Question

A machine gun of mass $10\, kg$ fires $20 \,g$ bullets at the rate of $180$ bullets per minute with a speed of $100 \,m s ^{-1}$ each The recoil velocity of the gun is

• A. $0.6\, m / s$
• B. $2.5\, m / s$
• C. $1.5 \,m / s$
• D. $0.02\, m / s$

Answer 1

The Correct Option is A

We are given the following data:

• Mass of the gun, \( M = 10 \, \text{kg} \)
• Mass of each bullet, \( m = 20 \, \text{g} = 0.02 \, \text{kg} \)
• Speed of each bullet, \( v = 100 \, \text{m/s} \)
• Rate of fire, \( R = 180 \, \text{bullets/min} = \frac{180}{60} = 3 \, \text{bullets/sec} \)

We need to find the recoil velocity of the gun.

The principle of conservation of momentum states that the total momentum before and after firing must be the same. Initially, both the gun and the bullets are at rest, so the initial momentum is zero.

The total momentum is the sum of the momentum of the bullets and the recoil momentum of the gun:

Total momentum = 0 Momentum of the bullets = (Number of bullets) × (Mass of one bullet) × (Velocity of the bullet) Recoil momentum of the gun = (Mass of the gun) × (Recoil velocity of the gun)

Since the total momentum must be zero:

0 = M × v(gun) + (m × v) × R

Solving for \( v_{\text{gun}} \) (recoil velocity of the gun):

M × v(gun) = -(m × v) × R v(gun) = - (m × v × R) / M

Substituting the given values:

v(gun) = - (0.02 × 100 × 3) / 10 v(gun) = - 6 / 10 v(gun) = - 0.6 m/s

Thus, the recoil velocity of the gun is \( 0.6 \, \text{m/s} \).