question_answer

Two identical billiard balls are in contact on a table. A third identical ball strikes them symmetrically and comes to rest after impact. The coefficient of restitution is

A) \[\frac{2}{3}\]

B) \[\frac{1}{3}\]

C) \[\frac{1}{6}\]

D) \[\frac{\sqrt{3}}{2}\]

Correct Answer: A

Solution :

[a] \[\sin \theta =\frac{r}{2r}=\frac{1}{2}\Rightarrow \theta =30{}^\circ \]

From conservation of linear momentum,

\[mu=2mv\,\,\cos 30{}^\circ \,\,\text{or}\,\,v=\frac{u}{\sqrt{3}}\]

Now  \[e=\frac{\text{Relative}\,\,\text{velocity}\,\,\text{of}\,\,\text{separation}}{\text{Relative}\,\,\text{velocity}\,\,\text{of}\,\,\text{approch}}\]

(in common normal direction)

Hence, \[e=\frac{v}{u\,\cos 30{}^\circ }=\frac{u/\sqrt{3}}{u\sqrt{3}/2}=\frac{2}{3}\]