In the non-relativistic regime, if the momentum, is increased by 100%, the percentage increase in kinetic energy is

Options

A. 100
B. 200
C. 300
D. 400

Explanation

To solve the problem of finding the percentage increase in kinetic energy when momentum is increased by 100%, we can follow these steps:

### Step-by-Step Solution:

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

2. **Define initial and final momentum**:
Let the initial momentum be \( p_i \) and the final momentum be \( p_f \). According to the problem, the momentum is increased by 100%, which means:
\[ p_f = p_i + 100\% \text{ of } p_i = p_i + p_i = 2p_i \]

3. **Calculate initial kinetic energy**:
The initial kinetic energy \( KE_i \) can be expressed as:
\[ KE_i = \frac{p_i^2}{2m} \]

4. **Calculate final kinetic energy**:
Substitute \( p_f = 2p_i \) into the kinetic energy formula to find the final kinetic energy \( KE_f \):
\[ KE_f = \frac{(p_f)^2}{2m} = \frac{(2p_i)^2}{2m} = \frac{4p_i^2}{2m} = \frac{2p_i^2}{m} \]

5. **Relate final kinetic energy to initial kinetic energy**:
Notice that:
\[ KE_f = 4 \cdot \frac{p_i^2}{2m} = 4 \cdot KE_i \]
This means that the final kinetic energy is four times the initial kinetic energy.

6. **Calculate the change in kinetic energy**:
The change in kinetic energy \( \Delta KE \) is given by:
\[ \Delta KE = KE_f - KE_i = 4KE_i - KE_i = 3KE_i \]

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 found:
\[ \text{Percentage Increase} = \left(\frac{3KE_i}{KE_i}\right) \times 100\% = 3 \times 100\% = 300\% \]

### Final Answer:
The percentage increase in kinetic energy is **300%**.