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The speed of sound in air and sea-water are given to be $340\,m/s$ and $1440\,m/s$ respectively. A ship sends a strong signal straight down and detects its echo after $1.5$ seconds. The depth of the sea at that point is:
A) $2.16\,kms$
B) $1.08\,kms$
C) $0.51\,kms$
D) $0.255\,kms$

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Last updated date: 17th Jun 2024
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Answer
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Hint: You can solve this question by firstly completely neglecting the fact that the speed of sound is given because there is no use of it in the question. The main solution can be reached by recalling that echo is the sound we hear of ourselves after its reflection, hence, for us to hear an echo, the sound has to travel the distance between us and the reflector twice, first going towards the reflector, second when coming back to us.

Complete step by step answer:
We will be solving the question using the same approach or concept as mentioned in the hint section of the solution to the question.
Since the sound waves have to travel twice the actual distance between the wall or reflector and the source, we are pretty sure that sound waves will thus take twice the time as compared to the time it takes them to reach the wall or reflector. Hence, we can say that if the actual distance between the source and the reflector (sea bottom in the question) is $d$ , sound waves have to travel a distance $2d$ for us to hear the echo. If it is so, let us assume that it takes sound waves time $t$ to reach the reflector (sea bottom in our case), then the time taken for us to hear the echo would be $2t$ .
The value of the time taken for us to hear the echo is given as $2t = 1.5\,s$
This means that the actual time taken by the sound waves to travel from the source in the ship to the reflector, i.e. sea bottom, can be given as:
$
  2t = 1.5\,s \\
  t = 0.75\,s \\
 $
Now, we all know that speed is defined as the ratio of distance travelled upon time taken to travel the distance. Mathematically, we can write:
$v = \dfrac{d}{t}$
We need to find the value of the distance; hence, we can write:
$d = vt$
The value of the velocity of sound in seawater is given to us as:
$\Rightarrow v = 1440\,m/s$
Or, $\Rightarrow v = 1.44\,km/s$
We have already found out the value of time taken as $t = 0.75\,s$
Substituting in the values, we get:
$
\Rightarrow d = 1.44 \times 0.75\,kms \\
\Rightarrow d = 1.08\,kms \\
 $
This matches with the value in option (B). Hence, option (B) is the correct answer.

Note: A serious mistake that students do is that they use the time taken to hear the echo directly to find the distance, which is completely wrong. If you are doing so, keep in mind that the answer is actually half the value that you found out, so always divide it by two.