Assertion: In case of bullet fired from gun, the ration of kinetic energy of gun and bullet is equal to ration of mass of bullet and gun.
Reason: Kinetic energy `prop(1)/(mass)`, if momentum is constant.

Options

A. (a) If both Assertion and Reason are true and the Reason is correct explanation of the Assertion.
B. (b) If both Assertion and Reason are true but Reason is not the correct explanation of Assertion.
C. (c) If Assertion is true, but the Reason is false.
D. (d) If Assertion is false but the Reason is true.

Explanation

To solve the assertion and reason question, we will analyze the given statements step by step.

### Step 1: Understand the Assertion
The assertion states that "In case of a bullet fired from a gun, the ratio of kinetic energy of the gun and bullet is equal to the ratio of mass of bullet and gun."

### Step 2: Apply Conservation of Momentum
When a bullet is fired from a gun, the total momentum before and after firing must be conserved. Let:
- \( m_b \) = mass of the bullet
- \( m_g \) = mass of the gun
- \( v_b \) = velocity of the bullet after firing
- \( v_g \) = velocity of the gun after firing

According to the conservation of momentum:
\[ m_b v_b + m_g v_g = 0 \]
This implies:
\[ m_b v_b = -m_g v_g \]
Taking magnitudes, we have:
\[ m_b v_b = m_g v_g \]

### Step 3: Express Kinetic Energy
The kinetic energy (KE) of the bullet and the gun can be expressed as:
- Kinetic Energy of Bullet:
\[ KE_b = \frac{1}{2} m_b v_b^2 \]
- Kinetic Energy of Gun:
\[ KE_g = \frac{1}{2} m_g v_g^2 \]

### Step 4: Relate Kinetic Energies to Masses
Using the relation from the conservation of momentum, we can express \( v_g \) in terms of \( v_b \):
\[ v_g = \frac{m_b}{m_g} v_b \]
Now substituting \( v_g \) in the kinetic energy of the gun:
\[ KE_g = \frac{1}{2} m_g \left(\frac{m_b}{m_g} v_b\right)^2 = \frac{1}{2} m_g \frac{m_b^2}{m_g^2} v_b^2 = \frac{m_b^2}{2 m_g} v_b^2 \]

### Step 5: Find the Ratio of Kinetic Energies
Now, we can find the ratio of the kinetic energies:
\[ \frac{KE_g}{KE_b} = \frac{\frac{m_b^2}{2 m_g} v_b^2}{\frac{1}{2} m_b v_b^2} = \frac{m_b^2}{m_g m_b} = \frac{m_b}{m_g} \]
Thus, we have shown that:
\[ \frac{KE_g}{KE_b} = \frac{m_b}{m_g} \]
This confirms the assertion.

### Step 6: Analyze the Reason
The reason states that "Kinetic energy is inversely proportional to mass if momentum is constant." This is also true, as we derived that if momentum is constant, the kinetic energy is inversely proportional to mass.

### Conclusion
Both the assertion and reason are true, and the reason correctly explains the assertion.

### Final Answer
Both assertion and reason are true, and the reason is the correct explanation of the assertion.

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