Question

A ball is dropped from height $5 \, m$ . What is the time (in seconds) after which ball stops rebounding if the coefficient of restitution between ball and ground $e=\frac{1}{2}$ ?

Answer 1

$T=\frac{v}{g}+\frac{2 e v}{g}+\frac{2 e^{2} v}{g}+\ldots \ldots \infty$
$T=\frac{v}{g}\left[1+2 e+2 e^{2}+\ldots \ldots \ldots \infty\right]$
$T=\frac{v}{g}\left[2+2 e+2 e^{2}+\ldots \ldots \ldots \infty-1\right]$
$T=\frac{v}{g}\left[2\left(1+e+(e)^{2}+\ldots \ldots \ldots \infty\right)-1\right]$
$T=\frac{v}{g}\left[2 \times \frac{1}{1-e}-1\right] $
$\Rightarrow T=\frac{\sqrt{2 g h}}{g}\left[2 \times \frac{1}{1-\frac{1}{2}}-1\right] \Rightarrow T=3 s$