Serway/Jewett: PSE 8e Problems Set – Ch. 04  Serway/Jewett, Physics for Scientists and Engineers, 8/ePSE 8e – Chapter 04 Motion in Two Dimensions Questions and Problems Objective Questions denotes answer available in Student Solutions Manual/Study Guide 1. Figure OQ4.1 shows a bird’seye view of a car going around a highway curve. As the car moves from point 1 to point 2, its speed doubles. Which of the vectors (a) through (e) shows the direction of the car’s average acceleration between these two points? Figure OQ4.1 2. Entering his dorm room, a student tosses his book bag to the right and upward at an angle of 45 with the horizontal (Fig. OQ4.2). Air resistance does not affect the bag. The bag moves through point _{ } immediately after it leaves the student’s hand, through point _{ }at the top of its flight, and through point _{ } immediately before it lands on the top bunk bed. (i) Rank the following horizontal and vertical velocity components from the largest to the smallest. (a) v_{}_{x} (b) v_{}_{y} (c) v_{}_{x} (d) v_{}_{y} (e) v_{}_{y}. Note that zero is larger than a negative number. If two quantities are equal, show them as equal in your list. If any quantity is equal to zero, show that fact in your list. (ii) Similarly, rank the following acceleration components. (a) a_{}_{x} (b) a_{}_{y} (c) a_{}_{x} (d) a_{}_{y} (e) a_{}_{y}_{.} 3. In which of the following situations is the moving object appropriately modeled as a projectile? Choose all correct answers. (a) A shoe is tossed in an arbitrary direction. (b) A jet airplane crosses the sky with its engines thrusting the plane forward. (c) A rocket leaves the launch pad. (d) A rocket moves through the sky, at much less than the speed of sound, after its fuel has been used up. (e) A diver throws a stone under water. 4. An astronaut hits a golf ball on the Moon. Which of the following quantities, if any, remain constant as a ball travels through the vacuum there? (a) speed (b) acceleration (c) horizontal component of velocity (d) vertical component of velocity (e) velocity 5. A projectile is launched on the Earth with a certain initial velocity and moves without air resistance. Another projectile is launched with the same initial velocity on the Moon, where the acceleration due to gravity is onesixth as large. How does the range of the projectile on the Moon compare with that of the projectile on the Earth? (a) It is onesixth as large. (b) It is the same. (c) It is times larger. (d) It is 6 times larger. (e) It is 36 times larger. 6. A projectile is launched on the Earth with a certain initial velocity and moves without air resistance. Another projectile is launched with the same initial velocity on the Moon, where the acceleration due to gravity is onesixth as large. How does the maximum altitude of the projectile on the Moon compare with that of the projectile on the Earth? (a) It is onesixth as large. (b) It is the same. (c) It is times larger. (d) It is 6 times larger. (e) It is 36 times larger. 7. Does a car moving around a circular track with constant speed have (a) zero acceleration, (b) an acceleration in the direction of its velocity, (c) an acceleration directed away from the center of its path, (d) an acceleration directed toward the center of its path, or (e) an acceleration with a direction that cannot be determined from the given information? 8. A rubber stopper on the end of a string is swung steadily in a horizontal circle. In one trial, it moves at speed v in a circle of radius r. In a second trial, it moves at a higher speed 3v in a circle of radius 3r. In this second trial, is its acceleration (a) the same as in the first trial, (b) three times larger, (c) onethird as large, (d) nine times larger, or (e) oneninth as large? 9. A set of keys on the end of a string is swung steadily in a horizontal circle. In one trial, it moves at speed v in a circle of radius r. In a second trial, it moves at a higher speed 4v in a circle of radius 4r. In the second trial, how does the period of its motion compare with its period in the first trial? (a) It is the same as in the first trial. (b) It is 4 times larger. (c) It is onefourth as large. (d) It is 16 times larger. (e) It is onesixteenth as large. 10. A certain light truck can go around a curve having a radius of 150 m with a maximum speed of 32.0 m/s. To have the same acceleration, at what maximum speed can it go around a curve having a radius of 75.0 m? (a) 64 m/s (b) 45 m/s (c) 32 m/s (d) 23 m/s (e) 16 m/s 11. A sailor drops a wrench from the top of a sailboat’s vertical mast while the boat is moving rapidly and steadily straight forward. Where will the wrench hit the deck? (a) ahead of the base of the mast (b) at the base of the mast (c) behind the base of the mast (d) on the windward side of the base of the mast (e) None of the choices (a) through (d) is true. 12. A girl, moving at 8 m/s on inline skates, is overtaking a boy moving at 5 m/s as they both skate on a straight path. The boy tosses a ball backward toward the girl, giving it speed 12 m/s relative to him. What is the speed of the ball relative to the girl, who catches it? (a) (8 + 5 + 12) m/s (b) (8 5 12) m/s (c) (8 + 5 12) m/s (d) (8 5 + 12) m/s (e) (8 + 5 + 12) m/s 13. A student throws a heavy red ball horizontally from a balcony of a tall building with an initial speed v_{i}. At the same time, a second student drops a lighter blue ball from the balcony. Neglecting air resistance, which statement is true? (a) The blue ball reaches the ground first. (b) The balls reach the ground at the same instant. (c) The red ball reaches the ground first. (d) Both balls hit the ground with the same speed. (e) None of statements (a) through (d) is true. 14. A baseball is thrown from the outfield toward the catcher. When the ball reaches its highest point, which statement is true? (a) Its velocity and its acceleration are both zero. (b) Its velocity is not zero, but its acceleration is zero. (c) Its velocity is perpendicular to its acceleration. (d) Its acceleration depends on the angle at which the ball was thrown. (e) None of statements (a) through (d) is true. ^ 1. If you know the position vectors of a particle at two points along its path and also know the time interval during which it moved from one point to the other, can you determine the particle’s instantaneous velocity? Its average velocity? Explain. 2. Construct motion diagrams showing the velocity and acceleration of a projectile at several points along its path, assuming (a) the projectile is launched horizontally and (b) the projectile is launched at an angle with the horizontal. 3. A spacecraft drifts through space at a constant velocity. Suddenly, a gas leak in the side of the spacecraft gives it a constant acceleration in a direction perpendicular to the initial velocity. The orientation of the spacecraft does not change, so the acceleration remains perpendicular to the original direction of the velocity. What is the shape of the path followed by the spacecraft in this situation? 4. Describe how a driver can steer a car traveling at constant speed so that (a) the acceleration is zero or (b) the magnitude of the acceleration remains constant. 5. A projectile is launched at some angle to the horizontal with some initial speed v_{i}, and air resistance is negligible. (a) Is the projectile a freely falling body? (b) What is its acceleration in the vertical direction? (c) What is its acceleration in the horizontal direction? 6. An ice skater is executing a figure eight, consisting of two identically shaped, tangent circular paths. Throughout the first loop she increases her speed uniformly, and during the second loop she moves at a constant speed. Draw a motion diagram showing her velocity and acceleration vectors at several points along the path of motion. 7. Explain whether or not the following particles have an acceleration: (a) a particle moving in a straight line with constant speed and (b) a particle moving around a curve with constant speed. Problems The problems found in this chapter may be assigned online in Enhanced WebAssign. 1. denotes straightforward problem; 2. denotes intermediate problem; 3. denotes challenging problem 1. full solution available in the Student Solutions Manual/Study Guide 1. denotes problems most often assigned with Enhanced WebAssign; these provide students with targeted feedback and either a Master It tutorial or a Watch It solution video. QC denotes asking for quantitative and conceptual reasoning denotes symbolic reasoning problem denotes Master It tutorial available in Enhanced WebAssign denotes guided problem shaded denotes "paired problems" that develop reasoning with symbols and numeric values Section 4.1 The Position, Velocity, and Acceleration Vectors 1. A motorist drives south at 20.0 m/s for 3.00 min, then turns west and travels at 25.0 m/s for 2.00 min, and finally travels northwest at 30.0 m/s for 1.00 min. For this 6.00min trip, find (a) the total vector displacement, (b) the average speed, and (c) the average velocity. Let the positive x axis point east. 2. When the Sun is directly overhead, a hawk dives toward the ground with a constant velocity of 5.00 m/s at 60.0° below the horizontal. Calculate the speed of its shadow on the level ground. 3. A golf ball is hit off a tee at the edge of a cliff. Its x and y coordinates as functions of time are given by x = 18.0t and y = 4.00t – 4.90t^{2}, where x and y are in meters and t is in seconds. (a) Write a vector expression for the ball’s position as a function of time, using the unit vectors and. By taking derivatives, obtain expressions for (b) the velocity vector as a function of time and (c) the acceleration vector as a function of time. (d) Next use unit–vector notation to write expressions for the position, the velocity, and the acceleration of the golf ball at t = 3.00 s. 4. QC The coordinates of an object moving in the xy plane vary with time according to the equations x = –5.00 sin t and y = 4.00 – 5.00 cos t, where is a constant, x and y are in meters, and t is in seconds. (a) Determine the components of velocity of the object at t = 0. (b) Determine the components of acceleration of the object at t = 0. (c) Write expressions for the position vector, the velocity vector, and the acceleration vector of the object at any time t > 0. (d) Describe the path of the object in an xy plot. Section 4.2 TwoDimensional Motion with Constant Acceleration 5. The vector position of a particle varies in time according to the expression , where is in meters and t is in seconds. (a) Find an expression for the velocity of the particle as a function of time. (b) Determine the acceleration of the particle as a function of time. (c) Calculate the particle’s position and velocity at t = 1.00 s. 6. A particle initially located at the origin has an acceleration of m/s^{2} and an initial velocity of m/s. Find (a) the vector position of the particle at any time t, (b) the velocity of the particle at any time t, (c) the coordinates of the particle at t = 2.00 s, and (d) the speed of the particle at t = 2.00 s. 7. A fish swimming in a horizontal plane has velocity m/s at a point in the ocean where the position relative to a certain rock is m. After the fish swims with constant acceleration for 20.0 s, its velocity is m/s. (a) What are the components of the acceleration of the fish? (b) What is the direction of its acceleration with respect to unit vector ? (c) If the fish maintains constant acceleration, where is it at t = 25.0 s and in what direction is it moving? 8. Review. A snowmobile is originally at the point with position vector 29.0 m at 95.0 counterclockwise from the x axis, moving with velocity 4.50 m/s at 40.0. It moves with constant acceleration 1.90 m/s^{2} at 200. After 5.00 s have elapsed, find (a) its velocity and (b) its position vector. Section 4.3 Projectile Motion Note: Ignore air resistance in all problems and take g = 9.80 m/s^{2} at the Earth’s surface. 9. In a local bar, a customer slides an empty beer mug down the counter for a refill. The height of the counter is 1.22 m. The mug slides off the counter and strikes the floor 1.40 m from the base of the counter. (a) With what velocity did the mug leave the counter? (b) What was the direction of the mug’s velocity just before it hit the floor? 10. In a local bar, a customer slides an empty beer mug down the counter for a refill. The height of the counter is h. The mug slides off the counter and strikes the floor at distance d from the base of the counter. (a) With what velocity did the mug leave the counter? (b) What was the direction of the mug’s velocity just before it hit the floor? 11. A projectile is fired in such a way that its horizontal range is equal to three times its maximum height. What is the angle of projection? 12. To start an avalanche on a mountain slope, an artillery shell is fired with an initial velocity of 300 m/s at 55.0° above the horizontal. It explodes on the mountainside 42.0 s after firing. What are the x and y coordinates of the shell where it explodes, relative to its firing point? 13. Chinook salmon are able to move through water especially fast by jumping out of the water periodically. This behavior is called porpoising. Suppose a salmon swimming in still water jumps out of the water with velocity 6.26 m/s at 45.0 above the horizontal, sails through the air a distance L before returning to the water, and then swims the same distance L underwater in a straight, horizontal line with velocity 3.58 m/s before jumping out again. (a) Determine the average velocity of the fish for the entire process of jumping and swimming underwater. (b) Consider the time interval required to travel the entire distance of 2L. By what percentage is this time interval reduced by the jumping/swimming process compared with simply swimming underwater at 3.58 m/s? 14. QC _{ } A rock is thrown upward from level ground in such a way that the maximum height of its flight is equal to its horizontal range R. (a) At what angle is the rock thrown? (b) In terms of its original range R, what is the range R_{max} the rock can attain if it is launched at the same speed but at the optimal angle for maximum range? (c) What If? Would your answer to part (a) be different if the rock is thrown with the same speed on a different planet? Explain. 15. A firefighter, a distance d from a burning building, directs a stream of water from a fire hose at angle _{i} above the horizontal as shown in Figure P4.15. If the initial speed of the stream is v_{i}, at what height h does the water strike the building? Figure P4.15 16. A ball is tossed from an upperstory window of a building. The ball is given an initial velocity of 8.00 m/s at an angle of 20.0° below the horizontal. It strikes the ground 3.00 s later. (a) How far horizontally from the base of the building does the ball strike the ground? (b) Find the height from which the ball was thrown. (c) How long does it take the ball to reach a point 10.0 m below the level of launching? 17. A placekicker must kick a football from a point 36.0 m (about 40 yards) from the goal. Half the crowd hopes the ball will clear the crossbar, which is 3.05 m high. When kicked, the ball leaves the ground with a speed of 20.0 m/s at an angle of 53.0° to the horizontal. (a) By how much does the ball clear or fall short of clearing the crossbar? (b) Does the ball approach the crossbar while still rising or while falling? 18. A landscape architect is planning an artificial waterfall in a city park. Water flowing at 1.70 m/s will leave the end of a horizontal channel at the top of a vertical wall h = 2.35 m high, and from there it will fall into a pool (Fig. P4.18). (a) Will the space behind the waterfall be wide enough for a pedestrian walkway? (b) To sell her plan to the city council, the architect wants to build a model to standard scale, which is onetwelfth actual size. How fast should the water flow in the channel in the model? Figure P4.18 19. A playground is on the flat roof of a city school, 6.00 m above the street below (Fig. P4.19). The vertical wall of the building is h = 7.00 m high, forming a 1mhigh railing around the playground. A ball has fallen to the street below, and a passerby returns it by launching it at an angle of = 53.0° above the horizontal at a point d = 24.0 m from the base of the building wall. The ball takes 2.20 s to reach a point vertically above the wall. (a) Find the speed at which the ball was launched. (b) Find the vertical distance by which the ball clears the wall. (c) Find the horizontal distance from the wall to the point on the roof where the ball lands. Figure P4.19 20. A basketball star covers 2.80 m horizontally in a jump to dunk the ball (Fig. P4.20a). His motion through space can be modeled precisely as that of a particle at his center of mass, which we will define in Chapter 9. His center of mass is at elevation 1.02 m when he leaves the floor. It reaches a maximum height of 1.85 m above the floor and is at elevation 0.900 m when he touches down again. Determine (a) his time of flight (his “hang time”), (b) his horizontal and (c) vertical velocity components at the instant of takeoff, and (d) his takeoff angle. (e) For comparison, determine the hang time of a whitetail deer making a jump (Fig. P4.20b) with centerofmass elevations y_{i} = 1.20 m, y_{max} = 2.50 m, and y_{f} = 0.700 m. Figure P4.20 21. A soccer player kicks a rock horizontally off a 40.0mhigh cliff into a pool of water. If the player hears the sound of the splash 3.00 s later, what was the initial speed given to the rock? Assume the speed of sound in air is 343 m/s. 22. The motion of a human body through space can be modeled as the motion of a particle at the body’s center of mass as we will study in Chapter 9. The components of the displacement of an athlete’s center of mass from the beginning to the end of a certain jump are described by the equations where t is in seconds and is the time at which the athlete ends the jump. Identify (a) the athlete’s position and (b) his vector velocity at the takeoff point. (c) How far did he jump? 23. A student stands at the edge of a cliff and throws a stone horizontally over the edge with a speed of v_{i} = 18.0 m/s. The cliff is h = 50.0 m above a body of water as shown in Figure P4.23. (a) What are the coordinates of the initial position of the stone? (b) What are the components of the initial velocity of the stone? (c) What is the appropriate analysis model for the vertical motion of the stone? (d) What is the appropriate analysis model for the horizontal motion of the stone? (e) Write symbolic equations for the x and y components of the velocity of the stone as a function of time. (f) Write symbolic equations for the position of the stone as a function of time. (g) How long after being released does the stone strike the water below the cliff? (h) With what speed and angle of impact does the stone land? Figure P4.23 24. QC The record distance in the sport of throwing cowpats is 81.1 m. This record toss was set by Steve Uren of the United States in 1981. Assuming the initial launch angle was 45 and neglecting air resistance, determine (a) the initial speed of the projectile and (b) the total time interval the projectile was in flight. (c) How would the answers change if the range were the same but the launch angle were greater than 45? Explain. 25. A boy stands on a diving board and tosses a stone into a swimming pool. The stone is thrown from a height of 2.50 m above the water surface with a velocity of 4.00 m/s at an angle of 60.0° above the horizontal. As the stone strikes the water surface, it immediately slows down to exactly half the speed it had when it struck the water and maintains that speed while in the water. After the stone enters the water, it moves in a straight line in the direction of the velocity it had when it struck the water. If the pool is 3.00 m deep, how much time elapses between when the stone is thrown and when it strikes the bottom of the pool? 26. _{ } A projectile is fired from the top of a cliff of height h above the ocean below. The projectile is fired at an angle above the horizontal and with an initial speed v_{i}. (a) Find a symbolic expression in terms of the variable v_{i}, g, and for the time at which the projectile reaches its maximum height. (b) Using the result of part (a), find an expression for the maximum height h_{max} above the ocean attained by the projectile in terms of h, v_{i}, g, and . Section 4.4 Analysis Model: Particle in Uniform Circular Motion Note: Problems 6 and 13 in Chapter 6 can also be assigned with this section. 27. The athlete shown in Figure P4.27 rotates a 1.00kg discus along a circular path of radius 1.06 m. The maximum speed of the discus is 20.0 m/s. Determine the magnitude of the maximum radial acceleration of the discus. Figure P4.27 28. In Example 4.6, we found the centripetal acceleration of the Earth as it revolves around the Sun. From information on the endpapers of this book, compute the centripetal acceleration of a point on the surface of the Earth at the equator caused by the rotation of the Earth about its axis. 29. Review. The 20g centrifuge at NASA’s Ames Research Center in Mountain View, California, is a horizontal, cylindrical tube 58 ft long and is represented in Figure P4.29. Assume an astronaut in training sits in a seat at one end, facing the axis of rotation 29.0 ft away. Determine the rotation rate, in revolutions per second, required to give the astronaut a centripetal acceleration of 20.0g. Figure P4.29 30. An athlete swings a ball, connected to the end of a chain, in a horizontal circle. The athlete is able to rotate the ball at the rate of 8.00 rev/s when the length of the chain is 0.600 m. When he increases the length to 0.900 m, he is able to rotate the ball only 6.00 rev/s. (a) Which rate of rotation gives the greater speed for the ball? (b) What is the centripetal acceleration of the ball at 8.00 rev/s? (c) What is the centripetal acceleration at 6.00 rev/s? Section 4.5 Tangential and Radial Acceleration 31. _{ }A train slows down as it rounds a sharp horizontal turn, going from 90.0 km/h to 50.0 km/h in the 15.0 s it takes to round the bend. The radius of the curve is 150 m. Compute the acceleration at the moment the train speed reaches 50.0 km/h. Assume the train continues to slow down at this time at the same rate. 32. Figure P4.32 represents the total acceleration of a particle moving clockwise in a circle of radius 2.50 m at a certain instant of time. For that instant, find (a) the radial acceleration of the particle, (b) the speed of the particle, and (c) its tangential acceleration. Figure P4.32 33. (a) Can a particle moving with instantaneous speed 3.00 m/s on a path with radius of curvature 2.00 m have an acceleration of magnitude 6.00 m/s^{2}? (b) Can it have an acceleration of magnitude 4.00 m/s^{2}? In each case, if the answer is yes, explain how it can happen; if the answer is no, explain why not. 34. A ball swings in a vertical circle at the end of a rope 1.50 m long. When the ball is 36.9° past the lowest point on its way up, its total acceleration is (22.5 20.2) m/s^{2}. For that instant, (a) sketch a vector diagram showing the components of its acceleration, (b) determine the magnitude of its radial acceleration, and (c) determine the speed and velocity of the ball. Section 4.6 Relative Velocity and Relative Acceleration 35. A police car traveling at 95.0 km/h is traveling west, chasing a motorist traveling at 80.0 km/h. (a) What is the velocity of the motorist relative to the police car? (b) What is the velocity of the police car relative to the motorist? (c) If they are originally 250 m apart, in what time interval will the police car overtake the motorist? 36. _{ } A moving beltway at an airport has a speed v_{1} and a length L. A woman stands on the beltway as it moves from one end to the other, while a man in a hurry to reach his flight walks on the beltway with a speed of v_{2} relative to the moving beltway. (a) What time interval is required for the woman to travel the distance L? (b) What time interval is required for the man to travel this distance? (c) A second beltway is located next to the first one. It is identical to the first one but moves in the opposite direction at speed v1. Just as the man steps onto the beginning of the beltway and begins to walk at speed v2 relative to his beltway, a child steps on the other end of the adjacent beltway. The child stands at rest relative to this second beltway. How long after stepping on the beltway does the man pass the child? 37. An airplane maintains a speed of 630 km/h relative to the air it is flying through as it makes a trip to a city 750 km away to the north. (a) What time interval is required for the trip if the plane flies through a headwind blowing at 35.0 km/h toward the south? (b) What time interval is required if there is a tailwind with the same speed? (c) What time interval is required if there is a crosswind blowing at 35.0 km/h to the east relative to the ground? 38. A car travels due east with a speed of 50.0 km/h. Raindrops are falling at a constant speed vertically with respect to the Earth. The traces of the rain on the side windows of the car make an angle of 60.0° with the vertical. Find the velocity of the rain with respect to (a) the car and (b) the Earth. 39. QC A bolt drops from the ceiling of a moving train car that is accelerating northward at a rate of 2.50 m/s^{2}. (a) What is the acceleration of the bolt relative to the train car? (b) What is the acceleration of the bolt relative to the Earth? (c) Describe the trajectory of the bolt as seen by an observer inside the train car. (d) Describe the trajectory of the bolt as seen by an observer fixed on the Earth. 40. QC _{ }A river has a steady speed of 0.500 m/s. A student swims upstream a distance of 1.00 km and swims back to the starting point. (a) If the student can swim at a speed of 1.20 m/s in still water, how long does the trip take? (b) How much time is required in still water for the same length swim? (c) Intuitively, why does the swim take longer when there is a current? 41. QC _{ } A river flows with a steady speed v. A student swims upstream a distance d and then back to the starting point. The student can swim at speed c in still water. (a) In terms of d, v, and c, what time interval is required for the round trip? (b) What time interval would be required if the water were still? (c) Which time interval is larger? Explain whether it is always larger. 42. QC A farm truck moves due east with a constant velocity of 9.50 m/s on a limitless, horizontal stretch of road. A boy riding on the back of the truck throws a can of soda upward (Fig. P4.42) and catches the projectile at the same location on the truck bed, but 16.0 m farther down the road. (a) In the frame of reference of the truck, at what angle to the vertical does the boy throw the can? (b) What is the initial speed of the can relative to the truck? (c) What is the shape of the can’s trajectory as seen by the boy? An observer on the ground watches the boy throw the can and catch it. In this observer’s frame of reference, (d) describe the shape of the can’s path and (e) determine the initial velocity of the can. Figure P4.42 43. A science student is riding on a flatcar of a train traveling along a straight, horizontal track at a constant speed of 10.0 m/s. The student throws a ball into the air along a path that he judges to make an initial angle of 60.0° with the horizontal and to be in line with the track. The student’s professor, who is standing on the ground nearby, observes the ball to rise vertically. How high does she see the ball rise? 44. A Coast Guard cutter detects an unidentified ship at a distance of 20.0 km in the direction 15.0° east of north. The ship is traveling at 26.0 km/h on a course at 40.0° east of north. The Coast Guard wishes to send a speedboat to intercept and investigate the vessel. If the speedboat travels at 50.0 km/h, in what direction should it head? Express the direction as a compass bearing with respect to due north. Additional Problems 45. Why is the following situation impossible? A normally proportioned adult walks briskly along a straight line in the +x direction, standing straight up and holding his right arm vertical and next to his body so that the arm does not swing. His right hand holds a ball at his side a distance h above the floor. When the ball passes above a point marked as x = 0 on the horizontal floor, he opens his fingers to release the ball from rest relative to his hand. The ball strikes the ground for the first time at position x = 7.00h. 46. A particle starts from the origin with velocity 5 m/s at t = 0 and moves in the xy plane with a varying acceleration given by = (6), where is in meters per second squared and t is in seconds. (a) Determine the velocity of the particle as a function of time. (b) Determine the position of the particle as a function of time. 47. The “Vomit Comet.” In microgravity astronaut training and equipment testing, NASA flies a KC135A aircraft along a parabolic flight path. As shown in Figure P4.47, the aircraft climbs from 24 000 ft to 31 000 ft, where it enters a parabolic path with a velocity of 143 m/s nose high at 45.0° and exits with velocity 143 m/s at 45.0° nose low. During this portion of the flight, the aircraft and objects inside its padded cabin are in free fall; astronauts and equipment float freely as if there were no gravity. What are the aircraft’s (a) speed and (b) altitude at the top of the maneuver? (c) What is the time interval spent in microgravity? Figure P4.47 48. _{ } A boy throws a stone horizontally from the top of a cliff of height h toward the ocean below. The stone strikes the ocean at distance d from the base of the cliff. In terms of h, d, and g, find expressions for (a) the time t at which the stone lands in the ocean, (b) the initial speed of the stone, (c) the speed of the stone immediately before it reaches the ocean, and (d) the direction of the stone’s velocity immediately before it reaches the ocean. 49. Lisa in her Lamborghini accelerates at the rate of (3.00 – 2.00) m/s^{2}, while Jill in her Jaguar accelerates at (1.00 + 3.00) m/s^{2}. They both start from rest at the origin of an xy coordinate system. After 5.00 s, (a) what is Lisa’s speed with respect to Jill, (b) how far apart are they, and (c) what is Lisa’s acceleration relative to Jill? 50. A basketball player is standing on the floor 10.0 m from the basket as in Figure P4.50. The height of the basket is 3.05 m, and he shoots the ball at a 40.0^{o} angle with the horizontal from a height of 2.00 m. (a) What is the acceleration of the basketball at the highest point in its trajectory? (b) At what speed must the player throw the basketball so that the ball goes through the hoop without striking the backboard? Figure P4.50 51. A flea is at point _{ } on a horizontal turntable, 10.0 cm from the center. The turntable is rotating at 33.3 rev/min in the clockwise direction. The flea jumps straight up to a height of 5.00 cm. At takeoff, it gives itself no horizontal velocity relative to the turntable. The flea lands on the turntable at point_{}. Choose the origin of coordinates to be at the center of the turntable and the positive x axis passing through _{ } at the moment of takeoff. Then the original position of the flea is 10.0 cm. (a) Find the position of point _{ }when the flea lands. (b) Find the position of point _{ } when the flea lands. 52. _{ } As some molten metal splashes, one droplet flies off to the east with initial velocity v_{i} at angle _{i} above the horizontal, and another droplet flies off to the west with the same speed at the same angle above the horizontal as shown in Figure P4.52. In terms of v_{i} and _{i}, find the distance between the two droplets as a function of time. Figure P4.52 53. Why is the following situation impossible? Manny Ramírez hits a home run so that the baseball just clears the top row of bleachers, 24.0 m high, located 130 m from home plate. The ball is hit at 41.7 m/s at an angle of 35.0° to the horizontal, and air resistance is negligible. 54. A pendulum with a cord of length r = 1.00 m swings in a vertical plane (Fig. P4.54). When the pendulum is in the two horizontal positions = 90.0° and = 270°, its speed is 5.00 m/s. Find the magnitude of (a) the radial acceleration and (b) the tangential acceleration for these positions. (c) Draw vector diagrams to determine the direction of the total acceleration for these two positions. (d) Calculate the magnitude and direction of the total acceleration at these two positions. Figure P4.54 55. An astronaut on the surface of the Moon fires a cannon to launch an experiment package, which leaves the barrel moving horizontally. Assume the freefall acceleration on the Moon is onesixth of that on the Earth. (a) What must the muzzle speed of the package be so that it travels completely around the Moon and returns to its original location? (b) What time interval does this trip around the Moon require? 56. QC A projectile is launched from the point (x = 0, y = 0), with velocity (12.0 + 49.0) m/s, at t = 0. (a) Make a table listing the projectile’s distance  from the origin at the end of each second thereafter, for 0 ≤ t ≤ 10 s. Tabulating the x and y coordinates and the components of velocity v_{x} and v_{y} will also be useful. (b) Notice that the projectile’s distance from its starting point increases with time, goes through a maximum, and starts to decrease. Prove that the distance is a maximum when the position vector is perpendicular to the velocity. Suggestion: Argue that if is not perpendicular to , then  must be increasing or decreasing. (c) Determine the magnitude of the maximum displacement. (d) Explain your method for solving part (c). 57. QC A spring cannon is located at the edge of a table that is 1.20 m above the floor. A steel ball is launched from the cannon with speed v_{i} at 35.0 above the horizontal. (a) Find the horizontal position of the ball as a function of v_{i} at the instant it lands on the floor. We write this function as x(v_{i}). Evaluate x for (b) v_{i} = 0.100 m/s and for (c) v_{i} = 100 m/s. (d) Assume v_{i} is close to but not equal to zero. Show that one term in the answer to part (a) dominates so that the function x(v_{i}) reduces to a simpler form. (e) If v_{i} is very large, what is the approximate form of x(v_{i})? (f) Describe the overall shape of the graph of the function x(v_{i}). 58. An outfielder throws a baseball to his catcher in an attempt to throw out a runner at home plate. The ball bounces once before reaching the catcher. Assume the angle at which the bounced ball leaves the ground is the same as the angle at which the outfielder threw it as shown in Figure P4.58, but that the ball’s speed after the bounce is onehalf of what it was before the bounce. (a) Assume the ball is always thrown with the same initial speed and ignore air resistance. At what angle should the fielder throw the ball to make it go the same distance D with one bounce (blue path) as a ball thrown upward at 45.0° with no bounce (green path)? (b) Determine the ratio of the time interval for the onebounce throw to the flight time for the nobounce throw. Figure P4.58 59. A World War II bomber flies horizontally over level terrain with a speed of 275 m/s relative to the ground and at an altitude of 3.00 km. The bombardier releases one bomb. (a) How far does the bomb travel horizontally between its release and its impact on the ground? Ignore the effects of air resistance. (b) The pilot maintains the plane’s original course, altitude, and speed through a storm of flak. Where is the plane when the bomb hits the ground? (c) The bomb hits the target seen in the telescopic bombsight at the moment of the bomb’s release. At what angle from the vertical was the bombsight set? 60. A truck loaded with cannonball watermelons stops suddenly to avoid running over the edge of a washedout bridge (Fig. P4.60). The quick stop causes a number of melons to fly off the truck. One melon leaves the hood of the truck with an initial speed v_{i} = 10.0 m/s in the horizontal direction. A cross section of the bank has the shape of the bottom half of a parabola, with its vertex at the initial location of the projected watermelon and with the equation y^{2} = 16x, where x and y are measured in meters. What are the x and y coordinates of the melon when it splatters on the bank? 61. _{ }A hawk is flying horizontally at 10.0 m/s in a straight line, 200 m above the ground. A mouse it has been carrying struggles free from its talons. The hawk continues on its path at the same speed for 2.00 s before attempting to retrieve its prey. To accomplish the retrieval, it dives in a straight line at constant speed and recaptures the mouse 3.00 m above the ground. (a) Assuming no air resistance acts on the mouse, find the diving speed of the hawk. (b) What angle did the hawk make with the horizontal during its descent? (c) For what time interval did the mouse experience free fall? 62. An aging coyote cannot run fast enough to catch a roadrunner. He purchases on eBay a set of jetpowered roller skates, which provide a constant horizontal acceleration of 15.0 m/s^{2} (Fig. P4.62). The coyote starts at rest 70.0 m from the edge of a cliff at the instant the roadrunner zips past in the direction of the cliff. (a) Determine the minimum constant speed the roadrunner must have to reach the cliff before the coyote. At the edge of the cliff, the roadrunner escapes by making a sudden turn, while the coyote continues straight ahead. The coyote’s skates remain horizontal and continue to operate while he is in flight, so his acceleration while in the air is (15.0 9.80) m/s^{2}. (b) The cliff is 100 m above the flat floor of the desert. Determine how far from the base of the vertical cliff the coyote lands. (c) Determine the components of the coyote’s impact velocity. 63. _{ }A car is parked on a steep incline, making an angle of 37.0° below the horizontal and overlooking the ocean, when its brakes fail and it begins to roll. Starting from rest at t = 0, the car rolls down the incline with a constant acceleration of 4.00 m/s^{2}, traveling 50.0 m to the edge of a vertical cliff. The cliff is 30.0 m above the ocean. Find (a) the speed of the car when it reaches the edge of the cliff, (b) the time interval elapsed when it arrives there, (c) the velocity of the car when it lands in the ocean, (d) the total time interval the car is in motion, and (e) the position of the car when it lands in the ocean, relative to the base of the cliff. 64. QC Do not hurt yourself; do not strike your hand against anything. Within these limitations, describe what you do to give your hand a large acceleration. Compute an orderofmagnitude estimate of this acceleration, stating the quantities you measure or estimate and their values. 65. A fisherman sets out upstream on a river. His small boat, powered by an outboard motor, travels at a constant speed v in still water. The water flows at a lower constant speed v_{w}. The fisherman has traveled upstream for 2.00 km when his ice chest falls out of the boat. He notices that the chest is missing only after he has gone upstream for another 15.0 min. At that point, he turns around and heads back downstream, all the time traveling at the same speed relative to the water. He catches up with the floating ice chest just as he returns to his starting point. How fast is the river flowing? Solve this problem in two ways. (a) First, use the Earth as a reference frame. With respect to the Earth, the boat travels upstream at speed v – v_{ow} and downstream at v + v_{w}. (b) A second much simpler and more elegant solution is obtained by using the water as the reference frame. This approach has important applications in many more complicated problems; examples are calculating the motion of rockets and satellites and analyzing the scattering of subatomic particles from massive targets. ^ 66. _{ } Two swimmers, Chris and Sarah, start together at the same point on the bank of a wide stream that flows with a speed v. Both move at the same speed c (where c > v) relative to the water. Chris swims downstream a distance L and then upstream the same distance. Sarah swims so that her motion relative to the Earth is perpendicular to the banks of the stream. She swims the distance L and then back the same distance, with both swimmers returning to the starting point. In terms of L, c, and v, find the time intervals required (a) for Chris’s round trip and (b) for Sarah’s round trip. (c) Explain which swimmer returns first. 67. QC A skier leaves the ramp of a ski jump with a velocity of v = 10.0 m/s at = 15.0° above the horizontal as shown in Figure P4.67. The slope where she will land is inclined downward at = 50.0°, and air resistance is negligible. Find (a) the distance from the end of the ramp to where the jumper lands and (b) her velocity components just before the landing. (c) Explain how you think the results might be affected if air resistance were included. Figure P4.67 68. _{ } A person standing at the top of a hemispherical rock of radius R kicks a ball (initially at rest on the top of the rock) to give it horizontal velocity as shown in Figure P4.68. (a) What must be its minimum initial speed if the ball is never to hit the rock after it is kicked? (b) With this initial speed, how far from the base of the rock does the ball hit the ground? Figure P4.68 69. A divebomber has a velocity of 280 m/s at an angle below the horizontal. When the altitude of the aircraft is 2.15 km, it releases a bomb, which subsequently hits a target on the ground. The magnitude of the displacement from the point of release of the bomb to the target is 3.25 km. Find the angle . 70. _{ } A fireworks rocket explodes at height h, the peak of its vertical trajectory. It throws out burning fragments in all directions, but all at the same speed v. Pellets of solidified metal fall to the ground without air resistance. Find the smallest angle that the final velocity of an impacting fragment makes with the horizontal. 71. An enemy ship is on the east side of a mountain island as shown in Figure P4.71. The enemy ship has maneuvered to within 2 500 m of the 1 800mhigh mountain peak and can shoot projectiles with an initial speed of 250 m/s. If the western shoreline is horizontally 300 m from the peak, what are the distances from the western shore at which a ship can be safe from the bombardment of the enemy ship? Figure P4.71 72. _{ } In the What If? section of Example 4.5, it was claimed that the maximum range of a ski jumper occurs for a launch angle given by where is the angle the hill makes with the horizontal in Figure 4.14. Prove this claim by deriving the equation above.
