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Hugh D. Young, College Physics, 9th edition, Addison Wesley 2011
39. A rocket fires two engines simultaneously. One produces a thrust of 725 N directly forward, while the other gives a 513-N thrust at 32.4° above the forward direction. Find the magnitude and direction (relative to the forward direction) of the resultant force that these engines exert on the rocket.
42. A woman takes her dog Rover for a walk on a leash. To get the little pooch moving forward, she pulls on the leash with a force of 20.0 N at an angle of 37° above the horizontal. (a) How much force is tending to pull Rover forward? (b) How much force is tending to lift Rover off the ground?
46. A plane leaves Seattle, flies 85 mi at 22° north of east, and then changes direction to 48° south of east. After flying at 115 mi in this new direction, the pilot must make an emergency landing on a field. The Seattle airport facility dispatches a rescue crew. (a) In what direction and how far should the crew fly to go directly to the field? Use components to solve this problem. (b) Check the reasonableness of your answer with a careful graphical sum.
47. You’re hanging from a chinning bar, with your ;arms at right angles to each other. The magnitudes of the forces exerted by both your arms are the same, and together they exert just enough upward force to support your weight, 620 N. (a) Sketch the two force vectors for your arms, along with their
resultant, and (b) use components to find the magnitude of each of the two “arm” force vectors.
48. Three horizontal ropes are attached to a boulder and produce the pulls shown in Figure 1.25.(a) Find the x and y components of each pull. (b) Find the components of the resultant of the three pulls. (c) Find the magnitude and direction (the counterclockwise angle with the +x-axis) of the resultant pull. (d) Sketch a clear graphical sum to check your answer in part (c).
49. A disoriented physics professor drives 3.25 km north, then 4.75 km west, and then 1.50 km south. (a) Use components to find the magnitude and direction of the resultant displacement of this professor. (b) Check the reasonableness of your answer with a graphical sum.
50. A postal employee drives a delivery truck along the route shown in Figure 1.26. Use components to determine the magnitude and direction of the truck’s resultant displacement. Then check the reasonableness of your answer by sketching a graphical sum.
51. Baseball mass. Baseball rules specify that a regulation ball shall weigh no less than 5.00 ounces nor more than ounces. What are the acceptable limits, in grams, for a regulation ball?
52. As you eat your way through a bag of chocolate chip cookies, you observe that each cookie is a circular disk with a diameter of 8.50 cm and a thickness of 0.050 cm. Find (a) the volume of a single cookie and (b) the ratio of the diameter to the thickness, and express both in the proper number of significant figures.
53. Breathing oxygen. The density of air under standard laboratory conditions is and about 20% of that air consists of oxygen. Typically, people breathe about 1/2 L of air per breath. (a) How many grams of oxygen does a person breathe in a day?(b) If this air is stored uncompressed in a cubical tank, how long is each side of the tank?
54. The total mass of Earth’s atmosphere is about 5×1015 metric tonnes. Suppose you breathe in about 1/3 L of air with each breath, and the density of air at room temperature is about 1.2 kg/m3. About how many breaths of air does the entire atmosphere contain? How does this compare to the number of atoms in one breath of air (about 1.2 × 1022)? It’s sometimes said that every breath you take contains atoms that were also breathed by Albert Einstein, Confucius, and in fact anyone else who ever lived. Based on your calculation, could this be true?
55. How much blood in a heartbeat? A typical human contains 5.0 L of blood, and it takes 1.0 min for all of it to pass through the heart when the person is resting with a pulse rate of 75 heartbeats per minute. On the average, what volume of blood, in liters and cubic centimeters, does the heart pump during each beat?
56. Muscle attachment. When muscles attach to bones, they usually do so by a series of tendons, as shown in Figure 1.27. In the figure, five tendons attach to the bone. The uppermost tendon pulls at 20.0° from the axis of the bone, and each tendon is directed 10.0° from the one next to it. (a) If each tendon exerts a 2.75 N pull on the bone, use vector components to find the magnitude and direction of the resultant force on this bone due to all five tendons. Let the axis of the bone be the +x axis. (b) Draw a graphical sum to check your results from part (a).
57. Hiking the Appalachian Trail. The Appalachian Trail runs from Mt. Katahdin in Maine to Springer Mountain in Georgia, a total distance of 2160 mi. If you hiked for 8 h per day, estimate (a) how many steps it would take to hike this trail and b) how many days it would take to hike it.
61. While surveying a cave, a spelunker follows a passage180 m straight west, then 210 m in a direction 45° east of south, and then 280 m at 30.0° east of north. After a fourth unmeasured displacement, she finds herself back where she started. Use vector components to find the magnitude and direction of the fourth displacement. Then check the reasonableness of your answer with a graphical sum.
62. A sailor in a small sailboat encounters shifting winds. She sails 2.00 km east, then 3.50 km southeast, and then an additional distance in an unknown direction. Her final position is 5.80 km directly east of her starting point. Find the magnitude and direction of the third leg of the journey. Draw the vector addition diagram, and show that it is in qualitative agreement with your numerical solution.
63. Dislocated shoulder. A patient with a dislocated shoulder is put into a traction apparatus as shown in Figure 1.29. The pulls A and B have equal magnitudes and must combine to produce an outward traction force of 5.60 N on the patient’s arm. How large should these pulls be?
64. On a training flight, a student pilot flies from Lincoln, Nebraska to Clarinda, Iowa, then to St. Joseph, Missouri, and then to Manhattan, Kansas (Fig. 1.30). The directions are shown relative to north: 0° is north, 90° is east, 180° is south, and 270° is west. Use the method of components to find (a) the distance she has to fly from Manhattan to get back to Lincoln, and (b) the direction (relative to north) she must fly to get there. Illustrate your solutions with a vector diagram.
65. Bones and muscles. A patient in therapy has a forearm that weighs 20.5 N and lifts a 112.0 N weight. The only other significant forces on his forearm come from the biceps muscle (which acts perpendicularly to the forearm) and the force at the elbow. If the biceps produce a pull of 232 N when the forearm is raised 43° above the horizontal, find the magnitude and direction of the force that the elbow exerts on the forearm.
1. An ant is crawling along a straight wire, which we shall call the x axis, from A to B to C to D (which overlaps A), as shown in Figure 2.39. O is the origin. Suppose you take measurements and find that AB is 50 cm, BC is 30 cm, and AO is 5 cm. (a) What is the ant’s position at points A, B, C, and D? (b) Find the displacement of the ant and the distance it has moved over each of the following intervals: (i) from A to B, (ii) from B to C, (iii) from C to D, and (iv) from A to D.
2. A person is walking briskly in a straight line, which we shall call the x axis. Figure 2.40 shows a graph of the person’s position x along this axis as a function of time t. (a) What is the person’s displacement during each of the following time intervals: (i) between t = 1.0 s and t = 10.0 s (ii) between t = 3.0 s and t = 10.0 s (iii) between t = 2.0 s and t = 3.0 s and (iv) between t = 2.0 and t = 4.0 s? (b) What distance did the person move from (i) t = 0 to t = 4.0 s (ii) 2.0 s to t = 4.0 and (iii) t = 8.0 s and t = 10.0?
3. A dog runs from points A to B to C in 3.0 s. Find the dog’s average velocity and average speed over this 3-second interval.
4. In an experiment, a shearwater (a seabird) was taken from its nest, flown 5150 km away, and released. The bird found its way back to its nest 13.5 days after release. If we place the origin in the nest and extend the +x-axis to the release point, what was the bird’s average velocity in m/s (a) for the return flight, and (b) for the whole episode, from leaving the nest to returning?
5. Figure 2.42 shows the position of a moving object as a function of time. (a) Find the average velocity of this object from points A to B, B to C, and A to C. (b) For the intervals in part (a), would the average speed be less than, equal to, or greater than the values you found in that part? Explain your reasoning.
6. An object moves along the x axis. Figure 2.43 shows a graph of its position x as a function of
time. (a) Find the average velocity of the object from points A to B, B to C, and A to C. (b) For the intervals in part (a), would the average speed be less than, equal to, or greater than the values you found in that part? Explain your reasoning.
7. A boulder starting from rest rolls down a hill in a straight line, which we shall call the x axis. A graph of its position x as a function of time t is shown in Figure 2.44. Find (a) the distance the boulder rolled between the end of the first second and the end of the third second and (b) the boulder’s average speed during (i) the first second, (ii) the second, (iii) the third second, (iv) the fourth second, and (v) the first 4 seconds.
8. Each graph in Figure 2.45 shows the position of a running cat, called Mousie, as a function of time. In each case, sketch a clear qualitative (no numbers) graph of Mousie’s velocity as a function of time.
9. In 1954, Roger Bannister became the first human to run a mile in less than 4 minutes. Suppose that a runner on a straight track covers a distance of 1.00 mi in exactly 4.00 min. What is his average speed in (a) mi/h, (b) ft/s, and (c) m/s.
10. Hypersonic scramjet. On March 27, 2004, the United States successfully tested the hypersonic X-43A scramjet, which flew at Mach 7 (seven times the speed of sound) for 11 seconds. (A scramjet gets its oxygen directly from the air, rather than from fuel.) (a) At this rate, how many minutes would it take such a scramjet to carry passengers the approximately 5000 km from San Francisco to New York? (b) How many kilometers did the scramjet travel during its 11 second test?
11. Plate tectonics. The earth’s crust is broken up into a series of more-or-less rigid plates that slide around due to motion of material in the mantle below. Although the speeds of these plates vary somewhat, they are typically about 5 cm/yr. Assume that this rate remains constant over time. (a) If you and your neighbor live on opposite sides of a plate boundary at which one plate is moving northward at 5.0 cm/yr with respect to the other plate, how far apart do your houses move in a century? (b) Los Angeles is presently 550 km south of San Francisco, but is on a plate moving northward relative to San Francisco. If the velocity 5 cm/yr continues, how many years will it take before Los Angeles has moved up to San Francisco?
12. A runner covers one lap of a circular track 40.0 m in diameter in 62.5 s. (a) For that lap, what were her average speed and average velocity? (b) If she covered the first half-lap in 28.7 s, what were her average speed and average velocity for that half-lap?
13. Sound travels at a speed of about 344 m/s in air. You see a distant flash of lightning and hear the thunder arrive 7.5 seconds later. How many miles away was the lightning strike?
14. Ouch! Nerve impulses travel at different speeds, depending on the type of fiber through which they move. The impulses for touch travel at 76.2 m/s while those registering pain move at 0.610 m/s. If a person stubs his toe, find (a) the time for each type of impulse to reach his brain, and (b) the time delay between the pain and touch impulses. Assume that his brain is 1.85 m from his toe and that the impulses travel directly from toe to brain.
15. While driving on the freeway at 110 km/h, you pass a truck whose total length you estimate at 25 m. (a) If it takes you, in the driver’s seat, 5.5 s to pass from the rear of the truck to its front, what is the truck’s speed relative to the road? (b) How far does the truck travel while you’re passing it?
16. A mouse travels along a straight line; its distance x from the origin at any time t is given by the equation x = (8.5 cm/s)t – (2.5 cm/s)t2. Find the average velocity of the mouse in the interval from and in the interval from t = 0 to t = 1.0 s and in the interval from t = 0 to t = 4.0 s.
17. The freeway blues! When you normally drive the freeway between Sacramento and San Francisco at an average speed of 105 km/hr (65 mph), the trip takes 1.0 hr and 20 min. On a Friday afternoon, however, heavy traffic slows you down to an average of 70 km/hr (43 mph) for the same distance. How much longer does the trip take on Friday than on the other days?
18. Two runners start simultaneously at opposite ends of a 200.0 m track and run toward each other. Runner A runs at a steady 8.0 m/s and runner B runs at a constant 7.0 m/s. When and where will these runners meet?
19. A physics professor leaves her house and walks along the sidewalk toward campus. After 5 min, she realizes that it is raining and returns home. The distance from her house as a function of time is shown in Figure 2.46. At which of the labeled points is her velocity (a) zero? (b) constant and positive? (c) constant and negative? (d) increasing in magnitude? and (e) decreasing in magnitude?
20. A test car travels in a straight line along the x axis. The graph in Figure 2.47 shows the car’s position x as a function of time. Find its instantaneous velocity at points A through G.
21. Figure 2.48 shows the position x of a crawling spider as a function of time. Use this graph to draw a numerical graph of the spider’s velocity as a function of time over the same time interval.
22. The graph in Figure 2.49 shows the velocity of a motorcycle police officer plotted as a function of time. Find the instantaneous acceleration at times t = 3 s, at t = 7 s and at t = 11 s.
23. A test driver at Incredible Motors, Inc., is testing a new model car having a speedometer calibrated to read m/s rather than mi/h. The following series of speedometer readings was obtained during a test run. (a) Compute the average acceleration during each 2 s interval. Is the acceleration constant? Is it constant during any part of the test run? (b) Make a velocity–time graph of the data shown, using scales of 1 cm = 1 s horizontally and 1 cm = 2 m/s vertically. Draw a smooth curve through the plotted points. By measuring the slope of your curve, find the magnitude of the instantaneous acceleration at times t = 9 s, 13 s, and 15 s.
24. (a) The pilot of a jet fighter will black out at an acceleration greater than approximately 5g if it lasts for more than a few seconds. Express this acceleration in m/s2 and ft/s2. (b) The acceleration of the passenger during a car crash with an air bag is about 60g for a very short time. What is this acceleration in m/s2 and ft/s2. (c) The acceleration of a falling body on our moon is 1.67 m/s2. How many g’s is this? (d) If the acceleration of a test plane is 24.3 m/s2 how many g’s is it?
25. For each graph of velocity as a function of time in Figure 2.50, sketch a qualitative graph of the acceleration as a function of time.
26. A little cat, Bella, walks along a straight line, which we shall call the x axis, with the positive direction to the right. As an observant scientist, you make measurements of her motion and construct a graph of the little feline’s velocity as a function of time. (a) Find Bella’s velocity at t = 4.0 s and at t = 7.0 s (b) What is her acceleration at t = 3.0 s? at t= 6.0 s? at t = 7.0 s? (c) Sketch a clear graph of Bella’s acceleration as a function of time.
27. A car driving on the turnpike accelerates uniformly in a straight line from 88 ft/s (60 mph) to 110 ft/s (75 mph) in 3.50 s. (a) What is the car’s acceleration? (b) How far does the car travel while it accelerates?
28. Animal motion. Cheetahs, the fastest of the great cats, can reach 45 mph in 2.0 s starting from rest. Assuming that they have constant acceleration throughout that time, find (a) their acceleration (in ft/s2 and m/s2) and (b) the distance (in m and ft) they travel during that time.
29. A cat drops from a shelf 4.0 ft above the floor and lands on all four feet. His legs bring him to a stop in a distance of 12 cm. Calculate (a) his speed when he first touches the floor (ignore air resistance), (b) how long it takes him to stop, and (c) his acceleration (assumed constant) while he is stopping, in and g’s.
30. Blackout? A jet fighter pilot wishes to accelerate from rest at 5g to reach Mach 3 (three times the speed of sound) as quickly as possible. Experimental tests reveal that he will black out if this acceleration lasts for more than 5.0 s. Use 331 m/s for the speed of sound. (a) Will the period of acceleration last long enough to cause him to black out? (b) What is the greatest speed he can reach with an acceleration of 5g before blacking out?
31. A fast pitch. The fastest measured pitched baseball left the pitcher’s hand at a speed of 45.0 m/s. If the pitcher was in contact with the ball over a distance of 1.50 m and produced constant acceleration, (a) what acceleration did he give the ball, and (b) how much time did it take him to pitch it?
32. If a pilot accelerates at more than 4g, he begins to “gray out,” but not completely lose consciousness. (a) What is the shortest time that a jet pilot starting from rest can take to reach Mach 4 (four times the speed of sound) without graying out? (b) How far would the plane travel during this period of acceleration?
33. Air-bag injuries. During an auto accident, the vehicle’s air bags deploy and slow down the passengers more gently than if they had hit the windshield or steering wheel. According to safety standards, the bags produce a maximum acceleration of 60 g, but lasting for only 36 ms (or less). How far (in meters) does a person travel in coming to a complete stop in 36 ms at a constant acceleration of 60 g?
34. Starting from rest, a boulder rolls down a hill with constant acceleration and travels 2.00 m during the first second. (a) How far does it travel during the second? (b) How fast is it moving at the end of the first second? at the end of the second?
35. Faster than a speeding bullet! The Beretta Model 92S (the standard-issue U.S. army pistol) has a barrel 127 mm long. The bullets leave this barrel with a muzzle velocity of 335 m/s (a) What is the acceleration of the bullet while it is in the barrel, assuming it to be constant? Express your answer in m/s2 and in g’s. (b) For how long is the bullet in the barrel?
36. An airplane travels 280 m down the runway before taking off. Assuming that it has constant acceleration, if it starts from rest and becomes airborne in 8.00 s, how fast (in m/s) is it moving at takeoff?
37. Entering the freeway. A car sits in an entrance ramp to a freeway, waiting for a break in the traffic. The driver accelerates with constant acceleration along the ramp and onto the freeway. The car starts from rest, moves in a straight line, and has a speed of 20 m/s (45 mi/h) when it reaches the end of the 120-m-long ramp. (a) What is the acceleration of the car? (b) How much time does it take the car to travel the length of the ramp? (c) The traffic on the freeway is moving at a constant speed of 20 m/s. What distance does the traffic travel while the car is moving the length of the ramp?
38. The “reaction time” of the average automobile driver is about 0.7 s. (The reaction time is the interval between the perception of a signal to stop and the application of the brakes.) If an automobile can slow down with an acceleration of 12.0 ft/s2 compute the total distance covered in coming to a stop after a signal is observed (a) from an initial velocity of 15.0 mi/h (in a school zone) and (b) from an initial velocity of 55.0 mi/h.
39. According to recent typical test data, a Ford Focus travels 0.250 mi in 19.9 s, starting from rest. The same car, when braking from 60.0 mph on dry pavement, stops in 146 ft. Assume constant acceleration in each part of its motion, but not necessarily the same acceleration when slowing down as when speeding up. (a) Find this car’s acceleration while braking and while speeding up. (b) If its acceleration is constant while speeding up, how fast (in mph) will the car be traveling after 0.250 mi of acceleration? (c) How long does it take the car to stop while braking from 60.0 mph?
40. A subway train starts from rest at a station and accelerates at a rate of 1.60 m/s2 for 14.0 s. It runs at constant speed for 70.0 s and slows down at a rate of 3.50 m/s2 until it stops at the next station. Find the total distance covered.
41. If the radius of a circle of area A and circumference C is doubled, find the new area and circumference of the circle in terms of A and C.
42. In the redesign of a machine, a metal cubical part has each of its dimensions tripled. By what factor do its surface area and volume change?
43. You have two cylindrical tanks. The tank with the greater volume is 1.20 times the height of the smaller tank. It takes 218 gallons of water to fill the larger tank and 150 gallons to fill the other. What is the ratio of the radius of the larger tank to the radius of the smaller one?
44. A speedy basketball point guard is 5 ft 10 inches tall; the center on the same team is 7 ft 2 inches tall. Assuming their bodies are similarly proportioned, if the point guard weighs 175 lb, what would you expect the center to weigh?
45. Two rockets having the same acceleration start from rest, but rocket A travels for twice as much time as rocket B. (a) If rocket A goes a distance of 250 km, how far will rocket B go? (b) If rocket A reaches a speed of 350 m/s what speed will rocket B reach?
46. Two cars having equal speeds hit their brakes at the same time, but car A has three times the acceleration as car B. (a) If car A travels a distance D before stopping, how far (in terms of D) will car B go before stopping? (b) If car B stops in time T, how long (in terms of T) will it take for car A to stop?
47. Airplane A, starting from rest with constant acceleration, requires a runway 500 m long to become airborne. Airplane B requires a takeoff speed twice as great as that of airplane A, but has the same acceleration, and both planes start from rest. (a) How long must the runway be for airplane B? (b) If airplane A takes time T to travel the length of its runway, how long (in terms of T) will airplane B take to travel the length of its runway?
48. (a) If a flea can jump straight up to a height of 22.0 cm, what is its initial speed (in m/s) as it leaves the ground, neglecting air resistance? (b) How long is it in the air? (c) What are the magnitude and direction of its acceleration while it is (i) moving upward? (ii) moving downward? (iii) at the highest point?
49. A brick is released with no initial speed from the roof of a building and strikes the ground in 2.50 s, encountering no appreciable air drag. (a) How tall, in meters, is the building? (b) How fast is the brick moving just before it reaches the ground? (c) Sketch graphs of this falling brick’s acceleration, velocity, and vertical position as functions of time.
50. Loss of power! In December of 1989, a KLM Boeing 747 airplane carrying 231 passengers entered a cloud of ejecta from an Alaskan volcanic eruption. All four engines went out, and the plane fell from 27,900 ft to 13,300 ft before the engines could be restarted. It then landed safely in Anchorage. Neglecting any air resistance and aerodynamic lift, and assuming that the plane had no vertical motion when it lost power, (a) for how long did it fall before the engines were restarted, and (b) how fast was it falling at that instant? c) In reality, why would the plane not be falling nearly as fast?
51. A tennis ball on Mars, where the acceleration due to gravity is 0.379g and air resistance is negligible, is hit directly upward and returns to the same level 8.5 s later. (a) How high above its original point did the ball go? (b) How fast was it moving just after being hit? (c) Sketch clear graphs of the ball’s vertical position, vertical velocity, and vertical acceleration as functions of time while it’s in the Martian air.
52. Measuring g. One way to measure g on another planet or moon by remote sensing is to measure how long it takes an object to fall a given distance. A lander vehicle on a distant planet records the fact that it takes 3.17 s for a ball to fall freely 11.26 m, starting from rest. (a) What is the acceleration due to gravity on that planet? Express your answer in m/s2 and in earth g’s. b) How fast is the ball moving just as it lands?
53. That’s a lot of hot air! A hot air balloonist, rising vertically with a constant speed of 5.00 m/s releases a sandbag at the instant the balloon is 40.0 m above the ground. (See Figure 2.52.) After it is released, the sandbag encounters no appreciable air drag. (a) Compute the position and velocity of the sandbag at 0.250 s and 1.00 s after its release. (b) How many seconds after its release will the bag strike the ground? (c) How fast is it moving as it strikes the ground? (d) What is the greatest height above the ground that the sandbag reaches? (e) Sketch graphs of this bag’s acceleration, velocity, and vertical position as functions of time.
54. Look out below. Sam heaves a 16-lb shot straight upward, giving it a constant upward acceleration from rest of 45.0 m/s2 for 64.0 cm. He releases it 2.20 m above the ground. You may ignore air resistance. (a) What is the speed of the shot when Sam releases it? (b) How high above the ground does it go? (c) How much time does he have to get out of its way before it returns to the height of the top of his head, 1.83 m above the ground?
55. Astronauts on the moon. Astronauts on our moon must function with an acceleration due to gravity of 0.170g. (a) If an astronaut can throw a certain wrench 12.0 m vertically upward on earth, how high could he throw it on our moon if he gives it the same starting speed in both places? (b) How much longer would it be in motion (going up and coming down) on the moon than on earth?
56. A student throws a water balloon vertically downward from the top of a building. The balloon leaves the thrower’s hand with a speed of 15.0 m/s (a) What is its speed after falling freely for 2.00 s? (b) How far does it fall in 2.00 s? (c) What is the magnitude of its velocity after falling 10.0 m?
57. A rock is thrown vertically upward with a speed of 12.0 m/s from the roof of a building that is 60.0 m above the ground. (a) In how many seconds after being thrown does the rock strike the ground? (b) What is the speed of the rock just before it strikes the ground? Assume free fall.
58. Physiological effects of large acceleration. The rocket driven sled Sonic Wind No. 2, used for investigating the physiological effects of large accelerations, runs on a straight, level track that is 1080 m long. Starting from rest, it can reach a speed of 1610 km/h in 1.80 s. (a) Compute the acceleration in m/s2 and in g’s. (b) What is the distance covered in 1.80 s? (c) A magazine article states that, at the end of a certain run, the speed of the sled decreased from to zero in 1.40 s and that, during this time, its passenger was subjected to more than 40g. Are these figures consistent?
59. Two stones are thrown vertically upward from the ground, one with three times the initial speed of the other. (a) If the faster stone takes 10 s to return to the ground, how long will it take the slower stone to return? (b) If the slower stone reaches a maximum height of H, how high (in terms of H) will the faster stone go? Assume free fall.
60. Two coconuts fall freely from rest at the same time, one from a tree twice as high as the other. (a) If the coconut from the taller tree reaches the ground with a speed V, what will be the speed (in terms of V) of the coconut from the other tree when it reaches the ground? (b) If the coconut from the shorter tree takes time T to reach the ground, how long (in terms of T) will it take the other coconut to reach the ground?
66. At the instant the traffic light turns green, an automobile that has been waiting at an intersection starts ahead with a constant acceleration of 2.50 m/s2. At the same instant, a truck, traveling with a constant speed of 15 m/s overtakes and passes the automobile. (a) How far beyond its starting point does the automobile overtake the truck? (b) How fast is the automobile traveling when it overtakes the truck?
67. On a 20-mile bike ride, you ride the first 10 miles at an average speed of 8 mi/h. What must your average speed over the next 10 miles be to have your average speed for the total 20 miles be (a) 4 mi/h (b) 12 mi/h (c) Given this average speed for the first 10 miles, can you possibly attain an average speed of for the total 20-mile ride? Explain.
68. You and a friend start out at the same time on a 10-km run. Your friend runs at a steady 2.5 m/s. How fast do you have to run if you want to finish the run 15 minutes before your friend?
69. Two rocks are thrown directly upward with the same initial speeds, one on earth and one on our moon, where the acceleration due to gravity is one-sixth what it is on earth. (a) If the rock on the moon rises to a height H, how high, in terms of H, will the rock rise on the earth? (b) If the earth rock takes 4.0 s to reach its highest point, how long will it take the moon rock to do so?
70. Prevention of hip fractures. Falls resulting in hip fractures are a major cause of injury and even death to the elderly. Typically, the hip’s speed at impact is about 2.0 m/s. If this can be reduced to 1.3 m/s or less, the hip will usually not fracture. One way to do this is by wearing elastic hip pads. (a) If a typical pad is 5.0 cm thick and compresses by 2.0 cm during the impact of a fall, what acceleration (in m/s2 and in g’s) does the hip undergo to reduce its speed to 1.3 m/s (b) The acceleration you found in part (a) may seem like a rather large acceleration, but to fully assess its effects on the hip, calculate how long it lasts.
72. Raindrops. If the effects of the air acting on falling raindrops are ignored, then we can treat raindrops as freely falling objects. (a) Rain clouds are typically a few hundred meters above the ground. Estimate the speed with which raindrops would strike the ground if they were freely falling objects. Give your estimate in m/s, km/h, and mi/h (b) Estimate (from your own personal observations of rain) the speed with which raindrops actually strike the ground. (c) Based on your answers to parts (a) and (b), is it a good approximation to neglect the effects of the air on falling raindrops? Explain.
73. Egg drop. You are on the roof of the physics building of your school, 46.0 m above the ground. (See Figure 2.53.) Your physics professor, who is 1.80 m tall, is walking alongside the building at a constant speed of 1.20 m/s. If you wish to drop an egg on your professor’s head, where should the professor be when you release the egg, assuming that the egg encounters no appreciable air drag.
74. A 0.525 kg ball starts from rest and rolls down a hill with uniform acceleration, traveling 150 m during the second 10.0 s of its motion. How far did it roll during the first 5.0 s of motion?
75. A large boulder is ejected vertically upward from a volcano with an initial speed of 40.0 m/s. Air resistance may be ignored. (a) At what time after being ejected is the boulder moving at 20.0 m/s upward? (b) At what time is it moving at 20.0 m/s downward? (c) When is the displacement of the boulder from its initial position zero? (d) When is the velocity of the boulder zero? (e) What are the magnitude and direction of the acceleration while the boulder is (i) moving upward? (ii) Moving downward? (iii) At the highest point? (f) Sketch ay-t, vy-t and y-t graphs for the motion.
1. A meteor streaking through the night sky is located with radar. At point A its coordinates are (5.00 km, 1.20 km) and 1.14 s later it has moved to point B with coordinates (6.24 km, 0.925 km). Find (a) the x and y components of its average velocity between A and B and (b) the magnitude and direction of its average velocity between these two points.
2. At an air show, a jet plane has velocity components vx = 625 km/h and vy = 415 km/h at time 3.85 s and vx = 838 km/h and vy = 365 km/h at time 6.52 s. For this time interval, find (a) the x and y components of the plane’s average acceleration and (b) the magnitude and direction of its average acceleration.
3. A dragonfly flies from point A to point B along the path shown in Figure 3.34 in 1.50 s. (a) Find the x and y components of its position vector at point A. (b) What are the magnitude and direction of its position vector at A? (c) Find the x and y components of the dragonfly’s average velocity between A and B. (d) What are the magnitude and direction of its average velocity between these two points?
4. A coyote chasing a rabbit is moving 8.00 m/s due east at one moment and 8.80 m/s due south 4.00 s later. Find (a) the x and y components of the coyote’s average acceleration during that time and (b) the magnitude and direction of the coyote’s average acceleration during that time.
5. An athlete starts at point A and runs at a constant speed of 6.0 m/s around a round track 100 m in diameter, as shown in Figure 3.35. Find the x and y components of this runner’s average
velocity and average acceleration between points (a) A and B, (b) A and C, (c) C and D, and (d) A and A (a full lap). (e) Calculate the magnitude of the runner’s average velocity between A and B. Is his average speed equal to the magnitude of his average velocity? Why or why not? (f) How can his velocity be changing if he is running at constant speed?
6. A stone is thrown horizontally at 30.0 m/s from the top of a very tall cliff. (a) Calculate its horizontal position and vertical position at 2-s intervals for the first 10.0 s. (b) Plot your positions from part (a) to scale. Then connect your points with a smooth curve to show the trajectory of the stone.
7. A baseball pitcher throws a fastball horizontally at a speed of 42.0 m/s. Ignoring air resistance, how far does the ball drop between the pitcher’s mound and home plate, 60 ft 6 in away?
8. A physics book slides off a horizontal tabletop with a speed of 1.10 m/s. It strikes the floor in 0.350 s. Ignore air resistance. Find (a) the height of the tabletop above the floor, (b) the horizontal distance from the edge of the table to the point where the book strikes the floor, and (c) the horizontal and vertical components of the book’s velocity, and the magnitude and direction of its velocity, just before the book reaches the floor.
9. A tennis ball rolls off the edge of a tabletop 0.750 m above the floor and strikes the floor at a point 1.40 m horizontally from the edge of the table. (a) Find the time of flight of the ball. (b) Find the magnitude of the initial velocity of the ball. (c) Find the magnitude and direction of the velocity of the ball just before it strikes the floor.
10. A military helicopter on a training mission is flying horizontally at a speed of 60.0 m/s when it accidentally drops a bomb (fortunately, not armed) at an elevation of 300 m. You can ignore air resistance. (a) How much time is required for the bomb to reach the earth? (b) How far does it travel horizontally while falling? (c) Find the horizontal and vertical components of the bomb’s velocity just before it strikes the earth. (d) Draw graphs of the horizontal distance vs. time and the vertical distance vs. time for the bomb’s motion. (e) If the velocity of the helicopter remains constant, where is the helicopter when the bomb hits the ground?
11. Inside a starship at rest on the earth, a ball rolls off the top of a horizontal table and lands a distance D from the foot of the table. This starship now lands on the unexplored Planet X. The commander, Captain Curious, rolls the same ball off the same table with the same initial speed as on earth and finds that it lands a distance 2.76D from the foot of the table. What is the acceleration due to gravity on Planet X?
12. A daring 510 N swimmer dives off a cliff with a running horizontal leap, as shown in Figure 3.36. What must her minimum speed be just as she leaves the top of the cliff so that she will miss the ledge at the bottom, which is 1.75 m wide and 9.00 m below the top of the cliff?
13. Leaping the river, I. A 10,000 N car comes to a bridge during a storm and finds the bridge washed out. The 650 N driver must get to the other side, so he decides to try leaping it with his car. The side the car is on is 21.3 m above the river, while the opposite side is a mere 1.80 m above the river. The river itself is a raging torrent 61.0 m wide. (a) How fast should the car be traveling just as it leaves the cliff in order to clear the river and land safely on the opposite side? (b) What is the speed of the car just before it lands safely on the other side?
14. A football is thrown with an initial upward velocity component of 15.0 m/s and a horizontal velocity component of 18.0 m/s (a) How much time is required for the football to reach the highest point in its trajectory? (b) How high does it get above its release point? (c) How much time after it is thrown does it take to return to its original height? How does this time compare with what you calculated in part (b)? Is your answer reasonable? (d) How far has the football traveled horizontally from its original position?
15. A tennis player hits a ball at ground level, giving it an initial velocity of 24 m/s at 57° above the horizontal. (a) What are the horizontal and vertical components of the ball’s initial velocity? (b) How high above the ground does the ball go? (c) How long does it take the ball to reach its maximum height? (d) What are the ball’s velocity and acceleration at its highest point? (e) For how long a time is the ball in the air? (f) When this ball lands on the court, how far is it from the place where it was hit?
16. (a) A pistol that fires a signal flare gives it an initial velocity (muzzle velocity) of 125 m/s at an angle of 55.0° above the horizontal. You can ignore air resistance. Find the flare’s maximum height and the distance from its firing point to its landing point if it is fired (a) on the level salt flats of Utah, and (b) over the flat Sea of Tranquility on the moon, where g = 1.67 m/s2.
17. A major leaguer hits a baseball so that it leaves the bat at a speed of 30.0 m/s and at an angle of 36.9° above the horizontal. You can ignore air resistance. (a) At what two times is the baseball at a height of 10.0 m above the point at which it left the bat? (b) Calculate the horizontal and vertical components of the baseball’s velocity at each of the two times you found in part (a). (c) What are the magnitude and direction of the baseball’s velocity when it returns to the level at which it left the bat?
18. A balloon carrying a basket is descending at a constant velocity of 20.0 m/s. A person in the basket throws a stone with an initial velocity of 15.0 m/s horizontally perpendicular to the path of the descending balloon, and 4.00 s later this person sees the rock strike the ground. (See Figure 3.37.) (a) How high was the balloon when the rock was thrown out? (b) How far horizontally does the rock travel before it hits the ground? (c) At the instant the rock hits the ground, how far is it from the basket?
19. A batted baseball leaves the bat at an angle of 30.0° above the horizontal and is caught by an outfielder 375 ft from home plate at the same height from which it left the bat. (a) What was the initial speed of the ball? (b) How high does the ball rise above the point where it struck the bat?
20. A man stands on the roof of a 15.0-m-tall building and throws a rock with a velocity of magnitude 30.0 m/s at an angle of 33.0° above the horizontal. You can ignore air resistance. Calculate (a) the maximum height above the roof reached by the rock, (b) the magnitude of the velocity of the rock just before it strikes the ground, and (c) the horizontal distance from the base of the building to the point where the rock strikes the ground.
21. The champion jumper of the insect world. The froghopper, Philaenus spumarius, holds the world record for insect jumps. When leaping at an angle of 58.0° above the horizontal, some of the tiny critters have reached a maximum height of 58.7 cm above the level ground. (a) What was the takeoff speed for such a leap? (b) What horizontal distance did the froghopper cover for this world-record leap?
22. A grasshopper leaps into the air from the edge of a vertical cliff, as shown in Figure 3.38. Use information from the figure to find (a) the initial speed of the grasshopper and (b) the height of the cliff.
23. Firemen are shooting a stream of water at a burning building. A high-pressure hose shoots out the water with a speed of 25.0 m/s as it leaves the hose nozzle. Once it leaves the hose, the water moves in projectile motion. The firemen adjust the angle of elevation of the hose until the water takes 3.00 s to reach a building 45.0 m away. You can ignore air resistance; assume that the end of the hose is at ground level. (a) Find the angle of elevation of the hose. (b) Find the speed and acceleration of the water at the highest point in its trajectory. (c) How high above the ground does the water strike the building, and how fast is it moving just before it hits the building?
24. Show that a projectile achieves its maximum range when it is fired at 45° above the horizontal if y = y0.
25. A water balloon slingshot launches its projectiles essentially from ground level at a speed of 25.0 m/s. (a) At what angle should the slingshot be aimed to achieve its maximum range? (b) If shot at the angle you calculated in part (a), how far will a water balloon travel horizontally? (c) For how long will the balloon be in the air? (You can ignore air resistance.)
26. A certain cannon with a fixed angle of projection has a range of 1500 m. What will be its range if you add more powder so that the initial speed of the cannonball is tripled?
27. The nozzle of a fountain jet sits in the center of a circular pool of radius 3.50 m. If the nozzle shoots water at an angle of 65°, what is the maximum speed of the water at the nozzle that will allow it to land within the pool? (You can ignore air resistance.)
28. Two archers shoot arrows in the same direction from the same place with the same initial speeds but at different angles. One shoots at 45° above the horizontal, while the other shoots at 60.0°. If the arrow launched at 45° lands 225 m from the archer, how far apart are the two arrows when they land? (You can assume that the arrows start at essentially ground level.)
29. A bottle rocket can shoot its projectile vertically to a height of 25.0 m. At what angle should the bottle rocket be fired to reach its maximum horizontal range, and what is that range? (You can ignore air resistance.)
30. An airplane is flying with a velocity of at an angle of 23.0° above the horizontal. When the plane is 114 m directly above a dog that is standing on level ground, a suitcase drops out of the luggage compartment. How far from the dog will the suitcase land? You can ignore air resistance.
31. You swing a 2.2 kg stone in a circle of radius 75 cm. At what speed should you swing it so its centripetal acceleration will be 9.8 m/s2
32. Consult Appendix E. Calculate the radial acceleration (in and g’s) of an object (a) on the ground at the earth’s equator and (b) at the equator of Jupiter (which takes 0.41 day to spin once), turning with the planet.
33. Consult Appendix E and assume circular orbits. (a) What is the magnitude of the orbital velocity, in of the earth around the sun? (b) What is the radial acceleration, in of the earth toward the sun? (c) Repeat parts (a) and (b) for the motion of the planet Mercury.
34. A model of a helicopter rotor has four blades, each 3.40 m in length from the central shaft to the tip of the blade. The model is rotated in a wind tunnel at (a) What is the linear speed, in of the blade tip? (b) What is the radial acceleration of the blade tip, expressed as a multiple of the acceleration g due to gravity?
35. A wall clock has a second hand 15.0 cm long. What is the radial acceleration of the tip of this hand?
36. A curving freeway exit has a radius of 50.0 m and a posted speed limit of 35 mi/h. What is your radial acceleration (in m/s2) if you take this exit at the posted speed? What if you take the exit at a speed of 50 mi/h?
37. Dizziness. Our balance is maintained, at least in part, by the endolymph fluid in the inner ear. Spinning displaces this fluid, causing dizziness. Suppose a dancer (or skater) is spinning at a very high 3.0 revolutions per second about a vertical axis through the center of his head. Although the distance varies from person to person, the inner ear is approximately 7.0 cm from the axis of spin. What is the radial acceleration (in m/s2 and in g’s) of the endolymph fluid?
38. Pilot blackout in a power dive. A jet plane comes in for a downward dive as shown in Figure 3.39. The bottom part of the path is a quarter circle having a radius of curvature of 350 m. According to medical tests, pilots lose consciousness at an acceleration of 5.5g. At what speed (in and mph) will the pilot black out for this dive?
39. A canoe has a velocity of 0.40 m/s southeast relative to the earth. The canoe is on a river that is flowing 0.50 m/s east relative to the earth. Find the velocity magnitude and direction) of the canoe relative to the river.
40. Crossing the river, I. A river flows due south with a speed of 2.0 m/s. A man steers a motorboat across the river; his velocity relative to the water is 4.2 m/s due east. The river is 800 m wide. (a) What is his velocity (magnitude and direction) relative to the earth? (b) How much time is required for the man to cross the river? (c) How far south of his starting point will he reach the opposite bank?
41. Crossing the river, II. (a) In which direction should the motorboat in the previous problem head in order to reach a point on the opposite bank directly east from the starting point? (The boat’s speed relative to the water remains 4.2 m/s (b) What is the velocity of the boat relative to the earth? (c) How much time is required to cross the river?
42. You’re standing outside on a windless day when raindrops begin to fall straight down. You run for shelter at a speed of 5.0 m/s, and you notice while you’re running that the rain drops appear to be falling at an angle of about 30 from the vertical. What’s the vertical speed of the raindrops?
43. Bird migration. Canadian geese migrate essentially along a north–south direction for well over a thousand kilometers in some cases, traveling at speeds up to about 100 km/h. If one such bird is flying at relative to the air, but there is a 40 km/h wind blowing from west to east, (a) at what angle relative to the north–south direction should this bird head so that it will be traveling directly southward relative to the ground? (b) How long will it take the bird to cover a ground distance of 500 km from north to south? (Note: Even on cloudy nights, many birds can navigate using the earth’s magnetic field to fix the north–south direction.)
44. A test rocket is launched by accelerating it along a 200.0-m incline at 1.25 m/s2 starting from rest at point A (Figure 3.40.) The incline rises at 35.0° above the horizontal, and at the instant the rocket leaves it, its engines turn off and it is subject only to gravity (air resistance can be ignored). Find (a) the maximum height above the ground that the rocket reaches, and (b) the greatest horizontal range of the rocket beyond point A.
45. A player kicks a football at an angle of 40.0° from the horizontal, with an initial speed of 12.0 m/s. A second player standing at a distance of 30.0 m from the first (in the direction of the kick) starts running to meet the ball at the instant it is kicked. How fast must he run in order to catch the ball just before it hits the ground?
46. Dynamite! A demolition crew uses dynamite to blow an old building apart. Debris from the explosion flies off in all directions and is later found at distances as far as 50 m from the explosion. Estimate the maximum speed at which debris was blown outward by the explosion. Describe any assumptions that you make.
47. Fighting forest fires. When fighting forest fires, airplanes work in support of ground crews by dropping water on the fires. A pilot is practicing by dropping a canister of red dye, hoping to hit a target on the ground below. If the plane is flying in a horizontal path 90.0 m above the ground and with a speed of 64.0 m/s at what horizontal distance from the target should the pilot release the canister? Ignore air resistance.
48. An errand of mercy. An airplane is dropping bales of hay to cattle stranded in a blizzard on the Great Plains. The pilot releases the bales at 150 m above the level ground when the plane is flying at 75 m/s 55° above the horizontal. How far in front of the cattle should the pilot release the hay so that the bales will land at the point where the cattle are stranded?
49. A cart carrying a vertical missile launcher moves horizontally at a constant velocity of to the right. It launches a rocket vertically upward. The missile has an initial vertical velocity of 30.0 m/s to the right. It launches a rocket vertically upward. The missile has an initial vertical velocity of 40.0 m/s relative to the cart. (a) How high does the rocket go? (b) How far does the cart travel while the rocket is in the air? (c) Where does the rocket land relative to the cart?
50. The longest home run. According to the Guinness Book of World Records, the longest home run ever measured was hit by Roy “Dizzy” Carlyle in a minor-league game. The ball traveled 188 m (618 ft) before landing on the ground outside the ballpark. (a) Assuming that the ball’s initial velocity was 45° above the horizontal, and ignoring air resistance, what did the initial speed of the ball need to be to produce such a home run if the ball was hit at a point 0.9 m (3.0 ft) above ground level? Assume that the ground was perfectly flat. (b) How far would the ball be above a fence 3.0 m (10 ft) in height if the fence were 116 m (380 ft) from home plate?
51. A professional golfer can hit a ball with a speed of 70.0 m/s. What is the maximum distance a golf ball hit with this speed could travel on Mars, where the value of g is 3.71 m/s2.
52. A baseball thrown at an angle of 60.0° above the horizontal strikes a building 18.0 m away at a point 8.00 m above the point from which it is thrown. Ignore air resistance. (a) Find the magnitude of the initial velocity of the baseball (the velocity with which the baseball is thrown). (b) Find the magnitude and direction of the velocity of the baseball just before it strikes the building.
53. A boy 12.0 m above the ground in a tree throws a ball for his dog, who is standing right below the tree and starts running the instant the ball is thrown. If the boy throws the ball horizontally at 8.50 m/s (a) how fast must the dog run to catch the ball just as it reaches the ground, and (b) how far from the tree will the dog catch the ball?
54. Suppose the boy in the previous problem throws the ball upward at 60.0° above the horizontal, but all else is the same. Repeat parts (a) and (b) of that problem.
55. A firefighting crew uses a water cannon that shoots water at 25.0 m/s at a fixed angle of 53.0° above the horizontal. The firefighters want to direct the water at a blaze that is 10.0 m above ground level. How far from the building should they position their cannon? There are two possibilities; can you get them both?
56. A gun shoots a shell into the air with an initial velocity of 100.0 m/s, 60.0° above the horizontal on level ground. Sketch quantitative graphs of the shell’s horizontal and vertical velocity components as functions of time for the complete motion.
57. Look out! A snowball rolls off a barn roof that slopes downward at an angle of 40.0°. (See Figure 3.42.) The edge of the roof is 14.0 m above the ground, and the snowball has a speed of 7.00 m/s as it rolls off the roof. Ignore air resistance. How far from the edge of the barn does the snowball strike the ground if it doesn’t strike anything else while falling?
58. Spiraling up. It is common to see birds of prey rising upward on thermals. The paths they take may be spiral-like. You can model the spiral motion as uniform circular motion combined with a constant upward velocity. Assume a bird completes a circle of radius 8.00 m every 5.00 s and rises vertically at a rate of 3.00 m/s. Determine: (a) the speed of the bird relative to the ground; (b) the bird’s acceleration (magnitude and direction); and (c) the angle between the bird’s velocity vector and the horizontal.
59. A water hose is used to fill a large cylindrical storage tank of diameter D and height 2D The hose shoots the water at 45° above the horizontal from the same level as the base of the tank and is a distance 6D away (Fig. 3.43). For what range of launch speeds will the water enter the tank? Ignore air resistance, and express your answer in terms of D and g.
60. A world record. In the shot put, a standard track-and-field event, a 7.3 kg object (the shot) is thrown by releasing it at approximately 40° over a straight left leg. The world record for distance, set by Randy Barnes in 1990, is 23.11 m. Assuming that Barnes released the shot put at 40.0° from a height of 2.00 m above the ground, with what speed, in and mph, did he release it?
61. A Ferris wheel with radius 14.0 m is turning about a horizontal axis through its center, as shown in Figure 3.44. The linear speed of a passenger on the rim is constant and equal to 7.00 m/s. What are the magnitude and direction of the passenger’s acceleration as she passes through (a) the lowest point in her circular motion and (b) the highest point in her circular motion? (c) How much time does it take the Ferris wheel to make one revolution?
62. Leaping the river, II. A physics professor did daredevil stunts in his spare time. His last stunt was an attempt to jump across a river on a motorcycle. (See Figure 3.45.) The takeoff ramp was inclined at 53.0°, the river was 40.0 m wide, and the far bank was 15.0 m lower than the top of the ramp. The river itself
was 100 m below the ramp. You can ignore air resistance. (a) What should his speed have been at the top of the ramp for him to have just made it to the edge of the far bank? (b) If his speed was only half the value found in (a), where did he land?
63. A 76.0 kg boulder is rolling horizontally at the top of a vertical cliff that is 20.0 m above the surface of a lake, as shown in Figure 3.46. The top of the vertical face of a dam is located 100.0 m from the foot of the cliff, with the top of the dam level with the surface of the water in the lake. A level plain is 25.0 m below the top of the dam. (a) What must the minimum speed of the rock be just as it leaves the cliff so that it will travel to the plain without striking the dam? (b) How far from the foot of the dam does the rock hit the plain?
64. A batter hits a baseball at a speed of 35.0 m/s and an angle of 65.0o above the horizontal. At the same instant, an out fielder 70.0 m away begins running away from the batter in the line of the ball’s flight, hoping to catch it. How fast must the outfielder run to catch the ball? (Ignore air resistance, and assume the fielder catches the ball at the same height at which it left the bat.)
65. A shell is launched at 150 m/s 53° above the horizontal. When it has reached its highest point, it launches a projectile at a velocity of 100.0 m/s 30.0° above the horizontal relative to the shell. Find (a) the maximum height about the ground that the projectile reaches and (b) its distance from the place where the shell was fired to its landing place when it eventually falls back to the ground.
1. A warehouse worker pushes a crate along the floor, as shown in Figure 4.33, by a force of 10 N that points downward at an angle of 45° below the horizontal. Find the horizontal and vertical components of the push.
2. Two dogs pull horizontally on ropes attached to a post; the angle between the ropes is 60.0°. If dog A exerts a force of 270 N and dog B exerts a force of 300 N, find the magnitude of the resultant force and the angle it makes with dog A’s rope.
3. A man is dragging a trunk up the loading ramp of a mover’s truck. The ramp has a slope angle of 20.0°, and the man pulls upward with a force of magnitude 375 N whose direction makes an angle of 30.0° with the ramp. (See Figure 4.34.) Find the horizontal and vertical components of the force F
4. Jaw injury. Due to a jaw injury, a patient must wear a strap (see Figure 4.35) that produces a net upward force of 5.00 N on his chin. The tension is the same throughout the strap. To what tension must the strap be adjusted to provide the necessary upward force?
5. Workmen are trying to free an SUV stuck in the mud. To extricate the vehicle, they use three horizontal ropes, producing the force vectors shown in Figure 4.36. (a) Find the x and y components of each of the three pulls. (b) Use the components to find the magnitude and direction of the resultant of the three pulls.
6. A box rests on a frozen pond, which serves as a frictionless horizontal surface. If a fisherman applies a horizontal force with magnitude 48.0 N to the box and produces an acceleration of magnitude 3.00 m/s2. What is the mass of the box?
7. In outer space, a constant net force of magnitude 140 N is exerted on a 32.5 kg probe initially at rest. (a) What acceleration does this force produce? (b) How far does the probe travel in 10.0 s?
8. A 68.5 kg skater moving initially at 2.40 m/s on rough horizontal ice comes to rest uniformly in 3.52 s due to friction from the ice. What force does friction exert on the skater?
9. Animal dynamics. An adult 68 kg cheetah can accelerate from rest to (45 mph) in 2.0 s. Assuming constant acceleration, (a) find the net external force causing this acceleration. (b) Where does the force come from? That is, what exerts the force on the cheetah?
10. A hockey puck with mass 0.160 kg is at rest on the horizontal, frictionless surface of a rink. A player applies a force of 0.250 N to the puck, parallel to the surface of the ice, and continues to apply this force for 2.00 s. What are the position and speed of the puck at the end of that time?
11. A dock worker applies a constant horizontal force of 80.0 N to a block of ice on a smooth horizontal floor. The frictional force is negligible. The block starts from rest and moves 11.0 m in the first 5.00 s. What is the mass of the block of ice?
12. (a) What is the mass of a book that weighs 3.20 N in the laboratory? (b) In the same lab, what is the weight of a dog whose mass is 14.0 kg?
13. Superman throws a 2400-N boulder at an adversary. What horizontal force must Superman apply to the boulder to give it a horizontal acceleration of 12.0 m/s2?
14. (a) How many newtons does a 150 lb person weigh? (b) Should a veterinarian be skeptical if someone said that her adult collie weighed 40 N? (c) Should a nurse question a med ical chart which showed that an average-looking patient had a mass of 200 kg?
15. (a) An ordinary flea has a mass of 210 mg. How many newtons does it weigh? (b) The mass of a typical froghopper is 12.3 mg. How many newtons does it weigh? (c) A house cat typically weighs 45 N. How many pounds does it weigh and what is its mass in kilograms?
16. An astronaut’s pack weighs 17.5 N when she is on earth but only 3.24 N when she is at the surface of an asteroid. (a) What is the acceleration due to gravity on this asteroid? (b) What is the mass of the pack on the asteroid?
19. What does a 138 N rock weigh if it is accelerating (a) upward at 12 m/s2 (b) downward at 3.5 m/s2 (c) What would be the answers to parts (a) and (b) if the rock had a mass of 138 kg? (d) What would be the answers to parts (a) and (b) if the rock were moving with a constant upward velocity of 23 m/s?
20. At the surface of Jupiter’s moon Io, the acceleration due to gravity is 1.81 m/s2. If a piece of ice weighs 44.0 N at the surface of the earth, (a) what is its mass on the earth’s surface? (b) What are its mass and weight on the surface of Io?
21. A scientific instrument that weighs 85.2 N on the earth weighs 32.2 N at the surface of Mercury. (a) What is the acceleration due to gravity on Mercury? (b) What is the instrument’s mass on earth and on Mercury?
22. Planet X! When venturing forth on Planet X, you throw a 5.24 kg rock upward at 13.0 m/s and find that it returns to the same level 1.51 s later. What does the rock weigh on Planet X?
23. The driver of a 1750 kg car traveling on a horizontal road at 110 km/h suddenly applies the brakes. Due to a slippery pavement, the friction of the road on the tires of the car, which is what slows down the car, is 25% of the weight of the car. (a) What is the acceleration of the car? (b) How many meters does it travel before stopping under these conditions?
24. You drag a heavy box along a rough horizontal floor by a horizontal rope. Identify the reaction force to each of the following forces: (a) the pull of the rope on the box, (b) the friction force on the box, (c) the normal force on the box, and (d) the weight of the box.
25. Imagine that you are holding a book weighing 4 N at rest on the palm of your hand. Complete the following sentences: (a) A downward force of magnitude 4 N is exerted on the book by . (b) An upward force of magnitude is exerted on by the hand. (c) Is the upward force in part (b) the reaction to the downward force in part (a)? d) The reaction to the force in part (a) is a force of magnitude, exerted on by . Its direction is . (e) The reaction to the force in part (b) is a force of magnitude, exerted on by . Its direction is . (f) The forces in parts (a) and (b) are “equal and opposite” because of Newton’s law. (g) The forces in parts (b) and (e) are “equal and opposite” because of Newton’s law.
26. Suppose now that you exert an upward force of magnitude 5 N on the book in the previous problem. (a) Does the book remain in equilibrium? (b) Is the force exerted on the book by your hand equal and opposite to the force exerted on the book by the earth? (c) Is the force exerted on the book by the earth equal and opposite to the force exerted on the earth by the book? (d) Is the force exerted on the book by your hand equal and opposite to the force exerted on your hand by the book? Finally, suppose that you snatch your hand away while the book is moving upward. (e) How many forces then act on the book? (f) Is the book in equilibrium?
27. The upward normal force exerted by the floor is 620 N on an elevator passenger who weighs 650 N. What are the reaction forces to these two forces? Is the passenger accelerating? If so, what are the magnitude and direction of the acceleration?
28. A person throws a 2.5 lb stone into the air with an initial upward speed of 15 ft/s. Make a free-body diagram for this stone (a) after it is free of the person’s hand and is traveling upward, (b) at its highest point, (c) when it is traveling down ward, and (d) while it is being thrown upward, but is still in contact with the person’s hand.
29. The driver of a car traveling at 65 mph suddenly hits his brakes on a horizontal highway. (a) Make a free-body diagram of the car while it is slowing down. (b) Make a free-body diagram of a passenger in a car that is accelerating on a freeway entrance ramp.
30. A tennis ball traveling horizontally at 22 m/s suddenly hits a vertical brick wall and bounces back with a horizontal velocity of 18 m/s. Make a free-body diagram of this ball (a) just before it hits the wall, (b) just after it has bounced free of the wall, and (c) while it is in contact with the wall.
31. Two crates, A and B, sit at rest side by side on a frictionless horizontal surface. The crates have masses mA and mB . A horizontal force F is applied to crate A and the two crates move off to the right. (a) Draw clearly labeled free-body diagrams for crate A and for crate B. Indicate which pairs of forces, if any, are third-law action–reaction pairs. (b) If the magnitude of force F is less than the total weight of the two crates, will it cause the crates to move? Explain.
32. A ball is hanging from a long string that is tied to the ceiling of a train car traveling eastward on horizontal tracks. An observer inside the train car sees the ball hang motionless. Draw a clearly labeled free-body diagram for the ball if (a) the train has a uniform velocity, and (b) the train is speeding up uniformly. Is the net force on the ball zero in either case? Explain.
33. A person drags her 65 N suitcase along the rough horizontal floor by pulling upward at 30° above the horizontal with a 50 N force. Make a free-body diagram of this suitcase.
34. A factory worker pushes horizontally on a 250 N crate with a force of 75 N on a horizontal rough floor. A 135 N crate rests on top of the one being pushed and moves along with it. Make a free-body diagram of each crate if the friction force exerted by the floor is less than the worker’s push.
35. A dock worker pulls two boxes connected by a rope on a horizontal floor, as shown in Figure 4.37. All the ropes are horizontal, and there is some friction with the floor. Make a free-body diagram of each box.
36. A hospital orderly pushes horizontally on two boxes of equipment on a rough horizontal floor, as shown in Figure 4.38. Make a free-body diagram of each box.
37. A uniform 25.0 kg chain 2.00 m long supports a 50.0 kg chandelier in a large public building. Find the tension in (a) the bottom link of the chain, (b) the top link of the chain, and (c) the middle link of the chain.
38. An acrobat is hanging by his feet from a trapeze, while supporting with his hands a second acrobat who hangs below him. Draw separate free-body diagrams for the two acrobats.
39. A 275 N bucket is lifted with an acceleration of 2.50 m/s2 by a 125 N uniform vertical chain. Start each of the following parts with a free-body diagram. Find the tension in (a) the top link of the chain, (b) the bottom link of the chain, and (c) the middle link of the chain.
40. Human biomechanics. World-class sprinters can spring out of the starting blocks with an acceleration that is essentially horizontal and of magnitude 15 m/s2 (a) How much horizontal force must a 55-kg sprinter exert on the starting blocks during a start to produce this acceleration? (b) What exerts the force that propels the sprinter, the blocks or the sprinter himself?
41. A chair of mass 12.0 kg is sitting on the horizontal floor; the floor is not frictionless. You push on the chair with a force F = 40.0 N that is directed at an angle of 37 below the horizontal, and the chair slides along the floor. (a) Draw a clearly labeled free-body diagram for the chair. (b) Use your diagram and Newton’s laws to calculate the normal force that the floor exerts on the chair.
42. Human biomechanics. The fastest pitched baseball was measured at 46 m/s. Typically, a baseball has a mass of 145 g. If the pitcher exerted his force (assumed to be horizontal and constant) over a distance of 1.0 m, (a) what force did he produce on the ball during this record-setting pitch? (b) Make free-body diagrams of the ball during the pitch and just after it has left the pitcher’s hand.
43. You walk into an elevator, step onto a scale, and push the “up” button. You also recall that your normal weight is 625 N. Start each of the following parts with a free-body diagram. (a) If the elevator has an acceleration of magnitude 2.50 m/s2, what does the scale read? (b) If you start holding a 3.85 kg package by a light vertical string, what will be the tension in this string once the elevator begins accelerating?
44. A truck is pulling a car on a horizontal highway using a horizontal rope. The car is in neutral gear, so we can assume that there is no appreciable friction between its tires and the highway. As the truck is accelerating to highway speeds, draw a free-body diagram of (a) the car and (b) the truck. (c) What force accelerates this system forward? Explain how this force originates.
45. The space shuttle. During the first stage of its launch, a space shuttle goes from rest to 4973 km/h while rising a vertical distance of 45 km. Assume constant acceleration and no variation in g over this distance. (a) What is the acceleration of the shuttle? (b) If a 55.0 kg astronaut is standing on a scale inside the shuttle during this launch, how hard will the scale push on her? Start with a free-body diagram of the astronaut. (c) If this astronaut did not realize that the shuttle had left the launch pad, what would she think were her weight and mass?
46. A woman is standing in an elevator holding her 2.5-kg briefcase by its handles. Draw a free-body diagram for the briefcase if the elevator is accelerating downward at 1.50 m/s2 and calculate the downward pull of the briefcase on the woman’s arm while the elevator is accelerating.
47. An advertisement claims that a particular automobile can “stop on a dime.” What net force would actually be necessary to stop a 850-kg automobile traveling initially at 45.0 km/h in a distance equal to the diameter of a dime, which is 1.8 cm?
48. A rifle shoots a 4.20 g bullet out of its barrel. The bullet has a muzzle velocity of 965 m/s just as it leaves the barrel. Assuming a constant horizontal acceleration over a distance of 45.0 cm starting from rest, with no friction between the bullet and the barrel, (a) what force does the rifle exert on the
bullet while it is in the barrel? (b) Draw a free-body diagram of the bullet (i) while it is in the barrel and (ii) just after it has left the barrel. (c) How many g’s of acceleration does the rifle give this bullet? (d) For how long a time is the bullet in the barrel?
49. A parachutist relies on air resistance (mainly on her parachute) to decrease her downward velocity. She and her parachute have a mass of 55.0 kg, and at a particular moment air resistance exerts a total upward force of 620 N on her and her parachute. (a) What is the weight of the parachutist? (b) Draw a free-body diagram for the parachutist (see Section 4.6). Use that diagram to calculate the net force on the parachutist. Is the net force upward or downward? (c) What is the acceleration (magnitude and direction) of the parachutist?
50. A spacecraft descends vertically near the surface of Planet X. An upward thrust of 25.0 kN from its engines slows it down at a rate of 1.20 m/s2, but it speeds up at a rate of 0.80 m/s2 with an upward thrust of 10.0 kN. (a) In each case, what is the direction of the acceleration of the spacecraft? (b) Draw a free- body diagram for the spacecraft. In each case, speeding up or slowing down, what is the direction of the net force on the spacecraft? (c) Apply Newton’s second law to each case, slowing down or speeding up, and use this to find the spacecraft’s weight near the surface of Planet X.
51. A standing vertical jump. Basketball player Darrell Griffith is on record as attaining a standing vertical jump of 1.2 m (4 ft). (This means that he moved upward by 1.2 m after his feet left the floor.) Griffith weighed 890 N (200 lb). (a) What was his speed as he left the floor? (b) If the time of the part of the jump before his feet left the floor was 0.300 s, what were the magnitude and direction of his acceleration (assuming it to be constant) while he was pushing against the floor? (c) Draw a free-body diagram of Griffith during the jump. (d) Use Newton’s laws and the results of part (b) to calculate the force he applied to the ground during his jump.
52. You leave the doctor’s office after your annual checkup and recall that you weighed 683 N in her office. You then get into an elevator that, conveniently, has a scale. Find the magnitude and direction of the elevator’s acceleration if the scale reads (a) 725 N, (b) 595 N.
53. Human biomechanics. The fastest served tennis ball, served by “Big Bill” Tilden in 1931, was measured at 73.14 m/s. The mass of a tennis ball is 57 g, and the ball is typically in contact with the tennis racquet for 30.0 ms, with the ball starting from rest. Assuming constant acceleration, (a) what force did Big Bill’s tennis racquet exert on the tennis ball if he hit it essentially horizontally? (b) Make free-body diagrams of the tennis ball during the serve and just after it has moved free of the racquet.
54. Extraterrestrial physics. You have landed on an unknown planet, Newtonia, and want to know what objects will weigh there. You find that when a certain tool is pushed on a frictionless horizontal surface by a 12.0 N force, it moves 16.0 m in the first 2.00 s, starting from rest. You next observe that if you release this tool from rest at 10.0 m above the ground, it takes 2.58 s to reach the ground. What does the tool weigh on Newtonia and what would it weigh on Earth?
55. An athlete whose mass is 90.0 kg is performing weight lifting exercises. Starting from the rest position, he lifts, with constant acceleration, a barbell that weighs 490 N. He lifts the barbell a distance of 0.60 m in 1.6 s. (a) Draw a clearly labeled free-body force diagram for the barbell and for the athlete. (b) Use the diagrams in part (a) and Newton’s laws to find the total force that the ground exerts on the athlete’s feet as he lifts the barbell.
56. Jumping to the ground. A 75.0 kg man steps off a platform 3.10 m above the ground. He keeps his legs straight as he falls, but at the moment his feet touch the ground his knees begin to bend, and, treated as a particle, he moves an additional 0.60 m before coming to rest. (a) What is his speed at the instant his feet touch the ground? (b) Treating him as a particle, what are the magnitude and direction of his acceleration as he slows down if the acceleration is constant? (c) Draw a free-body diagram of this man as he is slowing down. (d) Use Newton’s laws and the results of part (b) to calculate the force the ground exerts on him while he is slowing down. Express this force in newtons and also as a multiple of the man’s weight. (e) What are the magnitude and direction of the reaction force to the force you found in part (c)?
57. An electron (mass = 9.11×10-31 kg) leaves one end of a TV picture tube with zero initial speed and travels in a straight line to the accelerating grid, which is 1.80 cm away. It reaches the grid with a speed of 3.00 × 108 m/s. If the accelerating force is constant, compute (a) the acceleration of the electron, (b) the time it takes the electron to reach the grid, and (c) the net force that is accelerating the electron, in newtons. (You can ignore the gravitational force on the electron.)
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