A sphere of mass 'm', moving with velocity '3u' collides head-on with another identical sphere at rest. If ' e ' is coefficient of restitution then what will be the ratio of velocity of the second sphere to that of first sphere after collision?

Let m be the mass of each sphere, with initial velocities $u_1 = 3u$ and $u_2 = 0$. After collision, their velocities are $v_1$ and $v_2$.

Conservation of momentum gives $m(3u) + m(0) = mv_1 + mv_2$, so $v_1 + v_2 = 3u$.

The coefficient of restitution provides $e = \frac{v_2 - v_1}{3u}$, yielding $v_2 - v_1 = 3eu$.

Adding these equations eliminates $v_1$: $2v_2 = 3u(1 + e)$, so $v_2 = \frac{3u(1 + e)}{2}$.

Subtracting gives $2v_1 = 3u(1 - e)$, so $v_1 = \frac{3u(1 - e)}{2}$.

The required ratio is $\frac{v_2}{v_1} = \frac{1 + e}{1 - e}$, matching option B.