A body of mass (4m) is lying in x - y plane at rest. It suddenly...

The total momentum of the system will be conserved before and after explosion.

Then, ${p}_{i} = {p}_{f}$
O = mv\overset{^}{i} + mv\overset{^}{j} + 2m\overset{\to}{v}
\overset{\to}{v} = - \dfrac{v}{2}\overset{^}{i} - \dfrac{v}{2}\overset{^}{j}
\left|\overset{\to}{v}\right| = \dfrac{v}{\sqrt{2}}$<br />Total kinetic energy generated due to explosion is$KE = \dfrac{v}{2}m{v}^{2} + \dfrac{v}{2}m{v}^{2} + \dfrac{v}{2}2m{\left(\dfrac{v}{\sqrt{2}}\right)}^{2} = m{v}^{2} + \dfrac{m{v}^{2}}{2} = \dfrac{3}{2}m{v}^{2}