question_answer

Body of mass m moving with a velocity\[v\]in the \[+\text{ }ve\]X direction collides with a body of mass M moving with a velocity V in the \[+\text{ }ve\]Y direction. The collision is perfectly inelastic. Mark out the correct statement(s) w.r.t. this situation.

A) The magnitude of momentum of the composite body is\[\sqrt{{{(mv)}^{2}}+{{(MV)}^{2}}}\]

B) The composite body moves in a direction making an angle \[\theta ={{\tan }^{-1}}\left( \frac{MV}{mv} \right)\]with \[+\text{ }ve\]X-axis.

C) The loss of kinetic energy due to collision

D) All of the above

Correct Answer: D

Solution :

[d] As no external force is acting momentum of the system remains conserved, i.e., \[{{\vec{P}}_{f}}=mv\hat{i}+mV\hat{j}\]

\[|{{\vec{P}}_{f}}|=\sqrt{{{(mv)}^{2}}+{{(MV)}^{2}}}\]

\[\Rightarrow \]\[\tan \theta =\frac{MV}{mv}\]

Loss in \[KE==\Delta K\]

\[=\frac{m{{v}^{2}}}{2}+\frac{M{{V}^{2}}}{2}-\frac{1}{2}\left[ \frac{{{(mv)}^{2}}+{{(MV)}^{2}}}{M+m} \right]\]

\[=\frac{Mm}{2\,(M+m)}\times ({{V}^{2}}+{{v}^{2}})\]