Percentage error in mass and momentum are 3% and 2% respectively. Maximum possible percentage error in the kinetic energy is

Options

A. `7%`
B. `2%`
C. `4%`
D. `3%`

Explanation

To find the maximum possible percentage error in kinetic energy given the percentage errors in mass and momentum, we can follow these steps:

### Step-by-Step Solution:

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

2. **Relate momentum to kinetic energy:**
The momentum \( p \) is defined as:
\[ p = mv \]
We can express kinetic energy in terms of momentum:
\[ K.E. = \frac{p^2}{2m} \]

3. **Determine the percentage errors:**
Given:
- Percentage error in mass \( \left(\frac{\Delta m}{m}\right) \) = 3%
- Percentage error in momentum \( \left(\frac{\Delta p}{p}\right) \) = 2%

4. **Calculate the maximum possible percentage error in kinetic energy:**
The formula for the relative error in kinetic energy when expressed in terms of momentum is:
\[ \frac{\Delta K.E.}{K.E.} = 2 \frac{\Delta p}{p} + \frac{\Delta m}{m} \]
Substituting the known percentage errors:
\[ \frac{\Delta K.E.}{K.E.} = 2 \times 2\% + 3\% \]

5. **Perform the calculations:**
\[ \frac{\Delta K.E.}{K.E.} = 4\% + 3\% = 7\% \]

6. **Conclusion:**
Therefore, the maximum possible percentage error in kinetic energy is:
\[ \text{Maximum possible percentage error in K.E.} = 7\% \]