First slide

Collisions

Question

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

Moderate

A

Momentum is $\frac{mvM}{M + m}$
B

Kinetic energy is $\frac{{mv}^{2}}{2}$
C

Momentum is $\frac{mv (M + m)}{M}$
D

Kinetic energy is $\frac{m^{2} v^{2}}{2 (M + m)}$

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 = $\frac{1}{2} (m + M) V^{2}$
$= \frac{1}{2} (m + M) {(\frac{mv}{M + m})}^{2} = \frac{1}{2} \frac{m^{2} v^{2}}{M + m}$

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