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 $\delta T=0.01$ seconds and he measures the depth of the well to be $L=20$ meters. Take the acceleration due to gravity $g=10m{s}^{-2}$ and the velocity of sound is $300m{s}^{-1}$. Then the fractional error in the measurement, $\delta L/L$, is closest to

• A. $2\%$
• B. $5\%$
• C. $1\%$
• D. $3\%$

Answer 1

The correct option is C

$1\%$

Here time taken for the stone to reach the surface
$T=\sqrt{\frac{2L}{g}}+\frac{L}{C}$
where $C$ is the speed of sound
Now, diffrentiating the above equation

$\frac{\delta t}{\delta L}=\sqrt{\frac{2}{g}}\times \frac{1}{2\sqrt{L}}+\frac{1}{C}$

$\delta L=\frac{\delta t}{\sqrt{\frac{2}{g}}\times \frac{1}{2\sqrt{L}}+\frac{1}{C}}$

here $\delta t=0.01$

$\Rightarrow$ $\delta L=$ $\frac{0.01}{\sqrt{\frac{2}{10}}\times \frac{1}{2\sqrt{20}}+\frac{1}{300}}$
$\delta L=\frac{3}{16}$

Then the fractional error in the measurement, $\delta L/L$, is $\frac{3}{16\times 20}\times 100=1\%$