Question

A ball falls from a height $h$ and rebounds after striking the floor. The coefficient of restitution is $e$. The maximum distance covered before it comes to rest is

• A. $\frac{\left(1-e^{2}\right) h}{e^{2}}$
• B. $\frac{\left(1+e^{2}\right) h}{e^{2}}$
• C. $\left(\frac{1+e^{2}}{1-e^{2}}\right) h$
• D. $\frac{e^{2} h}{1-e^{2}}$

Answer 1

Answer: (C)

Ball falls from height $h$ then formula for height covered by it in $n$ th rebound is given by
$h_{n}=h e^{2 n}$
where $e=$ coefficient of restitution
$n=$ number of rebound
Total distance travelled by particle before rebounding has stopped.
$H=h+2 h_{1}+2 h_{2}+2 h_{n}+\ldots$
$H=h+2 h e^{2}+2 h e^{4}+2 h e^{6}+2 h e^{8}+\ldots$
$=h+2 h\left(e^{2}+e^{4}+e^{6}+e^{8}+\ldots\right)$
$=h+2 h\left[\frac{e^{2}}{1-e^{2}}\right]$
$=h\left[1+\frac{2 e^{2}}{1-e^{2}}\right]$
$=h\left[\frac{1+e^{2}}{1-e^{2}}\right]$