The kinetic energy of a particle depends on the square of speed of the particle. If error in measurement of speed is 30%, the error in the measurement of kinetic energy will be

Options

A. `30%`
B. `60%`
C. `69%`
D. `15%`

Answer

To solve the problem, we need to determine how the error in the measurement of speed affects the error in the measurement of kinetic energy. The kinetic energy (KE) of a particle is given by the formula:

\[ KE = \frac{1}{2} m v^2 \]

where \( m \) is the mass of the particle and \( v \) is its speed.

### Step-by-Step Solution:

1. **Understanding the Relationship**:
The kinetic energy depends on the square of the speed. If the speed \( v \) has an error of 30%, we can express this mathematically.

2. **Calculating the New Speed**:
If the speed \( v \) is measured with a 30% error, the new speed can be calculated as:
\[ v_{new} = v + 0.3v = 1.3v \]

3. **Calculating the Kinetic Energies**:
- The initial kinetic energy \( KE_1 \) is:
\[ KE_1 = \frac{1}{2} m v^2 \]
- The new kinetic energy \( KE_2 \) with the new speed is:
\[ KE_2 = \frac{1}{2} m (1.3v)^2 = \frac{1}{2} m (1.69v^2) = 1.69 \left(\frac{1}{2} m v^2\right) = 1.69 KE_1 \]

4. **Finding the Percentage Change in Kinetic Energy**:
The percentage change in kinetic energy can be calculated using the formula:
\[ \text{Percentage Change} = \frac{KE_2 - KE_1}{KE_1} \times 100 \]
Substituting \( KE_2 = 1.69 KE_1 \):
\[ \text{Percentage Change} = \frac{1.69 KE_1 - KE_1}{KE_1} \times 100 = \frac{0.69 KE_1}{KE_1} \times 100 \]
Simplifying this gives:
\[ \text{Percentage Change} = 0.69 \times 100 = 69\% \]

5. **Conclusion**:
Therefore, the error in the measurement of kinetic energy due to a 30% error in speed measurement is **69%**.