A particle of mass $5 m$ at rest suddenly breaks on its own into three fragments. Two fragments of m

Question

A particle of mass $5 m$ at rest suddenly breaks on its own into three fragments. Two fragments of mass $m$ each move along mutually perpendicular direction with speed $v$ each. The energy released during the process is\

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(d) : Let the speed of the third fragment of mass $3 m$ be $v^{\prime}$.\ From law of conservation\ of linear momentum,\ $$ 3 m v^{\prime}=\sqrt{2} m v \Rightarrow v^{\prime}=\frac{\sqrt{2} v}{3} 	ag{i} $$ $	herefore \quad$ Energy released during the process is,\ K.E. $=2\left(\frac{1}{2} m v^{2}ight)+\frac{1}{2}(3 m) v^{\prime 2}=m v^{2}+\frac{1}{2}(3 m) \frac{2 v^{2}}{9}$\ (Using eqn. (i))\ $=m v^{2}+\frac{m v^{2}}{3}=\frac{4}{3} m v^{2}$.\