If linear momentum if increased by `50%` then kinetic energy will be increased by

Options

A. `50%`
B. `100%`
C. `125%`
D. `25%`

Answer

To solve the problem of how much the kinetic energy increases when linear momentum is increased by 50%, we can follow these steps:

### Step-by-Step Solution:

1. **Define Initial Momentum**:
Let the initial linear momentum be \( P_0 \).

2. **Calculate Increased Momentum**:
If the momentum is increased by 50%, the new momentum \( P \) can be calculated as:
\[ P = P_0 + 0.5 P_0 = 1.5 P_0 = \frac{3}{2} P_0 \]

3. **Relate Kinetic Energy to Momentum**:
The kinetic energy \( K \) of an object is given by the formula:
\[ K = \frac{P^2}{2m} \]
where \( P \) is the momentum and \( m \) is the mass of the object.

4. **Calculate Initial Kinetic Energy**:
The initial kinetic energy \( K_0 \) can be expressed as:
\[ K_0 = \frac{P_0^2}{2m} \]

5. **Calculate Final Kinetic Energy**:
The final kinetic energy \( K \) after the momentum increase is:
\[ K = \frac{P^2}{2m} = \frac{(1.5 P_0)^2}{2m} = \frac{(2.25 P_0^2)}{2m} = \frac{2.25}{2} \cdot \frac{P_0^2}{m} = \frac{1.125 P_0^2}{m} \]

6. **Find the Ratio of Final to Initial Kinetic Energy**:
The ratio of the final kinetic energy \( K \) to the initial kinetic energy \( K_0 \) is:
\[ \frac{K}{K_0} = \frac{\frac{1.125 P_0^2}{m}}{\frac{P_0^2}{2m}} = \frac{1.125}{\frac{1}{2}} = 1.125 \times 2 = 2.25 \]

7. **Calculate the Change in Kinetic Energy**:
The change in kinetic energy can be calculated as:
\[ \Delta K = K - K_0 = K_0 \left( \frac{K}{K_0} - 1 \right) = K_0 (2.25 - 1) = K_0 \times 1.25 \]

8. **Express Change in Percentage**:
The change in kinetic energy as a percentage is:
\[ \text{Percentage Increase} = \Delta K \times 100\% = 1.25 K_0 \times 100\% = 125\% \]

### Final Answer:
The kinetic energy will be increased by **125%**.