A gun of mass M fires a bullet of mass m with a Kinetic energy E. The velocity of recoil of the gun is

Options

A. `sqrt(2ME)/m`
B. `sqrt(2mE)/M`
C. `sqrt(2mE)/(M+m)`
D. `sqrt(2ME)/(M+m)`

Answer

To find the velocity of recoil of the gun when it fires a bullet, we can follow these steps:

### Step 1: Understand the given information
- Mass of the gun = \( M \)
- Mass of the bullet = \( m \)
- Kinetic energy of the bullet = \( E \)

### Step 2: Relate kinetic energy to the velocity of the bullet
The kinetic energy \( E \) of the bullet can be expressed using the formula:
\[ E = \frac{1}{2} m v^2 \]
where \( v \) is the velocity of the bullet.

### Step 3: Solve for the velocity of the bullet
Rearranging the kinetic energy formula to solve for \( v \):
\[ v^2 = \frac{2E}{m} \]
Taking the square root of both sides gives:
\[ v = \sqrt{\frac{2E}{m}} \tag{1} \]

### Step 4: Apply the conservation of momentum
According to the law of conservation of momentum, the total momentum before firing must equal the total momentum after firing. Initially, both the gun and bullet are at rest, so:
\[ 0 = mv + MV' \]
where \( V' \) is the recoil velocity of the gun.

### Step 5: Set up the momentum equation
From the conservation of momentum:
\[ 0 = mv + MV' \]
Rearranging gives:
\[ MV' = -mv \]
Thus, we can express the velocity of the gun as:
\[ V' = -\frac{m}{M} v \tag{2} \]

### Step 6: Substitute the expression for \( v \) from Step 3 into the momentum equation
Substituting equation (1) into equation (2):
\[ V' = -\frac{m}{M} \sqrt{\frac{2E}{m}} \]
This simplifies to:
\[ V' = -\sqrt{\frac{2Em}{M}} \tag{3} \]

### Step 7: Interpret the result
The negative sign indicates that the direction of the recoil velocity \( V' \) of the gun is opposite to the direction of the bullet's velocity.

### Final Result
The velocity of recoil of the gun is:
\[ V' = -\sqrt{\frac{2Em}{M}} \]