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         [CPMT 1989; Kerala PMT 2002]

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 :

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