Question

A ball is dropped from a height $h$ on a floor. The coefficient of restitution for the collision between the ball and the floor is $e$ . The total distance covered by the ball before it comes to the rest.

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

Answer 1

Answer: (B)

Height achieved in $n ^{\text {th }}$ collision is given by $h_{ n }= e ^{2 n } h$
Distance covered in first collision $=h$,
Distance covered in $2^{\text {nd }}$ collision $=2 e^{2} h$, and so on.
Total distance covered by the ball before it comes to rest:
$s=h+2 e^{2} h+2 e^{4} h+\ldots \ldots$
$\Rightarrow s=h 1+2 e^{2}+2 e^{4}+2 e^{6}+\ldots \ldots$
from the sum of infinite geometric progression:
$\Rightarrow s=h 1+\frac{2 e^{2}}{1-e^{2}} $
$\Rightarrow s=h \frac{1+e^{2}}{1-e^{2}}$