If the error in the measurement of momentum is 20 % , then the error in the calculation of kinetic energy is (assume the error in measurement of m as zero)

Options

A. `20%`
B. `44%`
C. `40%`
D. `200%`

Answer

To solve the problem, we need to find the error in the calculation of kinetic energy given that the error in the measurement of momentum is 20%. We will assume that the error in the measurement of mass (m) is zero.

### Step-by-Step Solution:

1. **Understand the relationship between momentum and kinetic energy**:
- The formula for kinetic energy (KE) is given by:
\[ KE = \frac{1}{2} mv^2 \]
- The momentum (p) is defined as:
\[ p = mv \]

2. **Express kinetic energy in terms of momentum**:
- We can express kinetic energy in terms of momentum by substituting \(v\) from the momentum formula:
\[ v = \frac{p}{m} \]
- Substituting this into the kinetic energy formula:
\[ KE = \frac{1}{2} m \left(\frac{p}{m}\right)^2 = \frac{p^2}{2m} \]

3. **Determine the error in kinetic energy**:
- We need to find the relative error in kinetic energy. The formula for the relative error in a function of variables is given by:
\[ \frac{\Delta KE}{KE} = \frac{\Delta p}{p} + \frac{\Delta m}{m} \]
- Since the error in mass (\(\Delta m\)) is zero (as given), the equation simplifies to:
\[ \frac{\Delta KE}{KE} = \frac{\Delta p}{p} \]

4. **Account for the exponent in the momentum term**:
- Since kinetic energy is proportional to the square of momentum, we need to account for this:
\[ \frac{\Delta KE}{KE} = 2 \frac{\Delta p}{p} \]

5. **Substituting the known error**:
- Given that the error in momentum measurement is 20%, we can write:
\[ \frac{\Delta p}{p} = 0.20 \]
- Therefore, substituting this into the equation gives:
\[ \frac{\Delta KE}{KE} = 2 \times 0.20 = 0.40 \]

6. **Convert to percentage**:
- To express this as a percentage, we multiply by 100:
\[ \text{Percentage error in KE} = 0.40 \times 100 = 40\% \]

### Final Answer:
The error in the calculation of kinetic energy is **40%**.