A machine gun fires `10` bullets per second in horizontal direction. Mass of each bullet is `80g` and muzzle velocity is `500m//s`. If the average horizontal force the shooter must apply to keep the machine gun stationary is `Fxx10^(2)N`, find numerical velue of `F`

Options

A. `4`
B. `5`
C. `8`
D. `12`

Explanation

To solve the problem, we need to determine the average horizontal force that the shooter must apply to keep the machine gun stationary while it fires bullets. Here’s the step-by-step solution:

### Step 1: Understand the given data
- **Number of bullets fired per second (n)** = 10 bullets/s
- **Mass of each bullet (m)** = 80 g = 0.08 kg (since 1 g = 0.001 kg)
- **Muzzle velocity (v)** = 500 m/s

### Step 2: Calculate the momentum of one bullet
The momentum (p) of one bullet can be calculated using the formula:
\[ p = m \times v \]

Substituting the values:
\[ p = 0.08 \, \text{kg} \times 500 \, \text{m/s} = 40 \, \text{kg m/s} \]

### Step 3: Calculate the total momentum change per second
Since the machine gun fires 10 bullets per second, the total change in momentum per second (ΔP) is:
\[ \Delta P = n \times p \]
\[ \Delta P = 10 \, \text{bullets/s} \times 40 \, \text{kg m/s} = 400 \, \text{kg m/s} \]

### Step 4: Calculate the average force
The average force (F) exerted can be calculated using the formula:
\[ F = \frac{\Delta P}{\Delta t} \]
Since the time interval (Δt) is 1 second (as we are considering the change per second):
\[ F = \frac{400 \, \text{kg m/s}}{1 \, \text{s}} = 400 \, \text{N} \]

### Step 5: Express the force in the required format
The problem states that the force can be expressed as \( F \times 10^2 \, \text{N} \). We have found that:
\[ 400 \, \text{N} = F \times 10^2 \]
To find F, we rearrange the equation:
\[ F = \frac{400}{100} = 4 \]

### Final Answer
The numerical value of \( F \) is **4**.

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