A ball is projected from the ground at some angle with horizontal. Coefficient of restitution between the ball and the ground is e. Let a, b and c be the ratio of times of flight, horizontal range and maximum height in two successive paths. Find a, b and c in terms of e.

Explanation

Let us assume that ball is projected with speed u at an angle `theta` with the horizontal. Then
Before first collision with the ground.
Time fo flight `T=(2u_(y))/g`
Horizontal range `R=(2u_(x)u_(y))/g`
Maximum Height `H_("max")=(u_(y)^(2))/(2g)` ....(1)

After striking the ground the component `u_(y)` is change into e `u_(y)`, so
Time of flight `T'=(2eu_(y))/g,R'=(2u_(x)(eu_(y)))/g`
`H_("max")=((eu_(y))^(2))/(2g)` ....(2)
from eq (1) & (2)
Now `T/(T')=a=1/e`
`R/(R')=b=1/e` , `(H_("max"))/(H'_("max"))=1/(e^(2))=c`