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

Answer 1

Answer: (B)

Height achieved in $n^{\text{th }}$ collision is given by $h_n=e^{2n}h$
Distance covered in first collision $=h$,
Distance covered in $2^{\text{nd }}$ collision $=2e^{2}h$, and so on.
Total distance covered by the ball before it comes to rest:
$s=h+2e^{2}h+2e^{4}h+\dots \dots$
$\Rightarrow s=h1+2e^2+2e^4+2e^6+\dots \dots$
from the sum of infinite geometric progression:
$\Rightarrow s=h1+\frac{2e^2}{1-e^2}$
$\Rightarrow s=h\frac{1+e^2}{1-e^2}$