Inelastic collision

Let the velocity of the bigger ball be ${ v }_{ 0 }$ , depicted in the problem, and $v$ be the velocity of smaller balls. After collision, the smaller balls move along the line of impact, depicted in the figure.

Conserving momentum along the $X-$ axis, we get: $m {v}_{0}$ $=$ $2mv \cos { \theta }$ or ${v}_{0}$ $=$ $2v\cos { \theta }$

Coefficient of restitution, $e$ , is defined as the Ratio of velocity of separation and velocity of approach, along the line of impact.

Therefore, $e$ $=$ $\frac{v}{{v}_{0}\cos { \theta } }$

From the diagram, we can easily deduce that $\cos { \theta }$ $=$ $\frac { 2\sqrt { 2 } }{ 3 }$ .

Solving from the above three, we get $e$ $=$ $\frac{9}{16}$ .