A bullet is fired from the gun. The gun recoils, the kinetic energy of the recoil shall be-

Options

A. K
B. more than K
C. less than K
D. `sqrt(K)`

Explanation

To solve the problem regarding the kinetic energy of the recoil of a gun when a bullet is fired, we can follow these steps:

### Step 1: Understand the Law of Conservation of Momentum
When the bullet is fired from the gun, the total momentum before firing is equal to the total momentum after firing. Initially, both the gun and the bullet are at rest, so the total momentum is zero. After firing, the momentum of the bullet and the recoil momentum of the gun must be equal in magnitude but opposite in direction.

### Step 2: Define the Variables
Let:
- \( m_b \) = mass of the bullet
- \( m_g \) = mass of the gun
- \( v_b \) = velocity of the bullet after firing
- \( v_g \) = recoil velocity of the gun

According to the conservation of momentum:
\[ m_b \cdot v_b + m_g \cdot (-v_g) = 0 \]
This simplifies to:
\[ m_b \cdot v_b = m_g \cdot v_g \]

### Step 3: Express Recoil Velocity
From the momentum equation, we can express the recoil velocity of the gun:
\[ v_g = \frac{m_b \cdot v_b}{m_g} \]

### Step 4: Calculate Kinetic Energies
The kinetic energy (KE) of the bullet and the gun can be expressed as:
- Kinetic energy of the bullet:
\[ KE_b = \frac{1}{2} m_b v_b^2 \]
- Kinetic energy of the gun:
\[ KE_g = \frac{1}{2} m_g v_g^2 \]

Substituting \( v_g \) from the previous step into the kinetic energy equation for the gun:
\[ KE_g = \frac{1}{2} m_g \left(\frac{m_b \cdot v_b}{m_g}\right)^2 = \frac{1}{2} m_g \cdot \frac{m_b^2 \cdot v_b^2}{m_g^2} = \frac{1}{2} \cdot \frac{m_b^2 \cdot v_b^2}{m_g} \]

### Step 5: Compare Kinetic Energies
Now, we can compare the kinetic energies of the bullet and the gun. Since \( m_g > m_b \), it follows that:
\[ KE_g < KE_b \]
This indicates that the kinetic energy of the recoil of the gun is less than the kinetic energy of the bullet.

### Conclusion
Thus, the kinetic energy of the recoil of the gun is less than that of the bullet.

### Final Answer
The kinetic energy of the recoil shall be less than that of the bullet.

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