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 $g =10\, ms ^{-2}$ and the velocity of sound is $300\, ms ^{-1}$. Then the fractional error in the measurement, $\delta L / L$, is closest to

• A. $0.2\%$
• B. $1\%$
• C. $3\%$
• D. $5\%$

Answer 1

Answer: (B)

$1\%$

Answer 2

The duration required for the stone to reach the ground.
$t=\sqrt{\frac{2 L}{g}}+\frac{L}{C}$
Here,C represents the speed of sound.

Now, differentiating the above equation
$\frac{d t}{d L}=\sqrt{\frac{L}{g}} \times \frac{1}{2 \sqrt{L}}+\frac{1}{C}$

$dL =\frac{ dt }{\frac{1}{\sqrt{2 gL }}+\frac{1}{ C }}$
We hat dt=0.01
$\Rightarrow \frac{ dL }{ L } \times 100=\left(\frac{ dt }{\frac{1}{\sqrt{2 gL }}+\frac{1}{ C }}\right) \frac{1}{ L } \times 100$
$=\frac{15}{16 \%} \approx 1 \%$