Keeping the momentum unchanged, if the mass of a body is doubled, then its kinetic energy:

Options

A. remains unchanged
B. gets doubled
C. becomes half
D. increases four times

Answer

To solve the problem, we need to analyze how the kinetic energy of a body changes when its mass is doubled while keeping its momentum constant.

### Step-by-Step Solution:

1. **Understand the Definitions**:
- Kinetic Energy (K.E) is given by the formula:
\[ K.E = \frac{1}{2} mv^2 \]
- Momentum (p) is defined as:
\[ p = mv \]

2. **Express Velocity in Terms of Momentum**:
- From the momentum formula, we can express velocity (v) as:
\[ v = \frac{p}{m} \]

3. **Substitute Velocity into the Kinetic Energy Formula**:
- Now, substitute \(v\) in the kinetic energy formula:
\[ K.E = \frac{1}{2} m \left(\frac{p}{m}\right)^2 \]
- This simplifies to:
\[ K.E = \frac{1}{2} m \cdot \frac{p^2}{m^2} = \frac{p^2}{2m} \]

4. **Analyze the Effect of Doubling the Mass**:
- Let’s denote the initial mass as \(m\) and the new mass as \(m_2 = 2m\).
- The new kinetic energy \(K.E_2\) when mass is doubled becomes:
\[ K.E_2 = \frac{p^2}{2m_2} = \frac{p^2}{2(2m)} = \frac{p^2}{4m} \]

5. **Relate the Initial and New Kinetic Energies**:
- The initial kinetic energy \(K.E_1\) is:
\[ K.E_1 = \frac{p^2}{2m} \]
- Now, we can find the ratio of the new kinetic energy to the initial kinetic energy:
\[ \frac{K.E_2}{K.E_1} = \frac{\frac{p^2}{4m}}{\frac{p^2}{2m}} = \frac{1/4}{1/2} = \frac{1}{2} \]

6. **Conclusion**:
- Therefore, the new kinetic energy when the mass is doubled while keeping momentum constant is half of the initial kinetic energy:
\[ K.E_2 = \frac{1}{2} K.E_1 \]

### Final Answer:
The kinetic energy will be halved, so the answer is:
\[ \text{Kinetic Energy is } \frac{1}{2} \text{ of the original kinetic energy.} \]