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:
$m\left(v-v_1\right)=M{v}_1$
$\therefore v_1=\frac{mv}{M+m}...(i)$
$m\left(v\cos 30^{\circ }-v_2\right)$
$\therefore v_2=\frac{\sqrt{3}mv}{2(M+m)}...(ii)$
From equations (1) and (2)
$\frac{v_1}{v_2}=\frac{2}{\sqrt{3}}$