A ball of mass m moving at a speed v makes a head-on collision with an identical ball at rest. The kinetic energy of the balls after the collision is 3/4^th of the original. If the coefficient of restitution is e, find the value of e².

As we have seen in the above discussion that under the given conditions:

${\mathrm{v}}_1=\left(\frac{1+\mathrm{e}}{2}\right)\mathrm{v}\text{ and }{\mathrm{v}}_2=\left(\frac{1-\mathrm{e}}{2}\right)\mathrm{v}$

Given that ${\mathrm{K}}_{\mathrm{f}}=\frac{3}{4}{\mathrm{K}}_{\mathrm{i}}$

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

Substituting the values, we get ${\left(\frac{1+\mathrm{e}}{2}\right)}^2+{\left(\frac{1-\mathrm{e}}{2}\right)}^2=\frac{3}{4}$

$\text{ or }(1+\mathrm{e}{)}^2+(1-\mathrm{e}{)}^2=3\text{ or }2+2{\mathrm{e}}^2=3$

$\text{ or }{\mathrm{e}}^2=\frac{1}{2}\text{ or }\mathrm{e}=\frac{1}{\sqrt{2}}$