Question

A smooth sphere of mass $M$ moving with velocity $u$ directly collides elastically with another sphere of mass $m$ at rest. After the collision, their final velocities are $V$ and $v$ respectively. The value of $v$ is:

• A. $\frac{2uM}{m}$
• B. $\frac{2um}{M}$
• C. $\frac{2u}{1+\frac{m}{M}}$
• D. $\frac{2u}{1+\frac{M}{m}}$

Answer 1

The correct option is C $\frac{2u}{1+\frac{m}{M}}$

Final velocity of mass $M$, $v_2=V$
Final velocity of mass $m$, $v_1=v$
We know that,
Coefficient of restitution,
$\frac{v_2-v_1}{u_2-u_1}=-e$
Given,
$u_2=u,v_2=V,u_1=0,v_1=v$
$\therefore \frac{V-v}{u-0}=-e$
For elastic collision,
$\therefore \frac{V-v}{u-0}=-1$
$\begin{array}{}V=v-u& \text{(1)}\end{array}$
Applying conservation of momentum just before and after the collision,
$Mu+0=MV+mv$
Using euqation $\left(1\right)$,
$\Rightarrow Mu=M(v-u)+mv$
$\Rightarrow v=\frac{2Mu}{M+m}=\frac{2u}{1+\frac{m}{M}}$