Question

Two billiard balls of the same size and mass are in contact on a billiard table. A third ball of the same size and mass strikes them symmetrically and remains at rest after the impact. The coefficient of restitution between the balls is

• A. $\frac{2}{3}$
• B. $\frac{2}{5}$
• C. $\frac{1}{7}$
• D. $\frac{1}{8}$

Answer 1

Answer: (A)

$m u=2 \,m\, v \cos \theta$
Since all the balls are identical, their centres at the moment of collision form an equilateral triangle. Hence $\theta=30^{\circ}$
$u=2 v \cos 30^{\circ}=\frac{2 v \sqrt{3}}{2}$
$u=v \sqrt{3}$
Using any one ball,
$=\frac{v-0}{u \cos 30^{\circ}-0}$
$=\frac{v}{u \frac{\sqrt{3}}{2}}=\frac{v}{v \sqrt{3} \cdot \frac{\sqrt{3}}{2}}$
$\Rightarrow e=\frac{2}{3}$.