A machine gun has a mass 5 kg. It fires 50 gram bullets at the rate of 30 bullets per minute at a speed of `400" m s"^(-1)`. What force is required to keep the gun in positon ?

Options

A. 10N
B. 5N
C. 15N
D. 3N

Answer

To solve the problem, we need to determine the force required to keep the machine gun in position while it is firing bullets. We will use the principle of conservation of momentum and the concept of thrust force.

### Step-by-Step Solution:

1. **Identify Given Data:**
- Mass of the machine gun, \( m_g = 5 \, \text{kg} \)
- Mass of each bullet, \( m_b = 50 \, \text{g} = 0.05 \, \text{kg} \)
- Rate of fire, \( R = 30 \, \text{bullets/min} \)
- Speed of bullets, \( v = 400 \, \text{m/s} \)

2. **Convert Rate of Fire to Bullets per Second:**
\[ R = \frac{30 \, \text{bullets}}{1 \, \text{min}} = \frac{30}{60} \, \text{bullets/s} = 0.5 \, \text{bullets/s} \]

3. **Calculate Mass Flow Rate:**
The mass flow rate \( \frac{dm}{dt} \) can be calculated as follows:
\[ \frac{dm}{dt} = R \cdot m_b = 0.5 \, \text{bullets/s} \times 0.05 \, \text{kg/bullet} = 0.025 \, \text{kg/s} \]

4. **Calculate Thrust Force:**
The thrust force \( F \) exerted by the bullets can be calculated using the formula:
\[ F = \frac{dm}{dt} \cdot v \]
Substituting the values we have:
\[ F = 0.025 \, \text{kg/s} \times 400 \, \text{m/s} = 10 \, \text{N} \]

5. **Conclusion:**
The force required to keep the gun in position, counteracting the thrust force due to the bullets being fired, is:
\[ \text{Force} = 10 \, \text{N} \]