First slide

Collisions

Question

A bag of sand of mass M is suspended by a string. A bullet of mass m is fired at it with velocity v and gets embedded into it. The loss of kinetic energy in this process is

Moderate

A

$\frac{1}{2} {mv}^{2}$
B

$\frac{1}{2} {mv}^{2} \times \frac{1}{M + m}$
C

$\frac{1}{2} {mv}^{2} \times \frac{M}{m}$
D

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

Solution

Initial kinetic energy of bullet = $\frac{1}{2} {mv}^{2}$
After inelastic collision system moves with velocity V
By the conservation of momentum
$mv + 0 = (m + M) V \Rightarrow V = \frac{mv}{m + M}$
Kinetic energy of system = $\frac{1}{2} (m + M) V^{2}$
= $\frac{1}{2} (m + M) {(\frac{mv}{m + M})}^{2}$
Loss of kinetic energy = $\frac{1}{2} {mv}^{2} - \frac{1}{2} (m + M) {(\frac{mv}{m + M})}^{2}$
= $\frac{1}{2} {mv}^{2} (\frac{M}{m + M})$

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