Let \(v_{1}\) and \(v_{2}\) be their velocities after collision.

According to momentum conservation, \(m u=m\left(v_{1}+v_{2}\right)\)

or \(u=v_{1}+v_{2} \dots(i)\)

By definition \(e=\frac{v_{2}-v_{1}}{u-0}\) or \(v_{2}-v_{1}=e u \ldots .(i i)\)

Solving Eqs. \((i)\) and \((ii),\) we have \(v_{1}=\frac{(1-e) u}{2}\)

and \(v_{2}=\left(\frac{1+e}{2}\right) u\)

\(\Rightarrow \frac{v_{1}}{v_{2}}=\frac{1-e}{1+e}\)