Physics - Center of Mass Question with Solution | TestHub

This is a perfectly inelastic collision. By conservation of momentum, $mv=(m+M)V$, where $V$ is the final velocity of the combined system. So, $V=\frac{mv}{m+M}$. The initial kinetic energy is $K{E}_i=\frac{1}{2}m{v}^2$. The final kinetic energy is $K{E}_f=\frac{1}{2}(m+M)V^2=\frac{1}{2}(m+M)(\frac{mv}{m+M}{)}^2=\frac{1}{2}(m+M)\frac{m^2{v}^2}{(m+M{)}^2}=\frac{1}{2}\frac{m^2{v}^2}{m+M}$. The loss in kinetic energy is $\Delta KE=K{E}_i-K{E}_f=\frac{1}{2}m{v}^2-\frac{1}{2}\frac{m^2{v}^2}{m+M}=\frac{1}{2}m{v}^2(1-\frac{m}{m+M})=\frac{1}{2}m{v}^2(\frac{m+M-m}{m+M})=\frac{1}{2}\frac{mM}{m+M}{v}^2$.