When the kinetic energy of a body is increased by there its momentum is increased by:

Options

A. 9 times
B. 3 times
C. `sqrt2` times
D. `sqrt3` times

Explanation

To solve the problem, we need to establish the relationship between kinetic energy (K) and momentum (p) of a body.

### Step-by-Step Solution:

1. **Understanding the Relationship**:
The kinetic energy (K) of a body is given by the formula:
\[ K = \frac{p^2}{2m} \]
where \( p \) is the momentum and \( m \) is the mass of the body.

2. **Expressing Momentum in Terms of Kinetic Energy**:
Rearranging the formula for kinetic energy, we can express momentum in terms of kinetic energy:
\[ p = \sqrt{2mK} \]

3. **Initial and Final Kinetic Energy**:
Let the initial kinetic energy be \( K_i \) and the final kinetic energy be \( K_f \). According to the problem, the final kinetic energy is increased by a factor of 3:
\[ K_f = 3K_i \]

4. **Finding Initial and Final Momentum**:
Using the expression for momentum:
- Initial momentum:
\[ p_i = \sqrt{2mK_i} \]
- Final momentum:
\[ p_f = \sqrt{2mK_f} = \sqrt{2m(3K_i)} = \sqrt{6mK_i} \]

5. **Finding the Ratio of Final Momentum to Initial Momentum**:
Now, we can find the ratio of final momentum to initial momentum:
\[ \frac{p_f}{p_i} = \frac{\sqrt{6mK_i}}{\sqrt{2mK_i}} = \sqrt{\frac{6mK_i}{2mK_i}} = \sqrt{3} \]

6. **Conclusion**:
Therefore, when the kinetic energy of the body is increased by a factor of 3, the momentum is increased by a factor of \( \sqrt{3} \).

### Final Answer:
The momentum is increased by \( \sqrt{3} \).

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