Question

A sphere of mass $m$ moving with a velocity $u$ hits another sphere of same mass at rest. If $e$ is the coefficient of restitution then the ratio of velocities of two spheres after collision will be .

• A. $\frac{1-e}{1+e}$
• B. $\frac{1+e}{1-e}$
• C. $\frac{e+1}{e-1}$
• D. $\frac{e-1}{e+1}$

Answer 1

Answer: (A)

$\frac{1-e}{1+e}$

REF.Image.

Solution

Before collision

$u_1\to$ $u_2\to$

$O^m$ $O^m$

$u_1=u$ and $u_2=0$

After collision

${}^m{O}^{v_1}\to$ ${}^m{O}^{\to v_2}$

By momentum conservation

$m{v}_2+m{v}_2=m{u}_1+m{u}_2$

$\Rightarrow v_2+v_1=u...\left(i\right)$

Also,

$e=\frac{v_2-v_1}{u_2-u_1}=\frac{v_2-v_1}{0-u}$

$\Rightarrow v_2-v_1=-eu$ ...(ii)

By (i) and (ii), we get

$v_2=\frac{u}{2}(1-e)$ and

$v_1=\frac{u}{2}(1+e)$

So, $v_2,v_1=(1-e)!(1+e)$

Option - A is correct.