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Dynamics_Part47

# Dynamics_Part47 - Problem Set 6 Problem 3 Problem A girl...

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Unformatted text preview: Problem Set 6: Problem 3. Problem: A girl throws a ball at an inclined wall from height h, with velocity U . After bouncing off the wall, the ball hits her on her head. Note that, because she has just bent down to tie a loose shoelace, her head is at height h when the ball hits her. For simplicity, assume the ball travels horizontally as it approaches the wall. (a) If the coefficient of restitution of the ball is e, determine the velocity of the ball, v , just after it bounces from the wall as a function of U , and e. (b) Ignoring effects of friction as the ball moves, determine the time at which the ball lands on the girl's head as a function of U , , e and gravitational acceleration, g. HINT: Make use of the fact that sin cos = 1 sin 2 to simplify your answer. 2 (c) The solution developed in Part (b) is possible only if exceeds a special angle min . Explain why and determine min for e = 0.7. Solution: In the first part of this problem we focus on the impact as the ball strikes the wall. Then, using the velocity of the ball just after the impact as the initial condition, we compute the trajectory of the ball until it lands on the girl's head. (a) The line of impact is normal to the wall as indicated below. . . .. . .... .... .. ... . ... ... ... .... ... ...... .. . ... ........ .. .... . ... . .... .. ... .. .. ... . ... ... .. .. .. ... . .. .. . . .. . . .. .. . .. .. ..... . ..... . .. . . ................... . .................. .. . . .. . .. . . .. . .. .. . .. .. . ... .. . ...... ... .. ....... . ..... ............... . ..... .............. . . ...... ..... . . .. . .. . .. .. .. .. .... . .. .. .. . ...... ........... ................... . . . . v t = i cos + k sin U n = i sin - k cos Tangential-Velocity Invariance. The tangential velocity component is unchanged by the impact. Hence, vt = U cos Impact Relation. The impact relation holds along the line of impact and the wall is immovable, wherefore vn - 0 = e(0 - U sin ) = vn = -eU sin ...
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