Question

A ball dropped from 'a' metres above a given surface. Each time the ball hits the surface after falling a distance h it rebounce rh, where r is a positive number less than 1. find the total distance the ball travels up and down.

Answer 1

Ball drops r, goes up rhr then drops rhr, up rhrhr, down rhrhr...

This is basically a geometric series:

r + 2rhr + 2r(hr)^2 +...or

-r + 2h (1 + r + r^2 +...which is

-r + (2h/(1-r)) (geometric series to infinity)

= (r^2 -r +2h)/1-r

So if h was 1 unit to start with and r was a half, then

( (1/4)-(1/2)+ 2)/1-(1/2) = 2 and a half units total distance or if

h was 10 with a half bounce then 39 and a half feet.