The momentum of a body is increased by 25%. The kinetic energy is increased by about

Options

A. `25%`
B. `5%`
C. `56%`
D. `38%`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Define Initial and Final Momentum
Let the initial momentum of the body be \( P_i \). According to the problem, the momentum is increased by 25%. Therefore, the final momentum \( P_f \) can be expressed as:
\[ P_f = P_i + 0.25 P_i = 1.25 P_i \]

### Step 2: Express Kinetic Energy in Terms of Momentum
The kinetic energy \( K \) of a body is related to its momentum \( P \) and mass \( m \) by the formula:
\[ K = \frac{P^2}{2m} \]
Thus, the initial kinetic energy \( K_i \) can be expressed as:
\[ K_i = \frac{P_i^2}{2m} \]
And the final kinetic energy \( K_f \) can be expressed as:
\[ K_f = \frac{P_f^2}{2m} \]

### Step 3: Substitute Final Momentum into Kinetic Energy Formula
Substituting the expression for \( P_f \) into the kinetic energy formula gives:
\[ K_f = \frac{(1.25 P_i)^2}{2m} = \frac{1.5625 P_i^2}{2m} \]

### Step 4: Relate Final Kinetic Energy to Initial Kinetic Energy
Now, we can express \( K_f \) in terms of \( K_i \):
\[ K_f = 1.5625 \cdot \frac{P_i^2}{2m} = 1.5625 K_i \]

### Step 5: Calculate the Change in Kinetic Energy
The change in kinetic energy \( \Delta K \) is given by:
\[ \Delta K = K_f - K_i = 1.5625 K_i - K_i = (1.5625 - 1) K_i = 0.5625 K_i \]

### Step 6: Calculate Percentage Increase in Kinetic Energy
The percentage increase in kinetic energy can be calculated using the formula:
\[ \text{Percentage Increase} = \left( \frac{\Delta K}{K_i} \right) \times 100\% \]
Substituting the change in kinetic energy:
\[ \text{Percentage Increase} = \left( \frac{0.5625 K_i}{K_i} \right) \times 100\% = 56.25\% \]

### Conclusion
Thus, the kinetic energy is increased by approximately **56%** when the momentum of the body is increased by 25%.

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