Question

Percentage error in the measurement of mass and speed are $2\%$ and $3\%$ respectively. The error in the estimation of kinetic energy obtained by measuring mass and speed will be

• A. $12\%$
• B. $10\%$
• C. $2\%$
• D. $8\%$

Answer 1

The Correct Option is D

The problem involves calculating the percentage error in the estimation of kinetic energy when both mass and speed have measurement errors. Let's break down the solution step-by-step:

Step 1: Understanding Kinetic Energy Formula

The formula for kinetic energy (KE) is given as:

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

where m is the mass and v is the speed of the object.

Step 2: Percentage Error in Variables

The percentage error in mass, \Delta m\% = 2\%, and the percentage error in speed, \Delta v\% = 3\%.

Step 3: Apply Error Propagation to Kinetic Energy

For a product of quantities, the relative error in the result is the sum of the relative errors in the quantities:

\frac{\Delta KE}{KE} = \frac{\Delta m}{m} + 2 \times \frac{\Delta v}{v}

Thus, the percentage error in kinetic energy is:

\Delta KE\% = \Delta m\% + 2 \times \Delta v\%

Step 4: Calculate the Total Percentage Error

Plug in the given percentage errors:

\Delta KE\% = 2\% + 2 \times 3\%

Simplifying this, we get:

\Delta KE\% = 2\% + 6\% = 8\%

Conclusion

The percentage error in the estimation of kinetic energy will be 8\%. This matches the correct answer choice given in the options.