EXSC 408L Fall '03 Problem Set #4 Projectile Motion. Projectile Motion. At club departure At Apex At contact with green x = x = x = y = y = y =

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1 Problems: 1. A golfer hits a ball from a tee located 2.00 m above a creek. The ball reaches its maximum height and then falls to a green located 4.00 m above the tee. Assuming the ball leaves the club head at a velocity of 30.0 m/s at an angle of 50.0 relative to the ground, determine the following: At club departure At Apex At contact with green x = x = x = y = y = y = v x = v x = v x = v y = v y = v y = Table 1. Kinematics data for a long jump. Event Frame # (30 Hz) TBCM position (x,y) in pixels Take-off 11 (388, 195) Apex 19 (476, 255) Landing 27 (555, 183) 225 pixels = 1 meter 2. From Table 1, calculate the following: a. Horizontal and vertical displacement in meters from take -off to apex (using kinematics) b. Horizontal and vertical displacement in meters from apex to landing (using kinematics) c. Time from take-off to apex in seconds (using kinematics) d. Average horizontal velocity (m/s) from take -off to apex (using kinematics) e. Vertical velocity (m/s) at landing (assuming projectile motion) 3. From Table 1, sketch the graphs that represent the following TBCM characteristics from take-off to landing: a. Vertical position vs. time b. Vertical velocity vs. time c. Acceleration vs. time Page 1 of 7

2 4. Using the following kinematics data, calculate the following: Event Frame (30Hz) TBCM position (x,y) in pixels Takeoff 5 (100, 50) Apex 13 (180, 98) Landing 23 (260, 45) 75 pixels = 1meter a. Total horizontal displacement in meters b. Height at the apex in meters c. Time from takeoff to apex 5. The vertical velocity at takeoff of the object in the previous problem was 3.00 m/s. If the initial height of the TBCM was 0.67 m, calculate the height at the apex using projectile motion equations. When compared with the height found in the previous problem, what was the digitizing error in pixels? 6. During high jump practice, an athlete reached a maximum height of 3.03 m, where the time to reach the apex was 0.58 s. The athlete s horizontal speed at take-off was 5.66 m/s with a takeoff angle of 45.0 from the right horizontal. Determine the magnitude and direction of the theoretical resultant velocity of the athlete just before landing. 7. On a different attempt during the same high jump practice, the resultant speed of the athlete at takeoff was 8.0 m/s. At the time of takeoff, the athlete s TBCM was 1.40 m from the ground with a takeoff angle of 50.0 from the right horizontal. Determine the maximum height of the jump. 8. A shot putter puts the shot into the air with a resultant velocity of 15.0 m/s at an angle of 40.0 relative to the right horizontal. What is the horizontal velocity of the shot s center of mass at its maximum height? 9. Using the information from the previous problem, how far does the shot travel in the air if its height at release was 1.70 m? 10. A golf ball is driven straight out from the tee box that is 2.0 m above the fairway. At the instant the club contacts the ball, another golfer drops a golf ball from a height of 2.0 m. Which ball will contact the ground first? Why? 11. If the resultant velocity of is 29.7 m/s and is directed at an angel of 22.0 to the right horizontal, what are the horizontal and vertical components of the velocity vector? 12. If the horizontal acceleration is m/s 2 and the vertical acceleration is m/s 2, what is the resultant acceleration? At what angle does the resultant acceleration act? 13. A softball is thrown with a velocity of 22.5 m/s at an angle of 56.0 from the right horizontal. If the initial height is 1.70 m, calculate the following: a. Vertical and horizontal velocity components b. Time to apex c. Height of the ball from the point of release to apex d. Total height of the ball at the apex e. Time from the apex to the ground f. Total time in the air g. Total horizontal displacement Page 2 of 7

3 14. At take-off from a spring board, the total body center of mass (TBCM) of a diver is located 1.50 m above the board. The diver leaves the spring board with a resultant TBCM velocity of 9.20 m/s acting at an angle of 20.0 clockwise from the vertical. Assuming no air resistance, determine the following: a. Horizontal position of the TBCM at the apex b. Horizontal velocity of the TBCM at the apex c. Time from the apex to the time the hands enter the water. Note: the water is 2.0 meters below the board and the TBCM at the time of entry is 1.5 meters above the water with respect to the hands. d. Distance of the TBCM from the board in the horizontal direction at water entry e. Resultant velocity of the TBCM at the time the hands enter the water 15. A javelin was released at a horizontal velocity of 5.00 m/s and a vertical velocity of 3.25 m/s. What was the resultant velocity? 16. In projectile motion, what factors determine time in the air? a. Resultant velocity of total body center of mass b. Vertical velocity of total body center of mass c. Vertical position of total body center of mass d. a and c e. b and c 17. A volleyball player takes off to hit a ball with a TBCM velocity of 6.0 m/s, at a 65.0 angle from the horizontal plane. His TBCM at take off is 1.15 m from the floor. When he lands, his TBCM is 0.85 m from the floor. Assume the same level of TBCM at contact. a. How long will this volleyball player stay in the air? b. What is the vertical position of his TBCM when he reaches the apex? c. What is his vertical velocity at contact? d. How far away from the net should the volleyball player take off in order to avoid hitting the net? 18. During a long jump, an athlete jumped a maximum length of 1.9 m. It took him 0.23 s to reach his maximum vertical height. His horizontal speed when he left the ground was 8.3 m/s with a takeoff angle of 15.0 from the right horizontal. Determine the resultant velocity of the athlete just before landing assuming he takes-off and lands at the same height. 19. A 20.0 kg impacting missile was dropped from a height of 0.80 m onto a gymnastic mat. Assuming a frictionless environment, what was the magnitude of the missile s vertical velocity at contact with the mat? 20. A boy is sitting on a dock throwing stones into the harbor. The dock is 5.00 m above the water, and the boy is sitting 1.00 m behind the edge of the dock. If the initial resultant velocity of the stone is 20.0 m/s, 60.0 relative to the horizontal, calculate the following: At departure from dock At Apex At contact with water x = x = x = y = y = y = v x = v x = v x = v y = v y = v y = 21. An athlete s center of mass displaced 40.0 in. during a vertical jump. His time in the air from take-off to apex was 0.56 s. What was the athlete s vertical velocity at takeoff? Page 3 of 7

4 Answers: 1. At departure: (0.00, 2.00) m v x1 = 19.3 m/s v y1 = 23.0 m/s At apex: (45.3, 29.0) m v x2 = 19.3 m/s v y2 = 0 m/s At contact: (87.2, 6.00) m v x3 = 19.3 m/s v y3 = m/s 2. a. x = 0.391m; y = m b. x = 0.351m, y = m c s d m/s e m/s or 2.64 m/s 4. a m b m c s m, 13.5 to 14 pixels m/s, (-0.79 radians) from right horizontal m m/s m 11. v x = 27.5 m/s; v y = 11.1 m/s m/s 2 ; 89.6 counter-clockwise from the horizontal 13. a. v y = 18.7 m/s; v x = 12.6 m/s b s c m d m e s f s g m 14. a m b. 3.15m/s c. 1.09s d. 6.12m e m/s, from right horizontal m/s 16. e 17. a. 1.1 s b m c m/s d m m/s m/s 20. At departure: (-1.0,5. 0)m v x1 = 10.0 m/s v y1 = 17.3 m/s At apex: (16.7, 20.3)m v x2 = 10.0 m/s v y2 = 0.0 m/s At contact with the water: (37.1, 0.0)m v x3 = 10.0 m/s m/s v y3 = m/s Page 4 of 7

5 Bonus Problems: For questions 1 4: A ball leaves a soccer player s foot with a TBCM vertical velocity of 4.0 m/s and a horizontal velocity of 3.0 m/s. 1. Determine the angle of the resultant TBCM velocity vector from a right horizontal line passing through the TBCM of the ball. a b c d e Determine the vertical displacement of the ball from departure from the foot to the apex of the flight. a m b m c m d m e m 3. Determine the total flight time. Assume the time to apex is the same as time to the ground. a s b s c s d s e s 4. Determine the horizontal displacement from take-off to the apex. a) 0.93 m b) 1.23 m c) 1.86 m d) 2.46 m e) 2.70 m Page 5 of 7

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7 TBCM = (1.2,1) 5. Given that, at toe-off, the sprinter s TBCM vertical velocity is 1.20 m/s and his horizontal velocity is 7.00 m/s, use projectile motion equations to calculate his step length. Assume the vertical position of TBCM at foot strike is the same as at toe off and the horizontal position is meters behind the foot. a m b m c m d m e m 550N CP = (0, 0) 1200N Page 7 of 7