Question

A gun of weight of $10kg$ fires a shot of $0.5g$ with a velocity $230m/s$. Velocity of the recoil gun is,
A. $1.51cm/\sec $
B. $1.15cm/\sec $
C. $1.5cm/\sec $
D. $1.10cm/\sec $

Answer 1

Hint: Recoil velocity is the velocity of a body after the launch of the item from the body. In our case, the body is the gun and the item is the bullet. The recoil velocity can be obtained by applying the law of conservation to the gun-bullet system. The negative of the recoil velocity indicates its backward direction.

Formula used:
${v_{recoil}} = - \dfrac{{{m_b}{v_b}}}{{{m_g}}}$

Complete answer:
According to Newton’s 3rd law, “every action has an equal and opposite reaction”. In the case of guns, the shooting of a bullet is the action. This generates a force in the opposite direction, i.e., on the gun in the opposite direction. This is seen as a backward jerk in the shooter.
The velocity due to this force is known as recoil velocity.
Here, the system of the gun and the bullet is closed. So, according to the law of conservation of momentum, the momentum of the system is conserved.
The initial momentum ‘${p_i}$’ of the gun-bullet system is ${p_i} = 0$.
Once the bullet is shot, the final momentum of the system is
${p_f} = {m_b}{v_b} + {m_g}{v_{recoil}}$
Here
${p_f}$ is the final momentum of the gun-bullet system
${m_b}$ is the mass of the bullet
${v_b}$ is the velocity of the bullet
${m_g}$ is the mass of the gun
${v_{recoil}}$ is the recoil velocity of the gun.
From the law of conservation of the momentum, we have
 $\eqalign{
  & {p_i} = {p_f} \cr
  & \Rightarrow 0 = {m_b}{v_b} + {m_g}{v_{recoil}} \cr
  & \Rightarrow {v_{recoil}} = - \dfrac{{{m_b}{v_b}}}{{{m_g}}} \cr
  & \Rightarrow {v_{recoil}} = - \dfrac{{\left( {0.5 \times {{10}^{ - 3}}kg} \right) \times 230m/s}}{{10kg}} \cr
  & \Rightarrow {v_{recoil}} = - 11.5 \times {10^{ - 3}}m = - 1.15cm \cr
  & \therefore {v_{recoil}} = - 1.15cm \cr} $
You must understand that, the negative sign indicates that the recoil velocity of the gun is opposite in direction to the velocity of the bullet, i.e., backwards.

Therefore, the correct option is B.

Note:
- The momentum of a system is given by the product of mass and velocity.
- Always remember that the law of conservation applies only to a closed system. Here the closed system means there should be no exchange of mass and no external force must be acted upon.