Calculating initial velocity of a bouncing object

I'm not sure how to find the initial velocity of a ball that is released from a height y and bounces through a distance x. The ball has an elasticity E and radius R but I don't know an equation that relates these and the number of bounces to the ball's initial velocity.

Mathematically speaking, the ball bounces infinitely many times. In real life it won't really do this, but seeing as how the math doesn't know that, I can't think of an easy way to relate the number of bounces to the ball's initial velocity. Maybe you could tell us more about the problem.

The problem is that a ball is released from y=10 m and must hit a target 40 m away which is 6 m high. The ball is released and bounces with its height dropping by elasticity squared (E^2) each time. The problem then states that it is possible to caluculate the number of bounces using the range x = 40m, the target height 6 m and E. Then using number of bounces and a calculation of the time to the first bounce, the time to reach the target can be found and subsequently the initial horizontal velocity can be found. I just don't know how to form the relevant equations using these values.

can you make a picture or something like that? i have an idea of how you could solve the problem but i'm not sure if i understand the specifical case.

pd: the solution i thought is with limits of geometrical series, maybe that helps you.

Last edited:
russ_watters
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I'm not sure what elasticity squared is, but what determines how high an object bounces isn't it's elasticity, but the elastic/inelastic ratio. If an object is 40% elastic, then it loses 60% of it's kinetic energy when it bounces. Plug that back into mgh to find height. How many bounces is then whatever you choose it to be - calculate the time for that number of balances and use that to find the forward velocity to get it to the target in that time.