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A person measures the depth of a well by measuring the time in

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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

0.2%

1%

3%

5%

Views: 6,278 students

Updated on: Jun 15, 2023

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Amelia Discussed

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

14 mins ago

Discuss this question LIVE

14 mins ago

Text solutionVerified

Here time taken for the stone to reach the surface

T = \frac{2 L}{g} + \frac{L}{C}

where

C

is the speed of sound
Now, differentiating the above equation

\frac{δ t}{δ L} = \frac{2}{g} \times \frac{1}{2 L} + \frac{1}{C}

δ L = \frac{δ t}{\frac{2}{9} \times \frac{1}{2 L} + \frac{1}{C}}

here

δ t = 0.01

\Rightarrow δ L = \frac{0.01}{\frac{2}{10} \times \frac{1}{2 20} + \frac{1}{300}}

δ L = \frac{3}{16}

Then the fractional error in the measurement,

δ L / L

, is

\frac{3}{16 \times 20} \times 100 = 1%

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Question Text	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
Updated On	Jun 15, 2023
Topic	Units and Measurements
Subject	Physics
Class	Class 11
Answer Type	Text solution:1 Video solution: 9
Upvotes	1076
Avg. Video Duration	10 min

Text solutionVerified

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