Question

A ball of mass $m$ makes an elastic collision with another identical ball at rest. Show that if the collision is oblique, the bodies go at right angles to each other after collision.

Answer 1

In head on elastic collision between two particles, hey exchange their velocities. In this case, the component of ball 1 along common tangent direction of both the particles remain unchanged. Thus, the components along common tangent and common normal direction of both the balls in tabular form are given in Table.

Ball	Component long common tangent direction		Component along common normal direction
	Before Collision	After Collision	Before Collision	After Collision
1	$\nu sin\theta$	$\nu sin\theta$	$\nu cos\theta$	$0$
2	$0$	$0$	$0$	$\nu cos\theta$

From Table and Fig we see that both the balls move at right angle after collision with velocity $\nu sin\theta$ and $\nu cos\theta$
Method 2.
$mu=m{\nu }_{1}cos{\theta }_1+m{\nu }_{2}cos{\theta }_2$ (i)
$0=m{\nu }_{1}sin{\theta }_1+m{\nu }_{2}sin{\theta }_2$ (ii)
$\frac{1}{2}m{u}^2=\frac{1}{2}m{\nu }_1^2+\frac{1}{2}m{\nu }_2^2$ (iii)
Squaring and adding Eqs. (i) and (ii), we get
$u^2={\nu }_1^2+{\nu }_2^2+2{\nu }_1{\nu }_{2}cos({\theta }_1+{\theta }_2)$
Using Eq. (iii), we have $n^2={\nu }_1^2+{\nu }_2^2$