A ball is dropped on a floor from a height `h`. If the coefficient of restitution is `e`, find the height to which the ball will rise after touching the floor and the time it will take to come to rest again.

Explanation

The ball reaches the floor with velocity `u=sqrt(2gh)`
if collision on the floor is elastic, the ball will rebound with the same vellcity `u` and it will reach the same height. But if the coefficient of restitution between the ball and the floor is `e`, the ball will rebound with a lesser vellocity `v`, given as
`v=eu`.......i
where `v` is the velolcity of spearation and `u` is th velocty of approach
in this case the ball will rebound to a height `h_(1)`: from equations it is given that
`h_(1)=v^(2)/(2g)=e^(2)s`............ii
In downward and upward motion the accelertion of the particle is `g` only, thus total time of motion is the time of downward motion plus the time of upward motion as
`t=sqrt((2h)/g)+sqrt((2e^(2)h)/g)=sqrt((2h)/g)+esqrt((2h)/g)`