A sphere A impinges directly on an identical sphere B at rest. If coefficient restitution is e, the ratio of velocities of A and B after collision is

${\mathrm{m}}_1={\mathrm{m}}_2=\mathrm{m}{\mathrm{v}}_2=0$
coefficient of restitution
$\frac{{\mathrm{v}}_2-{\mathrm{v}}_1}{{\mathrm{u}}_1-{\mathrm{u}}_2}=\mathrm{e}{\mathrm{v}}_2-{\mathrm{v}}_1={\mathrm{eu}}_1$…….(i)
from conservation of momentum
${\mathrm{mu}}_1={\mathrm{mv}}_1+{\mathrm{mv}}_2$
${\mathrm{u}}_1={\mathrm{v}}_1+{\mathrm{v}}_2$…………..(ii)
$\begin{array}{l}\frac{{\mathrm{v}}_2+{\mathrm{v}}_1}{{\mathrm{v}}_2-{\mathrm{v}}_1}=\frac{{\mathrm{ev}}_1}{{\mathrm{v}}_1}=\mathrm{e}\\ \frac{{\mathrm{v}}_2}{{\mathrm{v}}_1}=\frac{1+\mathrm{e}}{1-\mathrm{e}}\\ \frac{{\mathrm{v}}_1}{{\mathrm{v}}_2}=\frac{1-\mathrm{e}}{1+\mathrm{e}}\end{array}$