If the error in the measurement of momentum of a particle is (+ 100%), then the error in the measurement of kinetic energy is

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To solve the problem of finding the error in the measurement of kinetic energy given that the error in the measurement of momentum is +100%, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Relationship**:
We know that kinetic energy (KE) is related to momentum (p) by the formula:
\[ KE = \frac{p^2}{2m} \]
This indicates that kinetic energy is directly proportional to the square of momentum.

2. **Identifying the Error in Momentum**:
Given that the error in the measurement of momentum is +100%, this means that the final momentum (p_f) can be expressed as:
\[ p_f = p + (100\% \text{ of } p) = p + p = 2p \]
where \( p \) is the initial momentum.

3. **Calculating the Change in Kinetic Energy**:
Now, we will calculate the initial and final kinetic energy:
- Initial kinetic energy (KE_initial):
\[ KE_{initial} = \frac{p^2}{2m} \]
- Final kinetic energy (KE_final):
\[ KE_{final} = \frac{(p_f)^2}{2m} = \frac{(2p)^2}{2m} = \frac{4p^2}{2m} = \frac{2p^2}{m} \]

4. **Finding the Percentage Change in Kinetic Energy**:
The percentage change in kinetic energy can be calculated using the formula:
\[ \text{Percentage Change} = \frac{KE_{final} - KE_{initial}}{KE_{initial}} \times 100\% \]
Substituting the values we found:
\[ \text{Percentage Change} = \frac{\frac{2p^2}{m} - \frac{p^2}{2m}}{\frac{p^2}{2m}} \times 100\% \]

5. **Simplifying the Expression**:
First, simplify the numerator:
\[ KE_{final} - KE_{initial} = \frac{2p^2}{m} - \frac{p^2}{2m} = \frac{4p^2 - p^2}{2m} = \frac{3p^2}{2m} \]
Now plug this back into the percentage change formula:
\[ \text{Percentage Change} = \frac{\frac{3p^2}{2m}}{\frac{p^2}{2m}} \times 100\% = \frac{3p^2}{2m} \cdot \frac{2m}{p^2} \times 100\% = 3 \times 100\% = 300\% \]

6. **Conclusion**:
Therefore, the error in the measurement of kinetic energy is +300%.

### Final Answer:
The error in the measurement of kinetic energy is **+300%**.