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 related to its momentum (P) by the formula:
\[ KE = \frac{P^2}{2m} \]
where \( m \) is the mass of the body.

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

3. **Calculate the final momentum after a 200% increase**:
A 200% increase in momentum means the final momentum \( P_f \) is:
\[ P_f = P_i + 2 \times P_i = 3P_i \]

4. **Calculate the initial kinetic energy**:
Using the formula for kinetic energy:
\[ KE_i = \frac{P_i^2}{2m} \]

5. **Calculate the final kinetic energy**:
Substitute \( P_f \) into the kinetic energy formula:
\[ KE_f = \frac{(P_f)^2}{2m} = \frac{(3P_i)^2}{2m} = \frac{9P_i^2}{2m} \]

6. **Determine the change in kinetic energy**:
The change in kinetic energy \( \Delta KE \) is given by:
\[ \Delta KE = KE_f - KE_i = \frac{9P_i^2}{2m} - \frac{P_i^2}{2m} \]
Simplifying this gives:
\[ \Delta KE = \frac{9P_i^2 - P_i^2}{2m} = \frac{8P_i^2}{2m} = \frac{4P_i^2}{m} \]

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:
\[ \text{Percentage Increase} = \left( \frac{\frac{4P_i^2}{m}}{\frac{P_i^2}{2m}} \right) \times 100 = \left( \frac{4}{\frac{1}{2}} \right) \times 100 = 8 \times 100 = 800\% \]

### Final Answer:
The kinetic energy increases by **800%** when the momentum of a body increases by 200%.