question_answer

A bag (mass \[M\]) hangs by a long thread and a bullet (mass \[m\]) comes horizontally with velocity \[v\] and gets caught in the bag. Then for the combined (bag + bullet) system

A) Momentum is \[\frac{mvM}{M+m}\]

B)  Kinetic energy is \[\frac{m{{v}^{2}}}{2}\]

C)  Momentum is \[\frac{mv{{(M+m)}^{{}}}}{M}\]

D)  Kinetic energy is \[\frac{{{m}^{2}}{{v}^{2}}}{2(M+m)}\]

Correct Answer: D

Solution :

[d] Initial momentum =mv Final momentum = (m+M)V By conservation of momentum \[mv=(m+M)V\] \[\therefore \]Velocity of (bag +bullet) system \[V=\frac{mv}{M+m}\] \[\therefore \] Kinetic energy \[=\frac{1}{2}(m+M){{V}^{2}}\] \[=\frac{1}{2}(m+M){{\left( \frac{mv}{M+m} \right)}^{2}}=\frac{1}{2}\frac{{{m}^{2}}{{v}^{2}}}{(M+m)}\]