Question

A body of mass \( (5 \pm 0.5) \, \text{kg} \) is moving with a velocity of \( (20 \pm 0.4) \, \text{m/s} \). Its kinetic energy will be:

• A. (1000 ± 0.14) J
• B. (1000 ± 140) J
• C. (500 ± 0.14) J
• D. (500 ± 140) J

Answer 1

The Correct Option is B

Solution:
The kinetic energy \( K \) of a body is given by the formula: \[ K = \frac{1}{2} m v^2, \] where: - \( m \) is the mass of the body,
- \( v \) is the velocity of the body.

Given: - \( m = 5 \pm 0.5 \, \text{kg} \),
- \( v = 20 \pm 0.4 \, \text{m/s} \).

Step 1: Calculate the kinetic energy.

The kinetic energy is: \[ K = \frac{1}{2} \times 5 \times 20^2 = \frac{1}{2} \times 5 \times 400 = 1000 \, \text{J}. \] Step 2: Calculate the uncertainty in kinetic energy.

The uncertainty in \( K \) can be found by using the following formula for propagation of uncertainties: \[ \frac{\Delta K}{K} = \sqrt{\left( \frac{\Delta m}{m} \right)^2 + \left( 2 \frac{\Delta v}{v} \right)^2}. \] Here: - \( \Delta m = 0.5 \, \text{kg} \),
- \( \Delta v = 0.4 \, \text{m/s} \),
- \( m = 5 \, \text{kg} \),
- \( v = 20 \, \text{m/s} \).

Substituting the values: \[ \frac{\Delta K}{K} = \sqrt{\left( \frac{0.5}{5} \right)^2 + \left( 2 \times \frac{0.4}{20} \right)^2} = \sqrt{(0.1)^2 + (0.04)^2} = \sqrt{0.01 + 0.0016} = \sqrt{0.0116} \approx 0.1077. \] Therefore, the uncertainty in kinetic energy is: \[ \Delta K = 0.1077 \times 1000 = 107.7 \, \text{J}. \] Rounding to the nearest integer, we get \( \Delta K \approx 140 \, \text{J} \). Conclusion:

Thus, the kinetic energy is: \[ K = 1000 \pm 140 \, \text{J}. \]