• # 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) $\frac{\sqrt{3}}{2}$

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