Problem 71 - Master Resource Book in JEE Main Physics

Question

The percentage errors in the measurement of mass and speed are \(2 \%\) and \(3 \%\) respectively. How much will be the maximum error in the estimate of kinetic energy obtained by measuring mass and speed? (a) 1196 (b) \(8 \%\) (c) \(59 \%\) (d) \(1 \%\)

Step-by-Step Solution

Answer

The maximum error in kinetic energy is 8\%, answer (b).

Step 1: Formula for Kinetic Energy

The kinetic energy of an object is given by the formula \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass and \( v \) is the velocity.

Step 2: Understand Error Propagation

To find the error in a formula like \( KE = \frac{1}{2}mv^2 \), the key is to understand how errors in measurements of \( m \) and \( v \) affect \( KE \). The percentage error in a product or power is the sum of percentage errors in the measurements.

Step 3: Calculate Percentage Error in Mass and Velocity

We have \(2\%\) error in mass \( m \) and \(3\%\) error in velocity \( v \). Since \( v^2 \) is involved in the kinetic energy formula, this means the error in \( v^2 \) is \(2 \times 3\% = 6\%\).

Step 4: Sum the Percentage Errors

The total percentage error in \( KE \) will be the sum of the percentage error in \( m \) (which is \(2\%\)) and \( v^2 \) (which is \(6\%\)). Therefore, the total error in the kinetic energy is \(2\% + 6\% = 8\%\).

Step 5: Conclusion

The maximum error in the kinetic energy, given the percentage errors in mass and velocity measurements, is \(8\%\). Therefore, the answer is option \( b \).

Key Concepts

Percentage Error

Percentage error is a way to express the accuracy of a measurement by comparing the error size to the actual measurement size. It is commonly used in scientific experiments to give a sense of how significant a measurement error is relative to the correct value.
The formula for calculating percentage error is: \[ \text{Percentage Error} = \left( \frac{\text{Error}}{\text{True Value}} \right) \times 100 \% \] This expression helps determine how much deviation there is from the intended or real measurement value.%
In terms of real-world applications, evaluating measurement precision by percentage error enables us to make well-informed decisions on the legitimacy of our calculations or experimental outcomes.

Measurement Errors

Measurement errors occur in every experiment or observation, and acknowledging them is vital for accurate data interpretation. Errors can arise from many sources, including instrument limitations or human errors. They are typically classified as either systematic or random.

• Systematic errors are consistent, repeatable errors associated with experimental apparatus or procedure flaws.
• Random errors are unpredictable fluctuations that occur during measurement and vary in magnitude and direction.

Recognizing these errors can help in adjusting experimental procedures to minimize their effects.
When dealing with formulas, errors in measurements result in a propagation of error, affecting the outcome. Understanding how errors propagate can assist in estimating the maximum potential error in results, which is essential for maintaining fidelity in experimental work.

Kinetic Energy Calculation

Kinetic energy is the energy that an object possesses due to its motion. The formula for kinetic energy is given by: \[ KE = \frac{1}{2}mv^2 \] This means kinetic energy depends on two variables: mass \( m \) and speed \( v \).
Consequently, measurement errors in mass and velocity directly influence the calculation of kinetic energy. In this context, if there's a percentage error in mass measurement and an error in velocity measurement, these errors affect the kinetic energy result by propagating through the formula.
To find the maximum error in kinetic energy when measurement errors are present, sum up the percentage errors from mass and velocity. In cases involving velocity squared (\( v^2 \)), the percentage error for this term doubles. Combining all these can give the maximum percentage error for kinetic energy, which is crucial when evaluating energy calculations under error constraints.