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A pebble is dropped freely in a well from its top. It takes 20 s for the pebble to reach the water surface in the well. Taking $g = 10m{s^{ - 2}}$ and speed of sound = $330m{s^{ - 1}}$, find:
i) the depth of water surface from the top
ii) the time when echo is heard after the pebble is dropped?

A. (i) 1000m, (ii) 26.1 s
B. (i) 2000m, (ii) 27 s
C. (i) 200m, (ii) 26.1 s
D. (i) 2000m, (ii) 26.1 s

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Last updated date: 20th Jun 2024
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Answer
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Hint: Understanding the principles of echo is fundamental for solving this problem. Echo is defined as the reflection of the sound wave that the listener perceives clearly, after the sound waves bounces off a hard surface. Whenever the sound waves are produced, they do reflect across objects around us and our ears do receive the sound. But our human ear cannot perceive the reflection of the sound and hence, does not classify as echo.

Complete step by step answer:
Our ears can perceive the echo sound if it is reflecting back with a difference of 0.1 second. We know that the speed of sound in air is $330m{s^{ - 1}}$.
So, the minimum distance that the sound has to travel in order to hear the echo,
$D = s \times t$
$ \Rightarrow D = 330 \times 0.1$
$ \Rightarrow D = 33m$
This distance includes the to and fro movement of the sound waves in the echo.
Thus, the minimum distance that the obstacle has to be present to hear an echo,
$d = \dfrac{D}{2}$
$ \Rightarrow d = \dfrac{{34}}{2}$
$ \Rightarrow d = 17m$

i) Depth of water surface from the top
It is possible to calculate the depth of the water surface from the top using the formula for displacement –
$s = ut + \dfrac{1}{2}a{t^2}$
where u = initial velocity, t = time, a = acceleration.
When the pebble is dropped to the water surface, the initial velocity of the pebble is zero. Hence, $u = 0$
Given that the pebble undergoes free-fall, its acceleration will be equal to the acceleration due to gravity. Hence, $a = g = 10m{s^{ - 2}}$
The time taken for the pebble to touch the water surface, $t = 20\sec $
Substituting the values,
$s = ut + \dfrac{1}{2}a{t^2}$
$ \Rightarrow s = 0 \times 20 + \dfrac{1}{2} \times 10 \times {20^2}$
$ \Rightarrow s = \dfrac{1}{2} \times 10 \times 400 = 2000m$
Therefore, the depth of the water is equal to 2000 m.

ii) Time when echo is heard after the pebble is dropped
As explained above, the echo is heard if the minimum distance is greater than or equal to 17m. Since, the distance between the top surface of water and the top of the well is very higher than 17m, the echo is heard when the pebble is dropped.
If V is the velocity of sound in air and $s$ is the distance taken for the echo to travel, the time taken for the echo to reach from water surface of the well to top, is given by –
$t = \dfrac{s}{V}$
$ \Rightarrow t = \dfrac{{2000}}{{330}} = 6 \cdot 06\sec $
The actual time taken for hearing the echo is equal to sum of the time taken for pebble to hit the water surface and the time taken for echo to reach the surface.
Actual time, $T = 20 + 6 \cdot 06 = 26 \cdot 06 \sim 26 \cdot 1\sec $

Hence, the correct answer is option (D).

Note: The devices SONAR and RADAR, based on the principles of echo, are primarily used for measurement of distances of the obstacles in the ship/submarine and on air, respectively. In these devices, the distance between the sender and the obstacle is calculated by the time taken for the reflection of the sound to reach the sender, after the echo.