Question

A ball of mass '$M$' moving with a velocity $\overrightarrow{V}$ collides head on elastically with another body of the same mass '$M$' moving with a velocity $-\overrightarrow{V}$ in the opposite direction. After the collision :

• A. The velocities are exchanged by the two balls
• B. Both the balls come to rest
• C. Both of them move at right angles to the original line of motion
• D. One ball comes to rest and the other ball travels back with a velocity $2v$

Answer 1

The correct option is A The velocities are exchanged by the two balls

Since the collision is elastic, we know that when two bodies of same mass have an elastic collision their velocities get interchanged.
$m_1{u}_1+m_2{u}_2=m_1{v}_1+m_2{v}_2$

But $m_1=m_2=M,u_1=V,u_2=-V$

$\Rightarrow MV-MV=M{v}_1+M{v}_2$

$\Rightarrow v_1+v_2=0$.............(1)

The coefficient of restitution $\left(e\right)$ is given by,

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

But for elastic collision, $e=1$

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

$\Rightarrow v_2-v_1=2V$..............(2)

From (1) and (2), we get,
$v_2=V$
$v_1=-V$