A running man has  $\frac{1}{4}\mathrm{th}$ kinetic energy that a boy of half his mass. Then the velocity of the boy is

Mass of the man = m
Velocity of man = u
Mass of the boy  = $\frac{\mathrm{m}}{2}$
Velocity of boy = v
Kinetic energy of man  $=\frac{1}{4}\mathrm{x}$ kinetic energy of boy
$\begin{array}{l}\frac{1}{2}{\mathrm{mu}}^2=\frac{1}{4}\times \frac{1}{2}\left(\frac{\mathrm{m}}{2}\right){\mathrm{v}}^2\\ {\mathrm{u}}^2=\frac{{\mathrm{v}}^2}{8}\\ 8{\mathrm{u}}^2={\mathrm{v}}^2\\ \mathrm{v}=\sqrt{8}\mathrm{u}\\ \mathrm{v}=2\sqrt{2}\mathrm{u}\end{array}$