A ball of mass m moving at a speed v makes a head on inelastic collision with an identical ball at rest. The kinetic energy of the balls after the collision is $\frac{3}{4}$^th of the original. Find the coefficient of restitution.

Using law of conservation of momentum,

m(v₁+v₂)=mv

v₁+v₂=v ...(i)

Also , v₁−v₂=ev .......(ii)

After solving,

$v_1^{ }=\left(\frac{1+e}{2}\right)v\text{ and }{v}_2^{ }=\left(\frac{1-e}{2}\right)v$

Given that $\text{ Final K.E. }=\frac{3}{4}\text{ Initial K.E. }$

or $\frac{1}{2}{\mathrm{mv}}_1^2+\frac{1}{2}{\mathrm{mv}}_2^2=\frac{3}{4}\left(\frac{1}{2}{\mathrm{mv}}^2\right)$

Substituting the value, we get 

$\begin{array}{l}{\left(\frac{1+e}{2}\right)}^2+{\left(\frac{1-e}{2}\right)}^2=\frac{3}{4}\\ (1+e{)}^2+\left(1-e^2\right)=3\\ 2+2e^2=3\\ e^2=\frac{1}{2}\\ e=\frac{1}{\sqrt{2}}\end{array}$