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. `1%`
B. `2%`
C. `4%`
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 1: Understand the formula for kinetic energy
The kinetic energy (KE) of a particle is given by the formula:
\[ KE = \frac{1}{2} mv^2 \]

### Step 2: Identify the percentage errors
We are given:
- Percentage error in mass (\( \Delta m/m \)) = 2%
- Percentage error in velocity (\( \Delta v/v \)) = 1%

### Step 3: Use the formula for percentage error in kinetic energy
The formula for the percentage error in kinetic energy is derived from the kinetic energy formula:
\[ \frac{\Delta KE}{KE} = \frac{\Delta m}{m} + 2 \times \frac{\Delta v}{v} \]
Where:
- \( \Delta KE \) is the absolute error in kinetic energy,
- \( KE \) is the kinetic energy,
- \( \Delta m \) is the absolute error in mass,
- \( m \) is the mass,
- \( \Delta v \) is the absolute error in velocity,
- \( v \) is the velocity.

### Step 4: Substitute the values into the formula
Now, substituting the given percentage errors into the formula:
\[ \frac{\Delta KE}{KE} = \frac{2}{100} + 2 \times \frac{1}{100} \]

### Step 5: Calculate the total percentage error
Calculating the right-hand side:
\[ \frac{\Delta KE}{KE} = \frac{2}{100} + \frac{2 \times 1}{100} = \frac{2}{100} + \frac{2}{100} = \frac{4}{100} \]
Thus, the percentage error in kinetic energy is:
\[ \frac{\Delta KE}{KE} \times 100 = 4\% \]

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

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