A machine gun of mass `10kg` fires `20g` bullets at the rate of `180` bullets per minute with a speed of `100ms^(-1)` each.The recoil velocity of the gun is :

Options

A. `0.6m/s`
B. `2.5m/s`
C. `1.5m/s`
D. `0.02m/s`

Answer

To find the recoil velocity of the machine gun, we can use the principle of conservation of momentum. Here’s a step-by-step solution:

### Step 1: Determine the number of bullets fired per second
Given that the machine gun fires 180 bullets per minute, we first convert this to bullets per second.

\[ \text{Bullets per second} = \frac{180 \text{ bullets}}{60 \text{ seconds}} = 3 \text{ bullets/second} \]

### Step 2: Calculate the total mass of bullets fired per second
Each bullet has a mass of 20 grams. We convert this mass into kilograms for consistency in SI units.

\[ \text{Mass of one bullet} = 20 \text{ g} = 20 \times 10^{-3} \text{ kg} = 0.02 \text{ kg} \]

Now, we find the total mass of bullets fired per second:

\[ \text{Total mass of bullets per second} = 3 \text{ bullets/second} \times 0.02 \text{ kg/bullet} = 0.06 \text{ kg/s} \]

### Step 3: Use conservation of momentum
According to the conservation of momentum, the momentum before firing (which is zero since the gun is at rest) must equal the momentum after firing.

Let \( m_g \) be the mass of the gun (10 kg), \( v_g \) be the recoil velocity of the gun, \( m_b \) be the total mass of bullets fired per second (0.06 kg), and \( v_b \) be the velocity of the bullets (100 m/s).

The equation based on conservation of momentum is:

\[ 0 = m_g v_g + m_b v_b \]

Rearranging gives:

\[ m_g v_g = -m_b v_b \]

### Step 4: Solve for the recoil velocity of the gun
Substituting the known values into the equation:

\[ 10 \text{ kg} \cdot v_g = -0.06 \text{ kg/s} \cdot 100 \text{ m/s} \]

Calculating the right side:

\[ 10 v_g = -6 \text{ kg m/s} \]

Now, solving for \( v_g \):

\[ v_g = \frac{-6 \text{ kg m/s}}{10 \text{ kg}} = -0.6 \text{ m/s} \]

The negative sign indicates that the gun recoils in the opposite direction to the bullets.

### Final Answer
The recoil velocity of the gun is \( 0.6 \text{ m/s} \) in the opposite direction of the bullet's motion.

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