Question

A gun fires a shell and recoils horizontally. If the shell travels along the barrel with speed $v$, the ratio of speed with which the gun recoils if (i) the barrel is horizontal (ii) inclined at an angle of $30^{\circ}$ with horizontal is

• A. $1$
• B. $\frac{2}{\sqrt{3}}$
• C. $\frac{\sqrt{3}}{2}$
• D. $\frac{1}{2}$

Answer 1

Answer: (B)

Let mass of gun is $M$ and that of shell is $m$. The two cases are shown in the figure as below.

Here $v_{1}$ and $v_{2}$ are the recoil speeds of the gun in two cases. Using conservation of linear momentum in horizontal direction in two cases:
$m\left(v-v_{1}\right)=M v_{1} $
$\therefore v_{1}=\frac{m v}{M+m} \,\,\,...(i)$
$ m\left(v \cos 30^{\circ}-v_{2}\right) $
$\therefore v_{2}=\frac{\sqrt{3} m v}{2(M+m)}\,\,\,...(ii)$
From equations (1) and (2)
$\frac{v_{1}}{v_{2}}=\frac{2}{\sqrt{3}}$