When momentum of a body increases by `200%` its KE increases by

Options

A. `200%`
B. `300%`
C. `400%`
D. `800%`

Answer

To solve the problem of how much the kinetic energy (KE) increases when the momentum of a body increases by 200%, we can follow these steps:

### Step-by-Step Solution:

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

2. **Define the initial momentum**:
Let the initial momentum be \( p_i \).

3. **Calculate the final momentum**:
If the momentum increases by 200%, the final momentum \( p_f \) can be calculated as:
\[ p_f = p_i + 200\% \text{ of } p_i = p_i + 2p_i = 3p_i \]

4. **Calculate the initial kinetic energy**:
Using the formula for kinetic energy, the initial kinetic energy \( KE_i \) is:
\[ KE_i = \frac{p_i^2}{2m} \]

5. **Calculate the final kinetic energy**:
Now, substituting \( p_f \) into the kinetic energy formula, the final kinetic energy \( KE_f \) is:
\[ KE_f = \frac{(p_f)^2}{2m} = \frac{(3p_i)^2}{2m} = \frac{9p_i^2}{2m} \]

6. **Find the change in kinetic energy**:
The change in kinetic energy \( \Delta KE \) is given by:
\[ \Delta KE = KE_f - KE_i = \frac{9p_i^2}{2m} - \frac{p_i^2}{2m} = \frac{8p_i^2}{2m} \]

7. **Calculate the percentage increase in kinetic energy**:
The percentage increase in kinetic energy can be calculated using the formula:
\[ \text{Percentage Increase} = \left( \frac{\Delta KE}{KE_i} \right) \times 100 \]
Substituting the values we have:
\[ \text{Percentage Increase} = \left( \frac{\frac{8p_i^2}{2m}}{\frac{p_i^2}{2m}} \right) \times 100 = \left( \frac{8}{1} \right) \times 100 = 800\% \]

### Final Answer:
The increase in kinetic energy is **800%**.