Question

A man standing in front of a vertical wall at a certain distance beats a drum at regular intervals. The drumming rate is gradually increased and he finds that the echo is not heard distinctly when the drumming rate becomes 2 beats in 3 seconds. He then moves nearer to the wall by 90 m and finds again the echo is not heard when the drumming rate becomes 1 per second. From this data, the velocity of the sound must be:
A. 330 m/s
B. 340 m/s
C. 360 m/s
D. 180 m/s

Answer 1

Hint: If a sound is directed towards the wall, the reflected sound from the wall gives its echo. Echo is the delayed sound which when reached back at a time when the drummer gives out the next beat, will not be heard distinctly.

Complete step by step answer:
Let initially the drummer be standing at a separation of x meters from the wall. To hear an echo, the sound must travel a distance of 2x back to the drummer's ear. If the sound velocity is v m/s, it will take a time of:
$t = \dfrac{2x}{v}$
We are given that initially, the beats have a frequency of 2 beats per 3 seconds or we can say one beat is created 1.5 seconds apart. Now if one beat is released and its echo comes back at the time of the next beat, i.e., after 1.5 seconds, it will not be heard distinctly. So time for echo to reach the drummer's ear is the same as the beat separation time.
$\dfrac{2x}{v} = 1.5$s
We can get the value of x as:
$x = \dfrac{1.5 v}{2}$
When the drummer moves 90 m closer, the time for echo reflection reduces to
$\dfrac{2(x-90)}{v}$
The rate of drumming also now becomes 1 beat per second. So separation between two beats is now 1 s. So we can write:
$\dfrac{2(x-90)}{v} = 1$s
Substituting the value of x from previous result, we get the velocity as:
$(1.5v - 180) = 1v$
$0.5v = 180$
v = 360 m/s.
Therefore the correct answer is option (C).

Note:
One might think that the velocity of sound is around 330 m/s so option (A) is correct. These conclusions are not correct. Also, check the units properly as one might get confused between the frequency and time separation of the beats.