A ball A moving with a certain velocity collides, with another ball B of the same mass at rest. If the coefficient of restitution is e, the ratio of the velocities of A and B just after the collision is

$m_1{u}_1+m_2{u}_2=m_1{v}_1+m_2{v}_2$

               $m{u}_1=m_1{v}_1+m{v}_2$

               $u_1=v_1+v_2\to$ ……… (1)

               $e=\frac{v_2-v_1}{u_1-0}\Rightarrow e{u}_1=v_2-v_1$ …….(2)

               Solving (1) and (2)

               $v_1=\frac{1-e}{2}$                                  $v_2=\frac{1+e}{2}$

               $\frac{v_1}{v_2}=\frac{1-e}{1+e}$