Question

A sphere of mass $m$ moving with a constant velocity hits another stationary sphere of the same mass. If $e$ is the coefficient of restitution, the ratio of speed of the first sphere to the speed of the second sphere, after the head on collision will be,

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

Answer 1

The correct option is A $\left(\frac{1-e}{1+e}\right)$


Let after collision, velocity of $A$ and $B$ are $v_1$ and $v_2$ respectively

According to conservation of momentum principle,

$mu+m\times 0=m{v}_1+m{v}_2$

$\Rightarrow u=v_1+v_2-\left(1\right)$

Also, $e=\frac{v_2-v_1}{u_1-u_2}$

$e=\frac{v_2-v_1}{u-0}$

From $\left(1\right)$ we get,
$\Rightarrow \frac{v_2-v_1}{v_1+v_2}=e$

After componendo dividendo, we get,

$\frac{v_2}{v_1}=\frac{1+e}{1-e}$

$\therefore \frac{v_1}{v_2}=\frac{1-e}{1+e}$

Hence, $\left(A\right)$ is the correct answer.