Question

The minimum distance to hear a clear echo is ($V$ is the velocity of sound)
A) $\dfrac{2v}{5}$
B) $\dfrac{5V}{2}$
C) $\dfrac{V}{20}$
D) $5V$

Answer 1

Hint: Echo is the repetition of a sound produced by the reflection of sound waves from a reflector and can be heard differently from the source. The persistence of hearing time for a human brain is $\dfrac{1}{10}$ seconds. Hence, to hear an echo of a sound the interval between reaching the main sound and echo must be $\dfrac{1}{10}$ seconds.
We can use the formula, distance \[d = V \times t\].

Formula used:
\[d = V \times t\],
where $V$ = The velocity of sound, $t$=The persistence of hearing time
For a human brain $t= \dfrac{1}{10}$ seconds.

Complete step by step answer:
As we know that echo needs $\dfrac{1}{10}$ seconds to reach our ear.
Hence, the total distance covered by the sound from the point of generation to the reflector and back should be at least \[\dfrac{d}{2}\] (Minimum distance to hear a clear echo).
So,
 \[d = V \times t\]
\[or,d = V \times \dfrac{1}{{10}}\]
\[or,d = \dfrac{V}{{10}}\]
Hence, the minimum distance to hear a clear echo is,
\[\dfrac{d}{2} = \dfrac{V}{{10 \times 2}}\]
\[\dfrac{d}{2} = \dfrac{V}{{20}}\]

The minimum distance to hear a clear echo with a velocity $V$ is $\dfrac{V}{20}$. So, option (C) is correct.

Additional information:
In audio signal processing and acoustics, echo is a reflection of sound which arrives at the listener with a delay after the direct sound. The delay is directly proportional to the distance of the reflector from the source and the listener.

Note:
The velocity is the speed of sound in air which is equal to 343 m/s and time is 0.1 s as this the time required by the human to hear two sounds clearly.
so, the minimum distance is,
\[\therefore \dfrac{d}{2} = \dfrac{V}{{10 \times 2}}\]
\[\dfrac{d}{2} = \dfrac{{343}}{{20}}\]
\[\dfrac{d}{2} = 17.15\]
Therefore, the minimum distance to hear a clear echo is \[17.15\]m.