Two billiard balls of same size and mass are in contact on a billiard table, A third ball of same mass and size strikes them symmetrically and remains at rest after the impact. If the coefficient of restitution between the balls is $\frac{2}{\mathrm{P}}$ , the value of P is

Let the billiard ball is approaching with a speed of 'u'

And after collision the ball is coming to rest and the other two balls are moving with a speed 'v'

As the linear momentum is conserved

$2\mathrm{mvcos\theta }=\mathrm{mu}\Rightarrow \mathrm{v}=\frac{\mathrm{u}}{\sqrt{3}}$

velocity of approach = $\mathrm{ucos}{30}^{\mathrm{o}}=\frac{\sqrt{3}}{2}\mathrm{u}$

$\mathrm{e}=\frac{\text{ relative velocity of separation }}{\text{ relative velocity of approach }}$

$=\frac{\mathrm{v}}{\mathrm{ucos}30°}=\frac{\mathrm{u}/\sqrt{3}}{\mathrm{u}\sqrt{3}/2}=\frac{2}{3}$