# 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.

## Answers and Replies

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
Mentor
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.