A ball impinges directly upon another ball at rest and is itself brought to rest by the impact. If half of initial kinetic energy is destroyed in the collision, the coefficient of restitution is 

Here,  ${\mathrm{u}}_2=0 {\mathrm{v}}_1=0$
From conservation of momentum
${\mathrm{m}}_1{\mathrm{u}}_1={\mathrm{m}}_2{\mathrm{v}}_2 \left(1\right)$
Final kinetic energy  $\frac{1}{2}{\mathrm{m}}_2{\mathrm{v}}_2^2=\frac{1}{2}\left(\frac{1}{2}{\mathrm{m}}_1{\mathrm{u}}_1^2\right) \left(2\right)$
From eq = (1)   
$\frac{1}{2}{\mathrm{m}}_2{\mathrm{v}}_2.\mathrm{\vartheta v}=\frac{1}{2}\frac{1}{2}{\mathrm{m}}_1{\mathrm{u}}_1{\mathrm{u}}_1$
${\mathrm{v}}_2=\frac{1}{2}{\mathrm{u}}_1$
$\mathrm{e}=\frac{{\mathrm{v}}_2}{{\mathrm{u}}_1}=\frac{1}{2}=0.5$