A machine gun fires a steady stream of bullets at the rate of n per minute into a stationary target in which the bullets get embedded. If each bullet has a mass m and arrives at the target with a velocity v, the average force on the target is

Options

A. 60 mnv
B. `(60v)/(mn)`
C. `(mn v)/(60)`
D. `(mv)/(60n)`

Explanation

To solve the problem, we need to determine the average force exerted on a stationary target by a machine gun firing bullets at a steady rate. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Given Information
- The machine gun fires bullets at a rate of \( n \) bullets per minute.
- Each bullet has a mass \( m \).
- Each bullet arrives at the target with a velocity \( v \).

### Step 2: Convert the Rate of Fire to Seconds
Since the rate of fire is given in bullets per minute, we need to convert this to bullets per second:
\[ \text{Rate of fire in bullets per second} = \frac{n}{60} \]

### Step 3: Calculate the Momentum Change for One Bullet
The momentum \( p \) of one bullet when it hits the target is given by:
\[ p = mv \]
Since the bullets get embedded in the target, the change in momentum for each bullet (from \( mv \) to 0) is simply:
\[ \Delta p = mv - 0 = mv \]

### Step 4: Calculate the Total Change in Momentum per Second
If \( \frac{n}{60} \) bullets are fired per second, the total change in momentum per second (which is the total momentum change due to all bullets hitting the target in one second) is:
\[ \text{Total change in momentum per second} = \left(\frac{n}{60}\right) \times mv = \frac{n mv}{60} \]

### Step 5: Relate Change in Momentum to Force
The average force \( F \) exerted on the target is defined as the rate of change of momentum. Therefore, we can express the average force as:
\[ F = \frac{\text{Total change in momentum}}{\text{Time}} = \frac{n mv}{60} \]

### Final Answer
Thus, the average force on the target is:
\[ F = \frac{n mv}{60} \]