Question

A ball is dropped onto a fixed horizontal surface from height h₀. The coefficient of restitution is e. Find the total distance D travelled by the ball before it comes to rest on the surface.

• A. $\frac{h_0(1-e^2)}{(1+e^2)}$
• B. $\frac{h_0(1+e^2)}{(1-e)}$
• C. $\frac{h_0(1+e^2)}{(1+e)}$
• D. $\frac{h_0(1+e^2)}{(1-e^2)}$

Answer 1

The correct option is D

$\frac{h_0(1+e^2)}{(1-e^2)}$

Let $h_i(I=1,2,3,\cdots \cdots .)$ be the height of rebound after the ith impact.
Then $e=\frac{h_i}{h_i-1}$ which gives $h_n=e^{2n}{h}_0$ $(n=1.2.3\cdots \cdots \cdots .)$
$D=h_0+2(h_1+h_2+h_3+h_4+h_5+\cdots \cdots )$
$=h_0\{1^0+2(e^2+e^4+e^6+e^8+e^{10}+\cdots \cdots \cdots )\}D=h_0\left(\frac{1+e^2}{1-e^2}\right)$