Question

A gun of mass $M$ fires a bullet of mass $m$ with a horizontal speed $V$. The gun is fitted with a concave mirror of focal length $f$, facing towards the receding bullet. The speed of separation of the bullet and the image just after the gun was fired is:

• A. $\frac{(M+m)}{M}V$
• B. $\frac{(M+m)}{m}V$
• C. $\frac{2(M+m)}{M}V$
• D. $\frac{2(M+m)}{m}V$

Answer 1

The correct option is C $\frac{2(M+m)}{M}V$

From conservation of linear momentum,
$mV=M{V}^{\prime }$

$⟹V^{\prime }=$ Speed of recoil of gun $=\frac{mV}{M}$

Just after the bullet is fired, $u\to 0$

Magnification $=m=\frac{f}{f-u}=1$

$v_{image}=-m^2{v}_{object}$

$=-1\times (V+V^{\prime })$

$=-(1+\frac{m}{M})V$

Hence the speed of separation of bullet and image $=2\left(\frac{M+m}{M}\right)V$