Question

A person measures the depth of a well by measuring the time interval between dropping a stone and receiving the sound of impact with the bottom of the well. The error in his measurement of time is $\partial T$=0.01 seconds and he measures the depth of the well to be L=20metres. Take acceleration due to gravity g=10m/$s^2$ and the velocity of sound is 300m/s. Then the fractional error in the measurement, $\dfrac{{\partial L}}{L}$, is closest to:
A) 2%
B) 5%
C) 1%
D) 3%

Answer 1

Error: uncertainty in a measurement is called error. It is the difference between the measured value and true value.
In the above question we have to find the error in the measurement of length with the total time taken by the stone to reach the well and on receiving the sound.
Total time is given as:
$T = \sqrt {\dfrac{{2L}}{g}} + \dfrac{L}{V}$ (Where L is the length of the well, V is the velocity T is the total time taken and g is the acceleration due to gravity.)

Complete step by step solution:
Let’s do the calculations in order to find the error in measurement of length:
Let the time taken by sound to reach the man as t1 and time taken by stone to reach the well is taken as t2.
${t_1} = \sqrt {\dfrac{{2L}}{g}}$ And ${t_2} = \dfrac{L}{V}$
Total time is:
$T = \sqrt {\dfrac{{2L}}{g}} + \dfrac{L}{V}$
Change in time and length is written as:
$\Rightarrow \Delta T = \sqrt {\dfrac{2}{g}} \dfrac{{\vartriangle L}}{{2L}} + \dfrac{{\Delta L}}{V}$
$\Rightarrow \Delta T = \sqrt {\dfrac{1}{{2Lg}}} \vartriangle L + \dfrac{{\Delta L}}{V} \\\ \Rightarrow 0.01 = \dfrac{1}{{\sqrt {2 \times 10 \times 20} }}\Delta L + \dfrac{{\Delta L}}{{300}} \\\$ (Substituting the values of L, T and V)
$\Rightarrow 0.01 = \dfrac{{\Delta L}}{{20}} + \dfrac{{\Delta L}}{{300}} \\\ \Rightarrow 3 = 16\Delta L \\\ \Rightarrow \Delta L = .1875 \\\$(Rearranging the terms to calculate$\Delta L$)
Now, the percentage change is given as:
$\Rightarrow \dfrac{{\partial L}}{L} \times 100 \\\ \Rightarrow \dfrac{{.1875}}{{20}} \times 100 = .93 \\\ \Rightarrow .93 \approx 1\% \\\$

Thus, option 3 is correct.

Note: We have different types of error, which are stated below:
Constant errors: errors which keep on repeating every time are constant errors. Systematic errors: errors which occur according to a certain pattern or system and are classified into 4 parts: instrumental errors, personal errors, errors due to external sources, errors due to external sources or errors due to imperfection. Gross errors: errors which occur due to improper setting of the instrument, recording observations wrongly, not to take precautions into account, using some wrong value in calculations. Random errors: it is common experience that the repeated measurement of a quantity gives values which are slightly different from each other.