A kid at the window of a tall building is trying to throw a ball (of mass m) to

Question

A kid at the window of a tall building is trying to throw a ball (of mass m) to the window of another kid who lives one floor up in the same building. Rather than throw the ball up, he throws it down so that it bounces off the ground below and then back up. He throws the ball with an initial velocity of V_0 downward. When the ball bounces, its instantaneous vertical velocity changes sign (i.e. goes from negative to positive) and its magnitude drops to a fraction p of its value just before the bounce (p is a number between zero and one). Given the height off the ground of the two kids (y_1 for the thrower below, and y_2 for the receiver above), what is the minimum initial speed V_0 in order for the ball to reach the higher kid? Express your answer in terms of y_1, y_2, g, m, p, and any other quantities you need.

Explanation / Answer

distance to be travelled = d = Y2 - Y1

acceleration = a = - g

initial velocity = Vi = p Vo

final velocity = Vf = 0

using the equation

Vf2 = Vi2 + 2 a d

02 = p2 Vo2 + 2 (-g) (Y2 - Y1 )

Vo2 = 2 (-g) (Y2 - Y1 )/ p2

Vo = sqrt(2 (g) (Y2 - Y1 )/ p2 ) = sqrt(2 (g) (Y2 - Y1 )/p

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