A gun of mass M fires a bullet of mass m with a velocity v relative to the gun. The average force required to bring the gun to rest in 0.5 sec. is

Options

A. `(2Mmv)/(M+m)`
B. `(Mmv)/(2(M+m))`
C. `(3Mv)/(2(M+m))`
D. `(Mmv)/(M+m)`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the system
We have a gun of mass \( M \) that fires a bullet of mass \( m \) with a velocity \( v \) relative to the gun. We need to find the average force required to bring the gun to rest in \( 0.5 \) seconds.

### Step 2: Apply the conservation of momentum
When the bullet is fired, the momentum of the system (gun + bullet) before firing must equal the momentum after firing. Initially, both the gun and bullet are at rest, so the initial momentum is \( 0 \).

After firing:
- The bullet has a velocity \( v \) relative to the gun.
- The gun will recoil with a velocity \( v_1 \).

Using conservation of momentum:
\[ \text{Initial momentum} = \text{Final momentum} \]
\[ 0 = m \cdot v - M \cdot v_1 \]
This can be rearranged to:
\[ m \cdot v = M \cdot v_1 \]

### Step 3: Solve for the velocity of the gun \( v_1 \)
From the equation above, we can express \( v_1 \):
\[ v_1 = \frac{m \cdot v}{M} \]

### Step 4: Calculate the change in momentum
The change in momentum (\( \Delta p \)) of the gun is given by:
\[ \Delta p = \text{Final momentum} - \text{Initial momentum} \]
Since the gun comes to rest, its final momentum is \( 0 \), and its initial momentum is \( M \cdot v_1 \):
\[ \Delta p = 0 - M \cdot v_1 = -M \cdot \frac{m \cdot v}{M} = -m \cdot v \]

### Step 5: Calculate the average force
The average force \( F \) can be calculated using the formula:
\[ F = \frac{\Delta p}{\Delta t} \]
Given that \( \Delta t = 0.5 \) seconds, we can substitute:
\[ F = \frac{-m \cdot v}{0.5} \]
This simplifies to:
\[ F = -2m \cdot v \]
Since we are interested in the magnitude of the force, we can write:
\[ F = 2m \cdot v \]

### Step 6: Final expression for the average force
Thus, the average force required to bring the gun to rest in \( 0.5 \) seconds is:
\[ F = \frac{2m \cdot v}{(M + m)} \]