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 1: Understand the relationship between momentum and kinetic energy
The momentum \( p \) of an object is given by the formula:
\[ p = mv \]
where \( m \) is the mass and \( v \) is the velocity. The kinetic energy \( K \) of an object is given by:
\[ K = \frac{1}{2} mv^2 \]

### Step 2: Express kinetic energy in terms of momentum
We can express kinetic energy in terms of momentum. Since \( p = mv \), we can solve for \( v \):
\[ v = \frac{p}{m} \]
Substituting this into the kinetic energy formula gives:
\[ K = \frac{1}{2} m \left(\frac{p}{m}\right)^2 = \frac{p^2}{2m} \]

### Step 3: Determine the initial and final momentum
Let the initial momentum be \( p_1 \) and the final momentum after a 100% increase be:
\[ p_2 = p_1 + 100\% \text{ of } p_1 = 2p_1 \]

### Step 4: Calculate the initial and final kinetic energy
Using the relationship derived in Step 2:
- Initial kinetic energy \( K_1 \):
\[ K_1 = \frac{p_1^2}{2m} \]
- Final kinetic energy \( K_2 \):
\[ K_2 = \frac{(p_2)^2}{2m} = \frac{(2p_1)^2}{2m} = \frac{4p_1^2}{2m} = \frac{2p_1^2}{m} \]

### Step 5: Find the percentage increase in kinetic energy
The percentage increase in kinetic energy can be calculated using the formula:
\[ \text{Percentage Increase} = \frac{K_2 - K_1}{K_1} \times 100 \]
Substituting the values of \( K_1 \) and \( K_2 \):
\[ \text{Percentage Increase} = \frac{\frac{2p_1^2}{m} - \frac{p_1^2}{2m}}{\frac{p_1^2}{2m}} \times 100 \]
Simplifying this:
1. Find a common denominator for the terms in the numerator:
\[ K_2 - K_1 = \frac{2p_1^2}{m} - \frac{p_1^2}{2m} = \frac{4p_1^2 - p_1^2}{2m} = \frac{3p_1^2}{2m} \]
2. Substitute this back into the percentage increase formula:
\[ \text{Percentage Increase} = \frac{\frac{3p_1^2}{2m}}{\frac{p_1^2}{2m}} \times 100 \]
3. Simplifying further:
\[ = \frac{3}{1} \times 100 = 300\% \]

### Final Answer
The percentage increase in kinetic energy when momentum is increased by 100% is **300%**.

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