• # question_answer A body of mass M splits into two parts $\alpha$ and $(1-\alpha )$ M by an internal explosion which generates kinetic energy T. After explosion if the two parts move in the same direction as before, their relative speed will be - A) $\sqrt{\frac{T}{(1-\alpha )M}}$ B) $\sqrt{\frac{2T}{\alpha (1-\alpha )M}}$ C) $\sqrt{\frac{T}{2(1-\alpha )M}}$ D) $\sqrt{\frac{2T}{(1-a)M}}$

 [b] Let the speed of the body before explosion be u. After explosion, if the two parts move with velocities ${{u}_{1}}$ and ${{u}_{2}}$ in the same direction, then according to conservation of momentum, Or $M{{u}_{1}}+\left( 1-\alpha \right)M\,{{u}^{2}}=Mu$ The kinetic energy T liberated during explosion is given by $T=\frac{1}{2}\alpha Mu_{1}^{2}+\frac{1}{2}\left( 1-\alpha \right)Mu_{2}^{2}-\frac{1}{2}M{{u}^{2}}$ $=\frac{1}{2}\alpha M\,u_{1}^{2}+\frac{1}{2}\left( 1-\alpha \right)M\,u_{2}^{2}-\frac{1}{2M}$ ${{\left[ \alpha M{{u}_{1}}+\left( 1-\alpha \right)M{{u}_{2}} \right]}^{2}}$ $=\frac{1}{2}M\alpha \left( 1-\alpha \right)\left[ u_{1}^{2}+u_{2}^{2}-2{{u}_{1}}{{u}_{2}} \right]$ ${{\left( {{u}_{1}}-{{u}_{2}} \right)}^{2}}=\frac{2T}{\alpha \left( 1-\alpha \right)M}$ $\Rightarrow \left( {{u}_{1}}-{{u}_{2}} \right)=\sqrt{\frac{2T}{\alpha \left( 1-\alpha \right)M}}$