Question

A bullet of mass $m$ leaves a gun of mass $M$ kept on a smooth horizontal surface. If the speed of the bullet relative to the gun is $v$ , the magnitude of recoil speed of the gun will be

• A. $\frac{m}{M}v$
• B. $\frac{m}{M+m}v$
• C. $\frac{m}{M-m}v$
• D. $\frac{M}{m}v$

Answer 1

Answer: (B)

Speed of the bullet relative to ground ${\vec{v}}_b=\vec{v}+{\vec{v}}_r$,
where $v_r$ is recoil velocity of gun.
Now for gun-bullet system applying the conservation law of momentum, we get
$m{\vec{v}}_b+M{\vec{v}}_r=0$
or $m\left(\vec{v}+(\vec{v}{)}_r\right)+M(\vec{v}{)}_r=0$
$\Rightarrow {\vec{v}}_r=-\frac{m\vec{v}}{m+M}$
or $v_r=\frac{mv}{m+M}$