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Unformatted text preview: n v $4.12.:rr‘f-4‘WWH— ~— , . -1--MA- _.....-V‘ ...r ). .g—g .43; an; . 48 Chapter 3 Motion in Two and Three Dimensions 71. A basketball player is 15 ft horizontally from the center of the basket, which is 10 ft off the ground. At what angle should the player aim the ball if it is thrown from a height of 8.2 ft with a speed of 26 fits? 72. A jet is diving vertically downward at 1200 krnjh. If the pilot can withstand a maximum acceleration of 'Sg (i.e., 5 times Eard‘t‘s gravitational acceleration) before losing consciousness, at what height must the plane start a quarter turn to pull out of the dive? Assume the speed remains constant. 73. An alpine rescue team is using a slingshot to send an emergency medical packet to climbers stranded on a ledge, as shown in Fig. 3.24. What should be the launch speed from the slingshot? Haul-1.24 Problem 73 74. If you can throw a stone straight up to a height of 16 rn, how far could you throw it horizontally over level ground? Assume the same throwing speed and optimum launch angle. 75. In a conversion from milimry to peacetime use, a missile with a maximum’horizon’ml range of 180 km is being adapted for study- ing the upper atmosphere. What is the maximum altitudeit can achieve if launched vertically? 76. I can kick a soccer ball 28 m on level ground, giving it an initial velocity at 40° to the horizontal. At the same initial speed and angle to the horizontal, what horizontal distance can I kick the ball on a 15° upward slope? 77. A diver leaves a 34m board on a trajectory that takes her 2.5 m above the board and then into the water a horizontal distance of 2.8 In from the end of the board. At what speed and angle did she leave the board? 78. In your calculus class, you may have learned that you can find the maximum or minimum of a function by differentiating and set- ting the result to zero. Do this for Equation 3.15, differentiating with respect to H. and thus verify that the maximum range occurs for 9 = 45°. 79. A well~engineered ski jump is less dangerous than it looks be- cause skiers hit the ground with very small velocity components perpendicular to the ground. Skiers leave the Olympic ski jump in Lake Placid, New York, at 28 m/s at an angle of 9.5” below the horizontal. Their landing zone is a horizontal distance of 55 111 from the end of the jump. The ground at that point is contoured so skiers‘ trajectories make an angle of only 3.0D with the ground on landing, as suggested in Fig. 3.25. What is the slope of the ground in the landing zone? 80. 81. 82. 83. 84. 85. £15m Problem 79 Differentiate the trajectory Equation 3.14 to find its slope, ‘ tanB =dyldx, and show that the slope is in the direction of the projectile’s velocity, as given by Equations 3.10 and 3.111 A projectile has horizontal range R and maximum height it. Find an expression for its initial speed. (a) Show that the position of a particle on a circle of radius R with center at the origin (x=0, 32:0) is given in unit vectors i— = cos 6i+ sinGj, where B is the angle of the position vector. " the x axis. (b) If the particle moves with constant speed v and riod T, starting on the x axis an: :0, find an expression for 6 terms of the time and T. (c) Differentiate the position vector ' with respect to time to find the acceleration, and show than. the centripetal acceleration, whose magnitude is given by ' tion 3.16 and whose direction is toward the center of the ur' (Note that the unit vectors 3 and j are constants, indem' of time.) A friend is writing a science fiction screenplay about an on a collision course with Earth. Despite your admonitions other films with the same plot, she asks you to calculate ' numbers so her scenario will be correct. Astronauts will . i rocket engine to the asteroid in an attempt to divert it. The ' oid is moving at 21 kin/s. The rocket will provide an acce, of 0.035 km/s2 at a right angle to the original motion.‘ change the direction of the asteroid's motion by 22.6“ place it 5.36 x103 km, enough to miss Earth and save. tion. Your friend asks you whether a rocket firing l ' will be enough to give these results. How do you advise An archery coach at your college wishes to know mate speed with which arrows are fired. She tells you fires an arrow horizontally from a height of 1.54 m. hits the ground 23 In downrange from the archer. 7_ sistance, what speed do you report to the coach? Using your textbook as a guide to projectile mu. Internet for information on the range of baseballs What factors affect the range? The theoretical for maximum range is 45°, but in reality this is not angle gives the greatest range for a baseball 01‘ 7' is the potential range of a baseball greater in. Minneapolis? Wong Answer to Chapter Opening Question Assuming negligible air resistance, the penguin should leave the water at a 45D angle. Answers to GOT IT? Questions 3.1 (c). 3.2 ((1) only. 3.3 (c) gives the greatest change in speed; 3.4 az>a3>a4>a1. change in direction. ...
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