Question

The error in the measurement of the momentum of a particle is $\left( { + 100} \right)\%$, then the error in the measurement of kinetic energy is:
(A) $25\%$
(B) $50\%$
(C) $100\%$
(D) $300\%$

Answer 1

When we use instruments to measure a quantity or calculate the value of a quantity, the measured or calculated value will be different from its actual value. This difference between the actual value and the measured or calculated value is called an error.

Complete step by step answer :
The percentage error in the measurement of momentum is given as,
$\% p = 100\%$
i.e.
$\dfrac{{\Delta p}}{p} \times 100 = 100$
Dividing with $100$ on both sides, this will become
$\dfrac{{\Delta p}}{p} = 1$
Let ${p_i}$ be the initial momentum and ${p_f}$ be the final momentum, then $\Delta p$ can be written as,
$\Delta p = {p_f} - {p_i}$
Then the above equation can be written as,
$\dfrac{{{p_f} - {p_i}}}{{{p_i}}} = 1$
This can be written as,
${p_f} = 2{p_i}$
The initial kinetic energy of the particle in terms of momentum is given as,
$K = \dfrac{{{p_i}^2}}{{2m}}$
Where $p$ stands for the momentum of the particle and $m$stands for the mass of the particle.
The error in the measurement of momentum is,
$p = 100\%$
The final momentum will be,
${p_f} = 2{p_i}$
Therefore, the final kinetic energy of the particle will be
$K = \dfrac{{{{\left( {2{p_i}} \right)}^2}}}{{2m}}$
This will be,
$K' = \dfrac{{4{p_i}^2}}{{2m}}$
The percentage error in kinetic energy can be written as,
$\dfrac{{K' - K}}{K} \times 100 = \dfrac{{\dfrac{{4p_i^2}}{{2m}} - \dfrac{{p_i^2}}{{2m}}}}{{\dfrac{{p_i^2}}{{2m}}}} \times 100$
This can be written as,
$\dfrac{{K' - K}}{K} \times 100 = \dfrac{{3\dfrac{{p_i^2}}{{2m}}}}{{\dfrac{{p_i^2}}{{2m}}}} \times 100$
Canceling common terms,
$\dfrac{{K' - K}}{K} \times 100 = 3 \times 100 = 300\%$

The answer is: Option (D): $300\%$

Additional information:
Errors in measurements are broadly classified into random errors and systematic errors. Systematic errors are the type of errors that go in either a positive direction or in a negative direction.

Note:
The magnitude of the difference between the true value and the measured value of a physical quantity is called the absolute error of the measurement. The relative error or fractional error is defined as the ratio of mean absolute error to the true or mean value. The relative error expressed in percentage is called the percentage error.