Question

A machine gun of mass 10kg, fires 10gm bullets at the rate of two every second. Each bullet comes out with a velocity of $200 {m}/{s}$. The velocity of recoil of the gun at the end of the fourth second, after firing starts is

Answer 1

To solve this problem, find the momentum of the bullet first and then find the momentum of the gun. Then, apply the law of conservation of momentum and equate the momentum of the gun and momentum of the bullet. This will be the velocity of the gun after 1 sec. Similarly, find the velocity of recoil of the gun after 4 sec. This obtained velocity is due to one bullet. Multiply it by the total number of bullets in 4 second and calculate the velocity of recoil of the gun at the end of the fourth second, after firing starts.

Complete step by step answer:
Given: Mass of the gun (M) = 10 kg
Mass of the bullet (m)= 1gm = 0.01 kg
Velocity of bullet (v)= $200 {m}/{s}$
Let V be the velocity of the gun
Momentum of the bullet is given by,
$p = m \times v$
Substituting values in above equation we get,
$p = 0.01 \times 200$
$\Rightarrow p = 2 kg.{m}/{s}$
Momentum of the gun is given by,
$p = M \times V$
Substituting values in above equation we get,
$p = 10 \times V$
According to the conservation of momentum,
Momentum of the gun= Momentum of the bullet
Substituting the values we get,
$10V = 2$
$\Rightarrow V= 0.2 {m}/{s}$
After the ${2}^{nd}$ second, velocity of the recoil of the gun will be,
$-MV = mv + M{V}_{2}$
Substituting the values in above equation we get,
$-10 \times 0.2 = 0.01 \times 200 + 10 {V}_{2}$
$\Rightarrow {V}_{2}= -0.3 {m}/{s}$
Similarly, after ${4}^{th}$ second, the velocity of the recoil of the gun will be,
${V}_{4} = 0.5 {m}/{s}$
Every second 2 bullets are fired. Therefore, 8 bullets will be fired in 4 secs.
Therefore, the velocity of 8 bullets will be given by,
${V}_{0}= 8 {V}_{4}$
$\Rightarrow {V}_{0}= 8 \times 0.5$
$\Rightarrow {V}_{0}= 4 {m}/{s}$

Hence, the velocity of recoil of the gun at the end of the fourth second, after firing starts is $4 {m}/{s}$.

Note:
Conservation of momentum states that the momentum of the system remains constant unless an external force is applied on the system. Only the external forces have an effect on the momentum of the system. Internal forces do not contribute to the change in the momentum of the system. Thus, it becomes important to distinguish between internal forces and the external forces. No matter which interaction is going on within the system, it’s total momentum will be conserved or remain constant.