter a sandy patch of road 7.2 m across. When you leave the sandy patch, your speed has been reduced by 2.0 m/s to 6.4 m/s. (a) Assuming the sand causes a constant acceleration, what was the bicycle's acceleration in the sandy patch? Give both magnitude and direction.(b) How long did it take to cross the sandy patch? (c) Suppose you enter the sandy patch with a speed of only 5.4 m/s. Is your final speed in this case 3.4 m/s, more than 3.4 m/s, or less than 3.4 m/s? Explain.

61. Surviving a large deceleration. On July 13, 1977, while on a test drive at Britain’s Silverstone racetrack, the throttle on David Purley's car stuck wide open. The resulting crash subjected Purley to the greatest "g -force" ever survived by a human - he decelerated from 173 km/h to zero in a distance of only about 0.66 m. Calculate the magnitude of the acceleration experienced by Purley (assuming it to be constant), and express your answer in units of the acceleration of gravity, 9.8 m/s2.

62. A boat is cruising in a straight line at a constant speed of 2.6 m/s when it is shifted into neutral. After coasting 12 m the engine is engaged again, and the boat resumes cruising at the reduced constant speed of 1.6 m/s. Assuming constant acceleration while coasting, (a) how long did it take for the boat to coast the 12 m? (b) What was the boat's acceleration while it was coasting? (c) When the boat had coasted for 6.0 m, was its speed 2.1 m/s, more than 2.1 m/s, or less than 2.1 m/s? Explain.

63. A model rocket rises with constant acceleration to a height of 3.2 m, at which point its speed is 26.0 m/s. (a) How much time does it take for the rocket to reach this height? (b) What was the magnitude of the rocket’s acceleration? Find the height and speed of the rocket 0.10 s after launch.

64. The infamous chicken is dashing toward home plate with a speed of 5.8 m/s when he decides to hit the dirt. The chicken slides for 1.1 s, just reaching the plate as he stops (safe, of course). (a) What are the magnitude and direction of the chicken’s acceleration? (b) How far did the chicken slide?

65. A bicyclist is finishing his repair of a flat tire when a friend rides by with a constant speed of 3.5 m/s. Two seconds later the bicyclist hops on his bike and accelerates at 2.4 m/s2 until he catches his friend. (a) How much time does it take until he catches his friend? (b) How far has he traveled in this time? (c) What is his speed when he catches up?

66. A car in stop-and-go traffic starts at rest, moves forward 13 m in 8.0 s, then comes to rest again. The velocity-versus-time plot for this car is given in Figure 2-30. What distance does the car cover in (a) the first 4.0 seconds of its motion and (b) the last 2.0 seconds of its motion? (c) What is the constant speed V that characterizes the middle portion of its motion?

67. A car and a truck are heading directly toward one another on a straight and narrow street, but they avoid a head-on collision by simultaneously applying their brakes at t = 0. The resulting velocity-versus-time graphs are shown in Figure 2-31. What is the separation between the car and the truck when they have come to rest, given that at t = 0 the car is at x = 15 m and the truck is at x = -35 m?

68. In a physics lab, students measure the time it takes a small cart to slide a distance of 1.00 m on a smooth track inclined at an angle θ above the horizontal. Their results are given in the following table.

θ 10.0° 20.0° 30.0°

time, s 1.08 0.770 0.640

(a) Find the magnitude of the cart’s acceleration for each angle.

(b) Show that your results for part (a) are in close agreement with the formula, a = g sin θ.

69. At the edge of a roof you throw ball 1 upward with an initial speed v0; a moment later you throw ball 2 downward with the same initial speed. The balls land at the same time. Which of the following statements is true for the instant just before the balls hit the ground? A. The speed of ball 1 is greater than the speed of ball 2; B. The speed of ball 1 is equal to the speed of ball 2; C. The speed of ball 1 is less than the speed of ball 2.

70. Legend has it that Isaac Newton was hit on the head by a falling apple, thus triggering his thoughts on gravity. Assuming the story to be true, estimate the speed of the apple when it struck Newton.

71. The cartoon shows a car in free fall. Is the statement made in the cartoon accurate? Justify your answer.

72. Referring to the cartoon in Problem 71, how long would it take for the car to go from 0 to 30 mi/h?

73. Jordan’s Jump. Michael Jordan’s vertical leap is reported to be 48 inches. What is his takeoff speed? Give your answer in meters per second.

74. Gulls are often observed dropping clams and other shellfish from a height to the rocks below, as a means of opening the shells. If a seagull drops a shell from rest at a height of 14 m, how fast is the shell moving when it hits the rocks?

75. A volcano launches a lava bomb straight upward with an initial speed of 28 m/s. Taking upward to be the positive direction, find the speed and direction of motion of the lava bomb (a) 2.0 seconds and (b) 3.0 seconds after it is launched.

76. An Extraterrestrial Volcano. The first active volcano observed outside the Earth was discovered in 1979 on Io, one of the moons of Jupiter. The volcano was observed to be ejecting material to a height of about 2.00 × 105 m. Given that the acceleration of gravity on Io is 1.80 m/s2, find the initial velocity of the ejected material.

77. Measure Your Reaction Time. Here’s something you can try at home – an experiment to measure your reaction time. Have a friend hold a ruler by one end, letting the other end hang down vertically. At the lower end, hold your thumb and index finger on either side of the ruler without warning. Catch it as quickly as you can. If you catch the ruler 5.2 cm from the lower end, what is your reaction time?

78. A carpenter on the roof of a building accidentally drops her hammer. As the hammer falls it passes two windows of equal height, as shown in Figure 2-32. (a) Is the increase in speed of the hammer as it drops past window I greater than, less than, or equal to the increase in speed as it drops past window 2? (b) Choose the best explanation from among the following:

I. The greater speed at window 2 results in a greater increase in speed.

II. Constant acceleration means the hammer speeds up the same amount for each window.

III. The hammer spends more time dropping past window 1.

79. Predict/Explain. Figure 2-33 shows a v-versus-t plot for the hammer dropped by the carpenter in Problem 78. Notice that the times when the hammer passes the two windows are indicated by shaded areas. (a) Is the area of the shaded region corresponding to window 1 greater than, less than, or equal to the area of the shaded region corresponding to window 2? (b) Choose the best explanation from among the following:

I. The shaded area for window 2 is higher than the shaded area for window 1.

II. The windows are equally tal1.

III. The shaded area for window 1 is wider than the shaded area for window 2.

80. A ball is thrown straight upward with an initial speed v0. When it reaches the top of its flight at height h, a second ball is thrown straight upward with the same initial speed. Do the balls cross paths at height ½ h, above ½ h, or below ½ h.

81. Bill steps off a 3.0-m-high diving board and drops to the water below. At the same time, Ted jumps upward with a speed of 4.2 m/ s from a 1.0-m-high diving board. Choosing the origin to be at the water's surface, and upward to be the positive x direction, write x-versus-t equations of motion for both Bill and Ted.

82. Repeat the previous problem, this time with the origin 3.0 m above the water, and with downward as the positive x direction.

83. On a hot summer day in the state of Washington while kayaking, I saw several swimmers jump from a railroad bridge into the Snohomish River below. The swimmers stepped off the bridge, and I estimated that they hit the water 1.5 s later. (a) How high was the bridge? (b) How fast were the swimmers moving when they hit the water? (c) What would the swimmers’ drop time be if the bridge were twice as high?

84. Highest Water Fountain. The world's highest fountain of water is located, appropriately enough in Fountain Hills, Arizona. The fountain rises to a height of 560 ft (5 feet higher than the Washington Monument). (a) What is the initial speed of the water? (b) How long does it take for water to reach the top of the fountain?

85. Wrongly called for a foul, an angry basketball player throws the ball straight down to the floor. If the ball bounces straight up and returns to the floor 2.8 s after first striking it, what was the ball's greatest height above the floor?

86. To celebrate a victory, a pitcher throws her glove straight upward with an initial speed of 6.0 m/s. (a) How long does it take for the glove to return to the pitcher? (b) How long does it take for the glove to reach its maximum height?

87. Standing at the edge of a cliff 32.5 m high, you drop a ball. Later, you throw a second ball downward with an initial speed of 11.0 m/s. (a) Which ball has the greater increase in speed when it reaches the base of the cliff, or do both balls speed up by the same amount? (b) Verify your answer to part (a) with a calculation.

88. You shoot an arrow into the air. Two seconds later (2.00 s) the arrow has gone straight upward to a height of 30.0 m above its launch point. (a) What was the arrow's initial speed? b) How, long did it take for the arrow to first reach a height of 15.0 m above its launch point?

89. While riding on an elevator descending with a constant speed of 3.0 m/s, you accidentally drop a book from under your arm. (a) How long does it take for the book to reach the elevator floor, 1.2 m below your arm? (b) What is the book's speed relative to you when it hits the elevator floor?

90. A hot-air balloon is descending at a rate of 2 0 m/s when a passenger drops a camera lf the camera is 45 m above the ground when it is dropped, (a) how long does it take for the camera to reach the ground, and (b) what is its velocity just before it lands? Let upward be the positive direction for this problem.

91. Standing side by side, you and a friend step off a bridge at different times and fall for 1.6 s to the water below. Your friend goes first, and you follow after she has dropped a distance of 2.0 m. (a) When your friend hits the water' is the separation between the two of you 2.0 m, less than 2.0 m, or more than 2.0 m? (b) Verify your answer to part (a) with a calculation.

92. A model rocket blasts off and moves upward with an acceleration of 12 m/s2 until it reaches a height of 26 m at which point its engine shuts off and it continues its flight in free fall. (a) What is the maximum height attained by the rocket? (b) What is the speed of the rocket just before it hits the ground? (c) What is the total duration of the rocket's flight?

93. Hitting the “High Striker”. A young woman at a carnival steps up to the "high striker," a popular test of strength where the contestant hit one end of a lever with a mallet, propelling a small metal plug upward toward a bell. She gives the mallet a mighty swing and sends the plug to the top of the striker, where it rings the bell. Figure 2-34 shows the corresponding position-versus-time plot for the plug. Using the information given in the plot, answer the following questions:

(a) What is the average speed of the plug during its upward journey? (b) By how much does the speed of the plug decrease during its upward journey? (c) What is the initial speed of the plug?

94. While sitting on a tree branch 10.0 m above the ground, you drop a chestnut. When the chestnut has fallen 2.5 m you throw a second chestnut straight down. What initial speed must you give the second chestnut if they are both to reach the ground at the same time?

95. In a well-known Jules Verne novel, Phileas Fogg travels around the world in 80 days. What was Mr. Fogg's approximate average speed during his adventure?

96. An astronaut on the Moon drops a rock straight downward from a height of 1.25 m. If the acceleration of gravity on the Moon is 1.62 m/s2, what is the speed of the rock just before it lands?

97. You jump from the top of a boulder to the ground 1.5 m below. Estimate your deceleration on landing.

98. A Supersonic Waterfall. Geologists have learned of periods in the past when the Strait of Gibraltar closed off, and the Mediterranean Sea dried out and become a desert. Later, when the strait reopened, a massive saltwater waterfall was created. According to geologists, the water in this waterfall was supersonic; that is, it fell with speeds in excess of the speed of sound. Ignoring air resistance, what is the minimum height necessary to create a supersonic waterfall? (The speed of sound may be taken to be 340 m/s.)

99. At the edge of a roof you drop ball A from rest and then throw ball B downward with an initial velocity of v0. Is the increase in speed just before the balls land more for ball A, more for ball B, or the same for each ball?

100. Suppose the two balls described in Problem 99 are released at the same time, with ball A dropped from rest and ball B thrown downward with an initial speed v0. Identify which of the five plots shown in Figure 2-36 corresponds to (a) ball A and (b) ball B.

101. Astronauts on a distant planet throw a rock straight upward and record its motion with a video camera. After digitizing their video, they are able to produce the graph of height, y, versus time, t, shown in Figure 2-36. (a) What is the acceleration of gravity on this planet? (b) What was the initial speed of the rock?

102. Drop Tower. NASA operates a 2.2-second drop tower at the Glenn Research Center in Cleveland, Ohio. At this facility, experimental packages are dropped from the top of the 8th floor of the building. During their 2.2 seconds of free fall, experiments experience a microgravity environment similar to that of a spacecraft in orbit. (a) What is the drop distance of a 2.2-s tower? (b) How fast are the experiments travelling when they hit the air bags at the bottom of the tower? (c) If the experimental package comes to rest over a distance of 0.75 m upon hitting the air bags, what is the average stopping acceleration?

103. A youngster bounces straight up and down on a trampoline. Suppose she doubles her initial speed from 2.0 m/s to 4.0 m/s. (a) By what factor does her time in the air increase? (b) By what factor does her maximum height increase? (c) Verify yours answers to parts (a) and (b) with an explicit calculation.

104. At the l8th green of the U.S. Open you need to make a 20.5-ft putt to win the tournament. When you hit the ba1l, giving it an initial speed of 1.57 m/s, it stops 6.00 ft short of the hole. (a) Assuming the deceleration caused by the grass is constant, what should the initial speed have been to just make the putt? (b) What initial speed do you need to make the remaining 6.00-ft putt?

105. A popular entertainment at some carnivals is the blanket toss. (a) If a person is thrown to a maximum height of 28.0 ft above the blanket, how long does she spend in the air? (b) Is the amount of time the person is above a height of 14.0 ft more than, less than, or equal to the amount of time the person is below a height of 14.0 ft? Explain. (c) Verify your answer to part (b) with a calculation.

106. Referring to Conceptual Checkpoint 2-5, find the separation between the rocks at (a) t = 1.0 s, (b) t = 2.0 s, and (c) t = 3.0 s, where time is measured from the instant the second rock is dropped. (d) Verify that the separation increases linearly with time.

107. A glaucous-winged gull, ascending straight upward at 5.20 m/s, drops a she1l when it is 12.5 m above the ground. (a) What are the magnitude and direction of the shell's acceleration just after it is released (b) Find the maximum height above the ground reached by the shell. (c) How long does it

take for the shell to reach the ground? (d) What is the speed of the shell at this time?

108. A doctor preparing to give a patient an injection, squirts a small amount of liquid straight upward from a syringe. If the liquid emerges with a speed of 1.5 m/s, (a) how long does it take for it to return to the level of the syringe? (b) What is the maximum height of the liquid above the syringe?

109. A hot-air balloon has just lifted off and is rising at the constant rate of 2.0 m/s. Suddenly one of the passengers realizes she has left her camera on the ground. A friend picks it up and tosses it straight upward with an initial speed of 13 m/s. If the passenger is 2.5 m above her friend when the camera is tossed, how high is she when the camera reaches her?

110. In the previous problem, what is the minimum initial speed of the camera if it is to just reach the passenger? (Hint: When the camera is thrown with its minimum speed, its speed on reaching the passenger is the same as the speed of the passenger.)

111. Old Faithful. Watching Old Faithful erupt, you notice that it takes a time t for water to emerge from the base of the geyser and reach its maximum height. (a) What is the height of the geyser, and (b) what is the initial speed of the water? Evaluate your expressions for (c) the height and (d) the initial speed for a measured time of 1.65 s.

112. A ball is thrown upward with an initial speed v0. When it reaches the top of its flight, at a height h, a second ball is thrown upward with the same initial velocity. (a) Sketch an x-versus-t plot for each ball. (b) From your graph, decide whether the balls cross paths at ½ h, above ½ h, or below ½ h. (c) Find the height where the paths cross.

113. Weights are tied to each end of a 20.0-cm string. You hold one weight in your hand and let the other hang vertically a height h above the floor. When you release the weight in your hand, the two weights strike the ground one after the other with audible thuds. Find the value of h for which the time between release and the first thud is equal to the time between the first thud and the second thud.

114. A ball, dropped from rest, covers three-quarters of the distance to the ground in the last second of its fall. (a) From what height was the ball dropped? (b) What was the total time of fall?

115. A stalactite on the roof of a cave drips water at a steady rate to a pool 4.0 m below. As one drop of water hits the pool, a second drop is in the air, and a third is just detaching from the stalactite. (a) What are the position and velocity of the second drop when the first drop hits the pool? (b) How many drops per minute fall into the pool?

116. You drop a ski glove from a height h onto fresh snow, and it sinks to a depth d before coming to rest. (a) In terms of g and h, what is the speed of the glove when it reaches the snow? (b) What are the magnitude and direction of the glove's acceleration as it moves through the snow, assuming it to be constant? Give your answer in terms of g, h, and d.

117. To find the height of an overhead power line, you throw a ball straight upward. The ball passes the line on the way up after 0.75 s, and passes it again on the way down 1.5 s after it was tossed. What are tire height of the power line and the initial speed of the ball?

118. Suppose the first rock in Conceptual Checkpoint 2-5 drops through a height h before the second rock is released from rest. Show that the separation between the rocks, S, is given by the following expression:

S = h + √(2gh)t

In this result, the time t is measured from the time the second rock is dropped.

119. An arrow is fired with a speed of 20.0 m/s at a block of Styrofoam resting on a smooth surface. The arrow penetrates a certain distance into the block before coming to rest relative to it. During this process the arrow's deceleration has a magnitude of 1550 m/s2 and the block's acceleration has a magnitude of 450 m/s2. (a) How long does it take for the arrow, to stop moving with respect to the block? (b) What is the common speed of the arrow and block when this happens? (c) How far into the block does the arrow penetrate?

120. Sitting in a second-story apartment, a physicist notices a ball moving straight upward just outside her window. The ball is visible for 0.25 s as it moves a distance of 1.05 m from the bottom to the top of the window. (a) How long does it take before the ball reappears? (b) What is the greatest height of the ball above the top of the window?

**Chapter 3**

1. Suppose that each component of a certain vector is doubled. (a) By what multiplicative factor does the magnitude of the vector change? (b) By what multiplicative factor does the direction angle of the vector change?

2. Rank the vectors in Figure 3-31 in order of increasing magnitude.

3. Rank the vectors in Figure 3-31 in order of increasing value of their x component.

4. Rank the vectors in Figure 3-31 in order of increasing value of their y component.

5. The press box at a baseball park is 32.0 ft above the ground. A reporter in the press box looks at an angle of 15.0° below the horizontal to see second base. What is the horizontal distance from the press box to second base?

6. You are driving up a long, inclined road. After 1.2 miles you notice that signs along the roadside indicate that your elevation has increased by 530 ft. (a) What is the angle of the road above the horizontal? (b) How far do you have to drive to gain an additional 150 ft of elevation?

7. A One-Percent Grade. A road that rises 1 ft for every 100 ft traveled horizontally is said to have a 1 % grade. Portions of the Lewiston grade, near Lewiston, Idaho, have a 6 % grade. At what angle is this road inclined above the horizontal?

8. Find the x and y components of a position vector r of magnitude r = 75 cm, if its angle relative to the x axis is (a) 35.0° and (b) 65.0°.

9. A baseball “diamond” (Figure 3-32) is a square with sides 90 ft in length. If the positive x axis points from home plate to first base, and the positive y axis points from home plate to third base, find the displacement vector of a base runner who has just hit (a) a double, (b) a triple, or (c) a home run.

10. A lighthouse that rises 49 ft above the surface of the water sits on a rocky cliff that extends 19 ft from its base, as shown in Figure 3-33. A sailor on the deck of a ship sights the top of the lighthouse at an angle of 30.0° above the horizontal. If the sailor's eye level is 14 ft above the water, how far is the ship from the rocks?

11. H2O A water molecule is shown schematically in Figure 3-34. The distance from the center of the oxygen atom to the center of a hydrogen atom is 0.96 Å, and the angle between the hydrogen atoms is 104.5°. Find the center-to-center distance between the hydrogen atoms. (1 Å = 10-10 m.)

12. The x and y components of a vector r are rx = 14 m and ry = -9.5 m, respectively. Find (a) the direction and (b) the magnitude of the vector r. (c) If both rx and ry are doubled, how do you answers to parts (a) and (b) change?

13. The Longitude Problem. In 1755, John Harrison (1693-1776) completed his fourth precision chronometer, the H4, which eventually won the celebrated Longitude Prize. (For the human drama behind the Longitude Prize, see Longitude, by Dava Sobel.) When the minute hand of the H4 indicated 10 minutes past the hour, it extended 3.0 in the horizontal direction. (a) How long was the H4’s minute hand? (b) At 10 minutes past the hour, was the extension of the minute hand in the vertical direction more than, less than, or equal to 3.0 cm? Explain. (c) Calculate the vertical extension of the minute hand at 10 minutes past the hour.

14. You drive a car 680 ft to the east, then 340 ft to the north. (a) What is the magnitude of your displacement? (b) Using a sketch, estimate the direction of your displacement. (c) Verify your estimate in part (b) with a numerical calculation of the direction.

15. Vector A has a magnitude of 50 units and points in the positive x direction. A second vector, B, has a magnitude of 120 units and points at an angle of 70° below the x axis. Which vector has (a) the greater x component, and (b) the greater y component?

16. A treasure map directs you to start at a palm tree and walk due north for 15.0 m. You are then to turn 90° and walk 22.0 m; then turn 90° again and walk 5.00 m. Give the distance from the palm tree, and the direction relative to north, for each of the four possible locations of the treasure.

17. A whale comes to the surface to breathe and then dives at an angle 20.0° below the horizontal (Figure 3-35). If the whale continues in a straight line for 150 m, (a) how deep is it, and (b) how far has it travelled horizontally?

18. Consider the vectors A and B shown in Figure 3-36. Which of the other four vectors in the figure (C, D, E, and F) best represent the direction of (a) A + B, (b) A – B, and (c) B – A?

19. Refer to Figure 3-36 for the following questions: (a) Is the magnitude of A + D greater than, less than, or equal to the magnitude of A + E? (b) Is the magnitude of A + E greater than, less than, or equal to the magnitude of A + F?

20. A vector A has a magnitude of 40.0 m and points in a direction 20.0° below the positive x axis. A second vector, B, has a magnitude of 75.0 m and points in a direction 50.0° above the positive x axis. (a) Sketch the vectors A, B, and C = A + B. (b) Using the component method of vector addition, find the magnitude and direction of the vector C.

21. An air traffic controller observes two airplanes approaching the airport. The displacement from the control tower to plane 1 is given by the vector A, which has a magnitude of 220 km and points in a direction 32° north of west. The displacement from the control tower to plane 2 is given by the vector B, which has a magnitude of 140 km and points 65° east of north. (a) Sketch the vectors A, -B, and D = A - B. Notice that D is the displacement from plane 2 to plane 1. (b) Find the magnitude and direction of the vector D.

22. The initial velocity of the car, vi, is 45 km/h in the positive x direction. The final velocity of the car, vf, is 66 km/h in a direction that points 75° above the positive x axis. (a) Sketch the vectors -vi, vf, and Δv = Af - Ai. (b) Find the magnitude and direction of the change in velocity, Δv.

23. Vector A points in the positive x direction and has a magnitude of 75 m. The vector C = A + B points in the positive y direction and has a magnitude of 95 m. (a) Sketch A, B, and C. (b) Estimate the magnitude and direction of the vector B. (c) Verify your estimate in part (b) with a numerical calculation.

24. Vector A points in the negative x direction and has a magnitude of 22 units. The vector B points in the positive y direction. (a) Find the magnitude of B if A + B has a magnitude of 37 units. (b) Sketch A and B.

25. Vector A points in the negative y direction and has a magnitude of 5 units. Vector B has twice the magnitude and points in the positive x direction. Find the direction and magnitude of (a) A + B, (b) A – B, and (c) B – A.

26. A basketball player runs down the court, following the path indicated by the vectors A, B, and C in Figure 3-37. The magnitudes of these three vectors are A = 10.0 m, B = 20.0 m, and C = 7.0 m. Find the magnitude and direction of the net displacement of the player using (a) the graphical method and (b) the component method of vector addition. Compare your results.

27. A particle undergoes a displacement Δr of magnitude 54 m in a direction 42° below the x axis. Express Δr in terms of the unit vectors x and y.

28. A vector has a magnitude of 3.50 m and points in a direction that is 145° counterclockwise from the x axis. Find the x and y components of this vector.

29. A vector A has a length of 6.1 m and points in the negative x direction. Find (a) the x component and (b) the magnitude of the vector -3.7A.

30. The vector -5.2A has a magnitude of 34 m and points in the positive x direction. Find (a) the x component and (b) the magnitude of the vector A.

36. The blue curves shown in Figure 3-39 display the constant speed motion of two different particles in the x-y plane. For each of the eight vectors in Figure 3-39, state whether it is (a) a position vector, (b) a velocity vector, or (c) an acceleration vector for the particles.

37. Moving the Knight. Two of the allowed chess moves for a knight are shown in Figure3-40. (a) Is the magnitude of displacement 1 greater than, less than, or equal to the magnitude of displacement 2? Explain. (b) Find the magnitude and direction of the knight's displacement for each of the two moves. Assume that the checkerboard squares are 3.5 cm on a side.

38. In its daily prowl of the neighborhood, a cat makes a displacement of 120 m due north, followed by a 72-m displacement due west. (a) Find the magnitude and direction of the displacement required for the cat to return home. (b) If, instead, the cat had first prowled 72 m west and then 120 m north, how would this affect the displacement needed to bring it home? Explain.

39. If the cat in Problem 38 takes 45 minutes to complete the 120-m displacement and 17 minutes to complete the 72-m displacement, what are the magnitude and direction of its average velocity during this 62-minute period of time?

40. What are the direction and magnitude of your total displacement if you have traveled due west with a speed of 27 m/s for 125 s, then due south at 14 m/s for 66 s?

41. You drive a car 1500 ft to the east, then 2500 ft to the north. If the trip took 3.0 minutes, what were the direction and magnitude of your average velocity?

42. A jogger runs with a speed of 3.25 m/s in a direction 30.0° above the x axis. (a) Find the x and y components of the jogger’s velocity. (b) How will the velocity components found in part (a) change if the jogger’s speed is halved?

43. You throw a ball upward with an initial speed of 4.5 m/s. When it returns to your hand 0.92 s later, it has the same speed in the downward direction (assuming air resistance can be ignored). What was the average acceleration vector of the ball?

44. A skateboarder rolls from rest down an inclined ramp that is 15.0 m long and inclined above the horizontal at an angle of θ = 20.0°. When she reaches the bottom of the ramp 3.00 s later her speed is 1.0 m/s. Show that the average acceleration of the skateboarder is g sinθ, where g = 9.81 m/s2.

45. Consider a skateboarder who starts from rest at the top of a ramp that is inclined at an angle of 17.5° to the horizontal. Assuming that the skateboarder’s acceleration is g sin17.5°, find his speed when he reaches the bottom of the ramp in 3.25 s.

46. The position of the Moon. Relative to the center of the Earth, the position of the Moon can be approximated by

r = (3.84 × 108 m){cos[(2.46 × 10-6 radians/s)t]x + sin[(2.46 × 10-6 radians/s)t]y}

where t is measured in seconds. (a) Find the magnitude and direction of the Moon’s average velocity between t = 0 and t = 7.38 (b) Is the instantaneous speed of the Moon greater than, less than, or the same as the average speed found in part (a)? Explain.

47. The velocity of the Moon. The velocity of the Moon relative to the center of the Earth can be approximated by

v = (945 m/s ){-sin[(2.46 × 10-6 radians/s)t]x + cos[(2.46 × 10-6 radians/s)t]y}

where t is measured in seconds. To approximate the instantaneous acceleration of the Moon at t = 0, calculate the magnitude and direction of the average acceleration between the times (a) t = 0 and t = 0.001 days and (b) t = 0 and t = 0.0100.

48. The accompanying photo shows a KC-10A Extender using a boom to refuel an aircraft in flight. If the velocity of the KC-10A is 125 m/s due east relative to the ground, what is the velocity of the aircraft being refueled relative to (a) the ground, and (b) the KC-10A?

49. As an airplane taxies on the runway with a speed of 16.5 m/s, a flight attendant walks toward the tail of the plane with a speed of 1.22 m/s. What is the flight attendant's speed relative to the ground?

50. Referring to part (a) of Example 3-2, find the time it taes for the boat to reach the opposite shore if the river is 35 m wide.

51. As you hurry to catch your flight at the local airport, you encounter a moving walkway that is 85 m long and has a speed of 2.2 m/s relative to the ground. If it takes you 68 s to cover 85 m when walking on the ground, how long will it take you to cover the same distance on the walkway? Assume that you walk with the same speed on the walkway as you do on the ground.

52. In problem 51, how long would it take you to cover the 85-m length of the walkway if, once you get on the walkway, you immediately turn around and start walking in the opposite direction with a speed of 1.3 m/s relative to the walkway?

53. The pilot of an airplane wishes to fly due north, but there is a 65 km/h wind blowing toward the east. (a) In what direction should the pilot head her plane if its speed relative to the air is 340 km/h? (b) Draw a vector diagram that illustrates your result in part (a). (c) If the pilot decreases the air speed of the plane, but still wants to head due north, should the angle found in part (a) be increased or decreased?

54. A passenger walks from one side of a ferry to the other as it approaches a dock. If the passenger’s velocity is 1.50 m/s due north relative to the ferry, and 4.50 m/s at an angle of 30.0° west of north relative to the water, what are the direction and magnitude of the ferry’s velocity relative to the water?

55. You are riding on a Jet Ski at an angle of 35° upstream on a river flowing with a speed of 2.8 m/s. If your velocity relative to the ground is 9.5 m/s at an angle of 20.0° upstream, what is the speed of the Jet Ski relative to the water?

56. In problem 55, suppose the Jet Ski is moving at a speed of 12 m/s relative to the water. (a) At what angle must you point the Jet Ski if your velocity relative to the ground is to be perpendicular to the shore of the river? (b) If you increase the speed of the Jet Ski relative to the water, does the angle in part (a) increase, decrease, or stay the same? Explain.

57. Two people take identical Jet Skis across a river, traveling at the same speed relative to the water. Jet Ski A heads directly across the river and is carried downstream by the current before reaching the opposite shore. Jet Ski B travels in a direction that is 35° upstream, and arrives at the opposite shore directly across from the starting point. (a) Which Jet Ski reaches the opposite shore in the least amount of time? (b) Confirm your answer to part (a) by finding the ratio of the time it takes for the two Jet Skis to cross the river.

60. You slide a box up a loading ramp that is 10.0 ft long. At the top of the ramp the box has risen a height of 3.00 ft. What is the angle of the ramp above the horizontal?

61. Find the direction and magnitude of the vector 2A + B, where A = (12.1 m)x and B = (-32.2 m)y.

62. The components of a vector A satisfy Ax < 0 and Ay < 0. Is the direction angle of A between 0° and 90°, between 90° and 180°, between 180° and 270°, or between 270° and 360°?

63. The components of a vector B satisfy Bx > 0 and By < 0. Is the direction angle of B between 0° and 90°, between 90° and 180°, between 180° and 270°, or between 270° and 360°?

64. It is given that A – B = (-51.4 m)x, C = (62.2 m)x, and A + B + C = (13.8 m)x. Find the vectors A and B.

65. Two students perform an experiment with a train and a ball. Michelle rides on a flatcar pulled at 8.35 m/s by a train on a straight, horizontal track; Gary stands at rest, on the ground near the tracks. When Michelle throws the ball with an initial angle of 65.0° above horizontal, from her point of view, Gary sees the ball rise straight up and back down from a fixed point on the ground. (a) Did Michelle throw the ball toward the front of the train or toward the rear of the train? Explain. (b) What was the initial speed of Michelle's throw? (c) What was the initial speed of the ball as seen by Gary?

66. An off-roader explores the open desert in her Hummer. First she drives 25° west of north with a speed of 6.5 km/h for 15 minutes, then due east with a speed of 12 km/h for 7.5 minutes. She completes the final leg of her trip in 22 minutes. What are the direction and speed of travel on the final leg?

67. Find the x, y, and z components of the vector A shown in Figure 3-41, given that A = 65 m.

68. A football is thrown horizontally with an initial velocity of (16.6 m/s)x. Ignoring air resistance, the average acceleration of the football over any period of time is (-9.81 m/s2)y. (a) Find the velocity vector of the ball 1.75 s after it thrown. (b) Find the magnitude and direction of the velocity at this time.

69. As a function of time, the velocity of the football described in Problem 68 can be written as

v = (16.6 m/s)x - [(9.81 m/s2)t]y.

Calculate the average acceleration vector of the football for the time periods (a) t = 0 to t = 1.00 s, (b) t = 0 to t = 2.50 s, and (c) ) t = 0 to t = 5.00 s.

70. Two airplanes taxi as they approach the terminal. Plane 1 taxies with a speed of 12 m/s due north. Plane 2 taxies with a speed of 7.5 m/s in a direction 20° north of west. (a) What are the direction and magnitude of the velocity of plane 1 relative to plane 2? (b) What are the direction and magnitude of the velocity of plane 2 relative to plane 1?

71. A shopper at the supermarket follows the path indicated by vectors A, B, C, and D in Figure 3-42. Given that the vectors have the magnitudes A = 51 ft, B = 45 ft, C = 35 ft, and D = 13 ft, find the total displacement of the shopper using (a) the graphical method and (b) the component method of vector addition. Give the direction of displacement relative to the direction of vector A.

72. Initially, a particle is moving at 4.10 m/s at an angle of 33.5° above the horizontal. Two seconds later, its velocity is 6.05 m/s at an angle of 59.0° below the horizontal. What was the particle's average acceleration during these 2.00 seconds?

73. A passenger on a stopped bus notices that rain is falling vertically just outside the window. When the bus moves with constant velocity, the passenger observes that the falling raindrops are now making an angle of 15° with respect to the vertical. (a) What is the ratio of the speed of the raindrops to the speed of the bus? (b) Find the speed of the raindrops, given that the bus is moving with a speed of 18 m/s.

74. A Big Clock. The clock that rings the bell known as Big Ben has an hour hand that is 9.0 feet long and a minute hand that is 14 feet long, where the distance is measured from the center of the clock to the tip of each hand. What is the tip-to-tip distance between these two hands when the clock reads 12 minutes after four o'clock?

75. Suppose we orient the x axis of a two-dimensional coordinate system along the beach at Waikiki. Waves approaching the beach have a velocity relative to the shore given by v = (1.3 m/s)y. Surfers move more rapidly than the waves, but at an angle to the beach. The angle is chosen so that the surfers approach the shore with the same speed as the waves. (a) If a surfer has a speed of 7.2 m/s relative to the water, what is her direction of motion relative to the positive x axis? (b) What is the surfer’s velocity relative to the wave? (c) If the surfer’s speed is increased, will the angle in part (a) increase or decrease? Explain.

76. Referring to Example 3-2, (a) what heading must the boat have if it is to land directly across the river from its starting point? (b) How much time is required for this trip if the river is 25.0 m wide? (c) Suppose the speed of the boat is increased, but it is still desired to land directly across from the starting point. Should the boat's heading be more upstream, downstream, or the same as in part (a)? Explain

77. Vector A points in the negative x direction. Vector B points at an angle of 30.0° above the positive x axis. Vector C has a magnitude of 15 m and points in a direction 40.0° below the positive x axis. Given that A + B + C = 0, find the magnitudes of A and B.

78. As two boats approach the marina, the velocity of boat 1 relative to boat 2 is 2.15 m/s in a direction 47.0° east of north. If boat 1 has a velocity that is 0.775 m/s due north, what is the velocity (magnitude and direction) of boat 2?

79. What speed must the dragonfly have if the line of sight, which is parallel to the x axis initially, is to remain parallel to the x axis?

A. 0.562 m/s B. 0.664 m/s C. 1.00 m/s D. 1.13 m/s

80. Suppose the dragonfly now approaches its prey along a path with θ > 48.5°, but it still keeps the line of sight parallel to the x axis. Is the speed of the dragonfly in this new case greater than, less than, or equal to its speed in problem 79?

81. What is the correct “motion camouflage” speed of approach for dragonfly pursuing its prey at an angle θ = 68.5°?

A. 0.295 m/s B. 0698 m/s C. 0.806 m/s D. 2.05 m/s

82. If the dragonfly approaches its prey with a speed of 0.950 m/s, what angle θ is required to maintain a constant line of sight parallel to the x axis?

A. 37.9° B. 38.3° C. 51.7° D. 52.1°

83. Suppose the speed of the boat relative to the water is 7.0 m/s. (a) At what angle to the x axis must the boat be headed if it is to land directly across the river from its starting point? (b) If the speed of the boat relative to the water is increased, will the angle needed to go directly across the river increase, decrease, or stay the same? Explain.

84. Suppose the boat has a speed of 6.7 m/s relative to the water, and that the dock on the opposite shore of the river is at the location x = 55 m and y = 28 m relative to the starting point of the boat. (a) At what angle relative to the x axis must the boat be pointed in order to reach the other dock? (b) With the angle found in part (a), what is the speed of the boat relative to the ground?

**Chapter 4**

1. As you walk briskly down the street, you toss a small ball into the air. (a) If you want the ball to land in your hand when it comes back down, should you toss the ball straight upward, in a forward direction, or in a backward direction, relative to you body. (b) Choose the best explanation from among the following: I. If the ball is thrown straight up you will leave it behind. II. You have to throw the ball in the direction you are walking. III. The ball moves in the forward direction with you walking speed at all times.

2. A sailboat runs before the wind with a constant speed of 4.2 m/s in a direction of 32° north of west. How far (a) west and (b) north has the sailboat traveled in 25 min?

3. As you walk to class with a constant speed of 1.75 m/s, you are moving in a direction that is 18.0° north of east. How much time does it take to change your displacement by (a) 20.0 m east or (b) 30.0 m north?

4. Starting from rest, a car accelerates at 2.0 m/s2 up a hill that is inclined 5.5° above the horizontal. How far (a) horizontally and (b) vertically has the car traveled in 12 s?

5. A particle passes through the origin with a velocity of (6.2 m/s)y. If the particle's acceleration is (-4.2 m/s2)x, (a) what are its x and y positions after 5.0 s? (b) what are vx and vy at this time? (c) Does the speed of this particle increase with time, decrease with time, or increase and then decrease? Explain.

6. An electron in a cathode-ray tube is traveling horizontally at 2.10 × 109 cm/s when deflection plates give it an upward acceleration of 5.30 × 1017 cm/s2. (a) How long does it take for the electron to cover a horizontal distance of 6.20 cm? (b) What is its vertical displacement during this time?

7. Two canoeists start paddling at the same time and head toward a small island in a lake, as shown in Figure 4-12. Canoeist 1 paddles with a speed of1.35 m/s at an angle of 45° north of east. Canoeist 2 starts on the opposite shore of the lake, a distance of 1.5 km due east of canoeist 1. (a) In what direction relative to north must canoeist 2 paddle to reach the island? (b) What speed must canoeist 2 have if the two canoes are to arrive at the island at the same time?

8. Two divers run horizontally off the edge of a low cliff. Diver 2 runs with twice the speed of diver 1. (a) When the divers hit the water, is the horizontal distance covered by diver 2 twice as much, four times as much, or equal to the horizontal distance covered by diver 1? (b) Choose the best explanation from among the following:

I. The drop time is the same for both divers.

II. Drop distance depends on t2.

III. All divers in free fall cover the same distance.

9. Two youngsters dive off an overhang into a lake. Diver 1 drops straight down, and diver 2 runs off the cliff with an initial horizontal speed v0. (a) Is the splashdown speed of diver 2 greater than, less than, or equal to the splashdown speed of diver 1? (b) Choose the best explanation from among the following:

I. Both divers are in free fall, and hence they will have the same splashdown speed.

II. The divers have the same vertical speed at splashdown, but diver 2 has the greater horizontal speed.

III. The diver who drops straight down gains more speed than the one who moves horizontally.

10. An archer shoots an arrow horizontally at a target 15 m away. The arrow is aimed directly at the center of the target, but it hits 52 cm lower. What was the initial speed of the arrow?

11. Victoria Falls. The great, gray-green, greasy Zambezi River flows over Victoria Falls in south central Africa. The falls are approximately 108 m high. If the river is flowing horizontally at 3.60 m/s just before going over the falls, what is the speed of the water when it hits the bottom? Assume the water is in freefall as it drops.

12. A diver runs horizontally off the end of a diving board with an initial speed of 1.85 m/s. If the diving board is 3.00 m above the water, what is the diver's speed just before she enters the water?

13. An astronaut on the planet Zircon tosses a rock horizontally with a speed of 6.95 m/s. The rock falls through a vertical distance of 1.40 m and lands horizontal distance of 8.75 m from the astronaut. What is the acceleration of gravity on Zircon?

14. Pitcher's Mounds. Pitcher's mounds are raised to compensate for the vertical drop of the ball as it travels a horizontal distance of 18 m to the catcher. (a) If a pitch is thrown horizontally with an initial speed of 32 m/s, how far does it drop by the time it reaches the catcher? (b) If the speed of the pitch is increased, does the drop distance increase, decrease, or stay the same? Explain. (c) If this baseball game were to be played on the Moon, would the drop distance increase, decrease, or stay the same? Explain.

15. Playing shortstop, you pick up a ground ball and throw it to second base. The ball is thrown horizontally, with a speed of 22 m/s, directly toward point A (Figure 4-13).When the ball reaches the second baseman 0.45 s later, it is caught at point B. (a) How far were you from the second baseman? (b) What is the distance of vertical drop, AB?

16. A crow is flying horizontally with a constant speed of 2.70 m/s when it releases a clam from its beak (Figure 4-14). The clam lands on the rocky beach 2.10 s later. Just before the clam lands, what is (a) its horizontal component of velocity, and (b) its vertical component of velocity? (c) How would your answers to parts (a) and (b) change if the speed of the crow were increased? Explain.

17. A mountain climber jumps a 2.8-m-wide crevasse by leaping horizontally with a speed of 7.8 m/s. (a) If the climber's direction of motion on landing is -45°, what is the height difference between the two sides of the crevasse? (b) Where does the climber land?

18. A white-crowned sparrow flying horizontally with a speed of 1.80 m/s folds its wings and begins to drop in free fall. (a) How far does the sparrow fall after traveling a horizontal distance of 0.500 m? (b) If the sparrow's initial speed is increased, does the distance of fall increase, decrease, or stay the same?

19. Pumpkin Toss. In Denver, children bring their old jack-o-lanterns to the top of a tower and compete for accuracy in hitting a target on the ground (Figure 4-15). Suppose that the tower is 9.0 m high and that the bull’s-eye is a horizontal distance of 3.5 m from the launch point. If the pumpkin is thrown horizontally, what is the launch speed needed to hit the bull’s-eye?

20. If, in the previous problem, a jack-o-lantern is given an initial horizontal speed of 3.3 m/s, what are the direction and magnitude of its velocity (a) 0.75 s later, and (b) just before it lands?

21. Fairgoers ride a Ferris wheel with a radius of 5.00 m (Figure 4-16). The wheel completes one revolution every 32.0 s. (a) What is the average speed of a rider on this Ferris wheel? (b) If a rider accidentally drops a stuffed animal at the top of the wheel, where does it land relative to the base of the ride?

22. A swimmer runs horizontally off a diving board with a speed of 3.32 m/s and hits the water a horizontal distance of 1.78 m from the end of the board. (a) How high above the water was the diving board? (b) If the swimmer runs off the board with a reduced speed, does it take more, less or the same time to reach the water?

23. Baseball and the Washington Monument. On August 25, 1894, Chicago catcher William Schriver caught a baseball thrown from the top of the Washington Monument (555 ft, 898 steps). (a) If the ball was thrown horizontally with a speed of 5.00 m/s, where did it land? (b) What were the ball's speed and direction of motion when caught?

24. A basketball is thrown horizontally with an initial speed of 4.20 m/s (Figure 4-17). A straight line drawn from the release point to the landing point makes an angle of 30.0° with the horizontal. What was the release height?

25. A ball rolls off a table and falls 0.75 m to the floor, landing with a speed of 4.0 m/s. (a) What is the acceleration of the ball just before it strikes the ground? (b) What was the initial speed of the ball? (c) What initial speed must the ball have if it is to land with a speed of 5.0 m/s?

26. A certain projectile is launched with an initial speed v0. At its highest point its speed is 1/2v0. What was the launch angle of the projectile?

A. 300 B. 450 C. 600 D. 750

27. Three projectiles (A, B, and C) are launched with the same initial speed but with different launch angles, as shown in Figure 4-18. Rank the projectiles in order of increasing (a) horizontal component of initial velocity and (b) time of flight. Indicate ties where appropriate.

28. Three projectiles (A, B, and C) are launched with different initial speeds so that they reach the same maximum height, as shown in Figure 4-19. Rank the projectiles in order of increasing (a) initial speed and (b) time of flight. Indicate ties where appropriate.

29. A second baseman tosses the ball to the first baseman, who catches it at the same level from which it was thrown. The throw is made with an initial speed of 18.0 m/s at an angle of 37.5° above the horizontal. (a) What is the horizontal component of the ball's velocity just before it is caught? (b) How long is the ball in the air?

30. Referring to the previous problem, what are the y component of the balls velocity and its direction of motion just before it is caught?

31. A cork shoots out of a champagne bottle at an angle of 35.0° above the horizontal. If the cork travels a horizontal distance of 1.30 m in 1.25 s, what was its initial speed?

32. A soccer ball is kicked with a speed of 9.85 m/s at an angle of 35.0° above the horizontal? If the ball lands at the same level from which it was kicked, how long was it in the air?

33. In a game of basketball, a forward makes a bounce pass to the center. The ball is thrown with an initial speed of 4.3 m/s at an angle of 15° below the horizontal. It is released 0.80 m above the floor. What horizontal distance does the ball cover before bouncing?

34. Repeat the previous problem for a bounce pass in which the ball is thrown 15° above the horizontal.

35. Snowballs are thrown with a speed of 13 m/s from a roof 7.0 m above the ground. Snowball A is thrown straight downward; snowball B is thrown in a direction 25° above the horizontal. (a) Is the landing speed of snowball A greater than, less than, or the same as the landing speed of snowball B? Explain. (b) Verify your answer to part (a) by calculating the landing speed of both snowballs.

36. In the previous problem, find the direction of motion of the two snowballs just before they land.

37. A golfer gives a ball a maximum initial speed of 34.4 m/s. (a) What is the longest possible hole-in-one for this golfer? Neglect any distance the ball might roll on the green and assume that the tee and the green are at the same level. (b) What is the minimum speed of the ball during this hole-in-one shot?

38. What is the highest tree the ball in the previous problem could clear on its way to the longest possible hole-in-one?

39. The “hang time” of a punt is measured to be 4.50 s. If the ball was kicked at an angle of 63.0° above the horizontal and was caught at the same level from which it was kicked, what was its initial speed?

40. In a friendly game of handball l, you hit the ball essentially at ground level and send it toward the wall with a speed of 18 m/s at an angle of 32° above the horizontal. (a) How long does it take for the ball to reach the wall if it is 3.8 m away? (b) How high is the ball when it hits the wall?

41. In the previous problem, (a) what are the magnitude and direction of the balls velocity when it strikes the wall? (b) Has the ball reached the highest point of its trajectory at this time? Explain.

42. A passenger on the Ferris wheel described in Problem 21 drops his keys when he is on the way up and at the 10 o’clock position. Where do the keys land relative to the base of the ride?

43. On a hot summer day, a young girl swings on a rope above the local swimming hole (Figure 4-20). When she lets go of the rope her initial velocity is 2.25 m/s at an angle of 35.0° above the horizontal. If she is in flight for 0.616 s, how high above the water was she when she let go of the rope?

44. A certain projectile is launched with an initial speed v0. At its highest point its speed is v0/4. What was the launch angle?

45. Punkin Chunkin. In Sussex County, Delaware, a post-Halloween tradition is "Punkin Chunkin," in which contestants build cannons, catapults, trebuchets, and other devices to launch pumpkins and compete for the greatest distance. Though hard to believe, pumpkins have been projected a distance of 4086 feet in this contest. What is the minimum initial speed needed for such a shot?

46. A dolphin jumps with an initial velocity of 12.0 m/s at an angle of 40.0° above the horizontal. The dolphin passes through the center of the hoop before returning to the water. If the dolphin is moving horizontally when it goes through the hoop, how high above the water is the center of the hoop?

47. A player passes a basketball to another player who catches it at the same level from which it was thrown. The initial speed of the ball is 7.1 m/s, and it travels a distance of 4.6 m. What were (a) the initial direction of the ball and (b) its time of flight?

48. A golf ball is struck with a five iron on level ground. It lands 92.2 m away 4.30 s later. What were (a) the direction and (b) the magnitude of the initial velocity?

49. A Record Toss. Babe Didrikson holds the world record for the longest baseball throw (296 ft) by a woman. For the following questions, assume that the ball was thrown at an angle of 45.0° above the horizontal, that it traveled a horizontal distance of 296 ft, and that it was caught at the same level from which it was thrown. (a) What was the ball's initial speed? (b) How long was the ball in the air?

50. In the photograph to the left on page 87, suppose the cart that launches the ball is 11 cm high. Estimate (a) the launch speed of the ball and (b) the time interval between successive stroboscopic exposures.

51. You throw a ball into the air with an initial speed of 10 m/s at an angle of 60° above the horizontal. The ball returns to the level from which it was thrown in the time T. (a) Referring to Figure 4-21, which of the plots (A, B, or C) best represents the speed of the ball as a function of time? (b) Choose the best explanation from among the following:

I. Gravity causes the ball’s speed to increase during its flight.

II. The ball has zero speed at its highest point.

III. The ball’s speed decreases during its flight, but it doesn’t go to zero.

52. Volcanoes on Io. Astronomers have discovered several volcanoes on Io, a moon of Jupiter. One of them, named Loki, ejects lava to a maximum height of 2.00 × 105 m. (a) What is the initial speed of the lava? (The acceleration of gravity on Io is 1.80 m/s2) (b) If this volcano were on Earth, would the maximum height of the ejected lava be greater than, less than, or the same as on Io? Explain.

53. A soccer ball is kicked with an initial speed of 10.2 m/s in a direction 25.0° above the horizontal. Find the magnitude and direction of its velocity (a) 0.250 s and (b) 0.500 s after being kicked. (c) Is the ball at its greatest height before or after 0.500 s? Explain.

54. A second soccer is kicked with the same initial velocity as in Problem 53. After 0.750 s it is at its highest point. What was its initial direction of motion?

55. A golfer tees off on level ground, giving the ball an initial speed of 46.5 m/s and an initial direction of 37.5° above the horizontal. (a) How far from the golfer does the ball land? (b) The next golfer in the group hits a ball with the same initial speed but at an angle above the horizontal that is greater than 45.0°. If the second ball travels the same horizontal distance as the first ball, what was its initial direction of motion? Explain.

56. One of the most popular events at Highland games is the hay toss, where competitors use a pitchfork to throw a bale of hay over a raised bar. Suppose the initial velocity of a bale of hay is v = (1.12 m/s)x + (8.85 m/s)y. (a) After what minimum time is its speed equal to 5.00 m/s? (b) How long after the hay is tossed is it moving in a direction that is 45.0° below the horizontal? (c) if the bale of hay is tossed with the same initial speed, only this time straight upward, will its time in the air increase, decrease, or stay the same? Explain.

57. Child 1 throws a snowball horizontally from the top of a roof; child 2 throws a snowball straight down. Once in flight, is the acceleration of snowball 2 greater than, less than, or equal to the acceleration of snowball 1?

58. The penguin to the left in the accompanying photo is about to land on an ice floe. Just before it lands, is its speed greater than, less than, or equal to its speed when it left the water.

59. A person flips a coin into the air and it lands on the ground a few feet away. (a) If a person were to perform an identical coin flip on an elevator rising with constant speed, would the coin’s time of flight be greater than, less than, or equal to its time of flight when the person was at rest? (b) Choose the best explanation from among the following:

I. The floor of the elevator is moving upward, and hence it catches up with the coin in mid flight.

II. The coin has the same upward speed as the elevator when it is tossed, and the elevator’s speed doesn’t change during the coin’s flight.

III. The coin starts off with a greater upward speed because of the elevator, and hence it reaches a greater height.

60. Suppose the elevator in the previous problem is rising with a constant upward acceleration, rather than constant velocity. (a) In this case, would the coin’s time of flight be greater than, less than, or equal to its time of flight when the person was at rest? (b) Choose the best explanation from among the following:

I. The coin has the same acceleration once it is tossed, whether the elevator accelerates or not.

II. The elevator’s upward speed increases during the coin’s flight, and hence it catches up with the coin at a greater height than before.

III. The coin’s downward acceleration is less than before because the elevator’s upward acceleration partially cancels it.

61. A train moving with constant velocity travels 170 m north in 12 s and undetermined distance to the west. The speed of the train is 32 m/s. (a) Find the direction of the train’s motion relative to north. (b) How far west has the train traveled in this time?

62. Referring to Example 4-2, find (a) the x component and (b) the y component of the hummingbird’s velocity at the time t = 0.72 s (c) What is the bird’s direction of travel at this time, relative to the positive x axis?

63. A racket ball is struck in such a way that it leaves the racket with a speed of 4.87 m/s in the horizontal direction. When the ball hits the court, it is a horizontal distance of 1.95 m from the racket. Find the height of the racket ball when it left the racket.

64. A hot-air balloon rises from the ground with a velocity of (2.00 m/s)y. A champagne bottle is opened to celebrate takeoff, expelling the cork horizontally with a velocity of (5.00 m/s)x relative to the balloon. When opened, the bottle is 6.00 m above the ground. (a) What is the initial velocity of the cork, as seen by an observer on the ground? Give your answer in terms of the x and y unit vectors. (b) What are the speed of the cork and its initial direction of motion as seen by the same observer? (c) Determine the maximum height above the ground attained by the cork. (d) How long does the cork remain in the air?

65. Repeat the previous problem, this time assuming that the balloon is descending with a speed of 2.00 m/s.

66. A soccer ball is kicked from the ground with an initial speed of 14.0 m/s. After 0.275 s its speed is12.9 m/s. (a) Give a strategy that will allow you to calculate the ball’s initial direction of motion. (b) Use your strategy to find the initial direction.

67. A particle leaves the origin with an initial velocity v = (2.40 m/s)x, and moves with constant acceleration a = (-1.90 m/s2)x + (3.20 m/s2)y. (a) How far does the particle move in the x direction before turning around? (b) What is the particle’s velocity at this time? (c) Plot the particle’s position at t = 0.500 s, 1.00 s, 1.50 s, and 2.00 s. Use these results to sketch position versus time for the particle.

68. When the dried-up seed pod of a scotch broom plant bursts open, it shoots out a seed with an initial velocity of 2.62 m/s at an angle of 60.5° above the horizontal. If the seed pod is 0.455 m above the ground, (a) how long does it take for the seed to land. (b) What horizontal distance does it cover during its flight?

69. Referring to Problem 68, a second seed shoots out from the pod with the same speed but with a direction of motion 30.0° below the horizontal. (a) How long does it take for the second seed to land? (b) What horizontal distance does it cover during its flight?

70. A shot-putter throws the shot with an initial speed of 12.2 m/s from a height of 5.15 ft above the ground. What is the range of the shot if the launch angle is (a) 20.0°, (b) 30.0°, or (c) 40.0°?

71. Pararescue Jumpers. Coast Guard pararescue jumpers are trained to leap from helicopters into the sea to save boaters in distress. The rescuers at ten like to step off their helicopter when it is “ten and ten”, which means that it is ten feet above the water and moving forward horizontally knots. (a) What are (a) the speed and (b) the direction of motion as a pararescuer enters the water following a ten a ten jump?

72. A ball thrown straight upward returns to its original level in 2.75 s. A second ball is thrown at an angle of 40.0° above the horizontal. What is the initial speed of the second ball if it also returns to its original level in 2.75 s?

73. To decide who pays for lunch, a passenger on a moving train tosses a coin straight upward with an initial speed of 4.38 m/s and catches it again when it returns to its initial level. From the point of view of the passenger, then, the coin’s initial velocity is (4.38 m/s)y. The train’s velocity relative to the ground is (12.1 m/s)x. (a) What is the minimum speed of the coin relative to the ground during its flight? At what point in the coin’s flight does this minimum speed occur? Explain (b) Find the initial speed and direction of the coin as seen by an observer on the ground. (c) Use the expression for ymax derived in Example 4-7 to calculate the maximum height of the coin, as seen by an observer on the ground. (d) Calculate the maximum height of the coin from the point of view of the passenger, who sees only one-dimension motion.

74. A cannon is placed at the bottom of a cliff 61.5 m high. If the cannon is fired straight upward, the cannonball just reaches the top of the cliff. (a) What is the initial speed of the cannonball? (b) Suppose a second cannon is placed at the top of the cliff. This cannon is fired horizontally, giving its cannonballs the same initial speed found in part (a). Show that the range of this cannon is the same as the maximum range of the cannon at the base of the cliff.

75. Shot Put Record. The men’s world record for the shot put, 23.12 m, was set by Randy Barnes of the United States on May 20, 1990. If the shot was launched from 6.00 ft above the ground at an initial angle of 42.0°, what was its initial speed?

76. Referring to Conceptual Checkpoint 4-3, suppose the two snowballs are thrown from an elevation of 15 m with an initial speed of 12 m/s. What is the speed of each ball when it is 5.0 m above the ground?

77. A hockey puck just clears the 2.00-m-high boards on its way out of the rink. The base of the boards is 20.2 m from the point where the puck is launched. (a) Given the launch angle of the puck, θ, outline a strategy that you can use to find its initial speed, v0. (b) Use your strategy to find v0 for θ = 15.0°.

78. Referring to Active Example 4-2, suppose the ball is punted from an initial height of 0.750. What is the initial speed of the ball in this case?

79. A “Lob” Pass Versus a “Bullet”. A quarterback can throw a receiver a high, lazy “lob” pass or a low, quick “bullet” pass. These passes are indicated by curves 1 and 2, respectively, in Figure 4-22. (a) The lob pass is thrown with an initial speed of 21.5 m/s and its time of flight is 3.97 s. What is its launch angle? (b) The bullet pass is thrown with a launch angle of 25.0°. What is the initial speed of this pass? (c) What is the time of flight of the bullet pass?

80. Collision Course. A useful rule of thumb in boating is that if the heading from your boat to a second boat remains constant, the two boats are on a collision course. Consider the two boats shown in Figure 4-23. At time t = 0, boat 1 is at the location (X, 0) and moving in the positive y direction; boat 2 is at (), Y) and moving in the positive x direction. The speed of boat 1 is v1.(a) What sped must boat 2 have if the boats are to collide at the point (X, Y)? (b) Assuming boat 2 has the speed found in part (a), calculate the displacement from boat 1 to boat 2, Δr = r2 – r1. (c) Use your results from part (b) to show that (Δr)y/(Δr)x = -Y/X, independent of time. This shows that Δr = r2 – r1 maintains a constant direction until the collision, as specified in the rule of thumb.

81. As discussed in Example 4-7, the archerfish hunts by dislodging an unsuspecting insect from its resting place with a stream of water expelled from the fish’s mouth. Suppose the archerfish squirts water with a speed of 2.15 m/s at an angle of 52.0° above the horizontal, and aims for a beetle on a leaf 3.00 cm above the water’s surface. (a) At what horizontal distance from the beatle should the archerfish fire if it is to hit its target in the least time? (b) How much time will the beetle have to react?

82. (a) What is the greatest horizontal distance from which the archerfish can hit the beetle, assuming the same squirt speed and direction as in Problem 81? (b) How much time does the beetle have to react in this case?

83. Find the launch angle for which the range and maximum height of a projectile are the same.

84. A mountain climber jumps a crevasse of width W by leaping horizontally with speed v0. (a) If the height difference between the two sides of the crevasse is h, what is the minimum value of v0 for the climber to land safely on the other side? (b) In this case, what is the climber’s direction of motion on landing?

85. Prove that the landing speed of a projectile is independent of launch angle for a given height of launch.

86. Maximum height and range. Prove that the maximum height of a projectile, H, divided by the range of the projectile, R, satisfies the relation H/R = ¼ tan θ.

87. Landing on a different level. A projectile fired from y = 0 with initial speed v0 and initial angle θ lands on a different level, y = h. Show that the time of flight of the projectile is T = ½ T0(1 + √(1 – h/H)), where T0 is the time of flight for h = 0 and H is the maximum height of the projectile.

88. A mountain climber jumps a crevasse by leaping horizontally with speed v0. If the climber’s direction of motion on landing is θ below the horizontal, what is the height difference h between the two sides of the crevasse?

89. Referring to Problem 73, suppose the initial velocity of the coin tossed by the passenger is v = (-2.25 m/s)x + (4.38 m/s)y. The train’s velocity relative to the ground is still (12.1 m/s)x (a) What is the minimum speed of the coin relative to the ground during its flight? At what point in the coin’s flight does this minimum speed occurs? Explain. (b) Find the initial speed and direction of the coin as seen by an observer on the ground. (c) Use the expression for ymax derived in Example 4-7 to calculate the maximum height of the coin, as seen by an observer on the ground. (d) Repeat part (c) from the point of view of the passenger. Verify that both observers calculate the same maximum height.

90. Projectiles: Coming or Going? Most projectiles continually move farther from the origin during their flight, but this is not the case if the launch angle is greater than cos-1(1/3) = 70.5°. For example, the projectile shown in the figure 4-24 has a launch angle of 75.0° and an initial speed of 10.1 m/s. During the portion of its motion shown in red, it is moving closer to the origin-it is moving away on the blue portions. Calculate the distance from the origin to the projectile (a) at the start of the red portion, (b) at the end of the red portion, and (c) just before the projectile lands. Notice that the distance for part (b) is the smallest of the three.

When the twin Mars exploration rovers, Spirit and Opportunity, set down on the surface of the red planet in January of 2004, their method of landing was both unique and elaborate. After initial braking with retro rockets, the rovers began their long descent through the thin Martian atmosphere on a parachute until they reached an altitude of about 16.7 m. At that point a system of four air bags with six lobes each were inflated, additional retro rocket blasts brought the craft to a virtual standstill, and the rovers detached from their parachutes. After a period of free fall to the surface, with an acceleration of 3.72 m/s2, the rovers bounced about a dozen times before coming to rest. They then deflated their air bags, righted themselves, and began to explore the surface. Figure 4-25 shows a rover with its surrounding cushion of air bags making its first contact with the Martian surface. After a typical first bounce the upward velocity of a rover would be 9.92 m/s at an angle of 75.0° above the horizontal. Assume this is the case for the problems that follow.

91. What is the maximum height of a rover between its first and second bounces?

A. 2.58 m B. 4.68 m. C. 12.3 m D. 148 m

92. How much time elapses between the first and second bounces?

A. 1.38 s B. 2.58 s C. 5.15 s D. 5.33 s

94. What is the average velocity of a rover between its first and second bounces?

A. 0 B. 2.57 m/s in the x-direction C. 9.92 m/s at 75.0° above the x-axis D. 9.58 m/s in the y direction.

95. Referring to Example 4-5 (a) At what launch angle greater than 54.0° does the golf ball just barely miss the top of the tree in front of the green? Assume the ball has an initial speed of 13.5 m/s, and that the tree is 3.00 m high and is a horizontal distance of 14.0 m from the launch point. (b) Where does the ball land in the case described in part (a)? (c) At what launch angle less than 54.0° does the golf ball just barely miss the top of the tree in front of the green? (d) Where does the ball land in the case described in part (c)?

96. Referring to Example 4-5 Suppose that he golf ball is launched with a speed of 15.0 m/s at an angle of 57.5° above the horizontal, and that it lands on a green 3.50 m above the level where it was struck. (a) What horizontal distance does the ball cover during its flight? (b) What increase in initial speed would be needed to increase the horizontal distance in part (a) by 7.50 m? Assume everything else remains the same.

97. Referring to example 4-6. Suppose the ball is dropped at the horizontal distance of 5.50 m, but from a new height of 5.00 m. The dolphin jumps with the same speed of 12.0 m/s. (a) What launch angle must the dolphin have if it is to catch the ball? (b) At what height does the dolphin catch the ball in this case? (c) What is the minimum initial speed the dolphin must have to catch the ball before it hits the water?

98. Referring to example 4-6. Suppose we change the dolphin’s launch angle to 45.0°, but everything else remains the same. Thus, the horizontal distance to the ball is 5.50 m, the drop height is 4.10 m, and the dolphin’s launch speed is 12.0 m/s. (a) What is the vertical distance between the dolphin and the ball when the dolphin reaches the horizontal position of the ball? We refer to this as the “miss distance”. (b) If the dolphin’s launch speed is reduced, will the miss distance increase, decrease, or stay the same? (c) Find the miss distance for a launch speed of 10.0 m/s.

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