If the percentage errors in measuring mass and velocity of a particle are respectively `2%` and `1%` percentage error in measuring its kinetic energy isp

Options

A. (a)`1%`
B. (b)`2%`
C. (c)`4%`
D. (d)`8%`

Answer

To find the percentage error in measuring the kinetic energy of a particle given the percentage errors in mass and velocity, we can follow these steps:

### Step-by-Step Solution:

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

2. **Identify the given percentage errors**:
- The percentage error in measuring mass (\(m\)) is given as \(2\%\).
- The percentage error in measuring velocity (\(v\)) is given as \(1\%\).

3. **Determine the formula for percentage error in kinetic energy**:
The percentage error in a product or quotient can be calculated using the formula:
\[ \text{Percentage Error in } KE = \left(\frac{\Delta KE}{KE}\right) \times 100 \]
For kinetic energy, we can express the error as:
\[ \frac{\Delta KE}{KE} = \frac{1}{2} \left( \frac{\Delta m}{m} + 2 \frac{\Delta v}{v} \right) \]
Here, \(\Delta m\) and \(\Delta v\) are the absolute errors in mass and velocity, respectively.

4. **Substitute the given percentage errors**:
- The percentage error in mass \(\frac{\Delta m}{m}\) is \(2\%\).
- The percentage error in velocity \(\frac{\Delta v}{v}\) is \(1\%\).

Plugging these values into the formula gives:
\[ \text{Percentage Error in } KE = \frac{1}{2} \left( 2\% + 2 \times 1\% \right) \]

5. **Calculate the total percentage error**:
Simplifying the expression:
\[ \text{Percentage Error in } KE = \frac{1}{2} \left( 2\% + 2\% \right) = \frac{1}{2} \times 4\% = 2\% \]

6. **Final result**:
Therefore, the percentage error in measuring the kinetic energy is:
\[ \text{Percentage Error in } KE = 4\% \]

### Conclusion:
The percentage error in measuring the kinetic energy of the particle is **4%**.

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