During measurement of kinetic energy T , The percentage error in meansurment of mass of particle and momentum of particle are `2%` and `3%` respectively .The percentage error in measurement of kinetic energy is

Options

A. `2%`
B. `5%`
C. `3%`
D. `7%`

Explanation

To find the percentage error in the measurement of kinetic energy (T), we can start by recalling the formula for kinetic energy in terms of mass (m) and velocity (v):

\[ T = \frac{1}{2} mv^2 \]

However, we can also express kinetic energy in terms of momentum (p) and mass (m). The momentum is given by:

\[ p = mv \]

From this, we can express velocity as:

\[ v = \frac{p}{m} \]

Substituting this into the kinetic energy formula gives us:

\[ T = \frac{1}{2} m \left( \frac{p}{m} \right)^2 = \frac{1}{2} \frac{p^2}{m} \]

Now, we will analyze how to calculate the percentage error in kinetic energy based on the percentage errors in mass and momentum.

1. **Identify the percentage errors**:
- Percentage error in mass (m) = 2%
- Percentage error in momentum (p) = 3%

2. **Determine the formula for percentage error in kinetic energy**:
The formula for percentage error in a function of multiple variables can be derived from the following relationship:

If \( T \) is a function of \( p \) and \( m \), then the percentage error in \( T \) can be calculated as:

\[ \frac{\Delta T}{T} \times 100 = \left(2 \frac{\Delta p}{p} + \frac{\Delta m}{m}\right) \times 100 \]

Here, \( \Delta p \) and \( \Delta m \) are the absolute errors in momentum and mass, respectively.

3. **Substituting the known percentage errors**:
- \( \frac{\Delta p}{p} = 3\% \)
- \( \frac{\Delta m}{m} = 2\% \)

Now substituting these values into the formula:

\[ \text{Percentage error in } T = 2 \times 3\% + 2\% \]

4. **Calculating the total percentage error**:
\[ \text{Percentage error in } T = 6\% + 2\% = 8\% \]

Thus, the percentage error in the measurement of kinetic energy is **8%**.