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(C) The initial kinetic energy of the system is $K_i = \frac{1}{2} m u^2$. In a perfectly inelastic collision, the particles stick together and move w

Vedclass · Accepted Answer

Statement-$1$ is false, Statement-$2$ is true.

Vedclass · Answer

(C) The initial kinetic energy of the system is $K_i = \frac{1}{2} m u^2$. In a perfectly inelastic collision, the particles stick together and move with a common velocity $v = \frac{mu}{m+M}$. The final kinetic energy of the system is $K_f = \frac{1}{2} (m+M) v^2 = \frac{1}{2} (m+M) \left( \frac{mu}{m+M} \right)^2 = \frac{1}{2} \frac{m^2 u^2}{m+M} = \left( \frac{m}{m+M} \right) \left( \frac{1}{2} m u^2 \right)$. The energy loss is $\Delta K = K_i - K_f = \frac{1}{2} m u^2 - \left( \frac{m}{m+M} \right) \left( \frac{1}{2} m u^2 \right) = \left( 1 - \frac{m}{m+M} \right) \left( \frac{1}{2} m u^2 \right) = \left( \frac{M}{m+M} \right) \left( \frac{1}{2} m u^2 \right)$. Comparing this with the given expression $f \left( \frac{1}{2} m u^2 \right)$, we find $f = \frac{M}{m+M}$. Statement-$1$ claims $f = \frac{m}{M+m}$, which is incorrect. Statement-$2$ is correct because maximum energy loss occurs in a perfectly inelastic collision where particles stick together.

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