\[\begin {array} {1 1} (a)\;\frac{2}{3} & \quad (b)\; \frac{1}{2} \\ (c)\;\frac{1}{3} & \quad (d)\;\sqrt {\frac{2}{3}} \end {array}\]

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By symmetry both the billiard balls will move with the same velocity along their respectively lines of impact .

By momentum conservation along the line of impact.

$mu= 2 mv \cos 30$

$\qquad= 2 mv . \large\frac{\sqrt 3}{2}$

$u= \sqrt 3 v$

Newton's law of restitution along line of impact.

$\large\frac{V-0}{0- u \cos 30}$$=-e$

$\large\frac{V}{\sqrt 3 u/2}$$=e$

$e= \large\frac{2}{3}$

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