1. Tardigrade
  2. Question
  3. Physics
  4. 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 δ 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, δ L / L, is closest to

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Q. 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

JEE AdvancedJEE Advanced 2017Physical World, Units and Measurements

Solution:

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