A ball collides  directly on another ball at rest. The first ball is brought to rest after the impact. If half of the kinetic energy is lost by impact, the value of coefficient of restitution (e) is

$\therefore \mathrm{Initial} \mathrm{KE}, {\mathrm{K}}_{\mathrm{i}} = \frac{1}{2}{m_1{u}_1}^2+\frac{1}{2}{m_2{u}^2}_2 = \frac{1}{2}{m_1{u}^2}_1$

and final KE, ${\mathrm{K}}_{\mathrm{f}} = \frac{1}{2}{m_1{v}^2}_1+\frac{1}{2}{m_2{v}_2}^2 = \frac{1}{2}{m_2{v}^2}_2$

Loss in KE, $∆\mathrm{K} = {\mathrm{K}}_{\mathrm{i}}-{\mathrm{k}}_{\mathrm{f}} = \frac{1}{2}{m_1{u}^2}_1-\frac{1}{2}{m}_2{v_2}^2$

It is given $\frac{1}{2}\left(\frac{1}{2}{m_1{u}^2}_1\right) = \frac{1}{2}{m_1{u}^2}_1-\frac{1}{2}{m_2{v}^2}_2$

($\text{∴ Half of its KE is lost by impact}$)

or ${{\mathrm{m}}_1{\mathrm{u}}^2}_1 = 2{\mathrm{m}}_2{{\mathrm{v}}^2}_2 \cdots \cdots \left(1\right)$

$\mathrm{again} \mathrm{from} \mathrm{conservation} \mathrm{of} \mathrm{momentum}$

${\mathrm{m}}_1{\mathrm{u}}_1={\mathrm{m}}_2{\mathrm{v}}_2\cdots \cdots \left(2\right)$

$\mathrm{dividing} \left(1\right) \mathrm{by} \left(2\right)$

${\mathrm{u}}_1=2{\mathrm{v}}_2$

 coefficient of restitution, $\mathrm{e} = \left|\frac{{\mathrm{v}}_2-{\mathrm{v}}_1}{{\mathrm{u}}_1-{\mathrm{u}}_2}\right| = \frac{{\mathrm{v}}_2}{{\mathrm{u}}_1} = \frac{1}{2}$