Question

A man standing between two cliffs produces a sound and hears two successive echoes at intervals of 3s and 4s respectively. Calculate the distance between two cliffs. The speed of sound in the air is 330m/s

Answer 1

Hint: It is given in the question the question that the man is standing somewhere in between the cliff. Therefore sound created by him will hit the either cliffs and reach back to him and this is basically what an echo is. In the question it is given that he hears a successive echo at an interval of 3s and 4s. Hence we will use the distance formula to calculate the distance between the man and the cliffs as time taken for sound in air to reflect and come back to the man is given.

Complete answer:
To begin with let us first define what an echo means.
An echo is a repeated sound in a regular interval of time. In the above question a man hears two successive echo’s such that the interval between them is one sec. Given below is a diagram depicting the following.

In the above diagram we can see that the two cliffs are at a distance D apart. Let us say the man is standing at a distance x from the cliff A and at a distance y from cliff B. Hence we can write $D=x+y...(1)$. The sound produced by the man hits the cliff A and reverts back to him in a time interval of 2secs. The relation between distance (d), time(t) and speed (v)is given as $v=\dfrac{d}{t}$.
The sound travels twice the distance twice of x as it hits the cliff and comes back. Hence distance x is equal to,
$\begin{align}
  & v=\dfrac{2x}{t} \\
 & 330=\dfrac{2x}{3} \\
 & x=\dfrac{990}{2}=495m \\
\end{align}$
Similarly the sound takes 4secs to hit cliff b and reach back to the man. Hence distance y is equal to,
$\begin{align}
  & v=\dfrac{2y}{t} \\
 & 330=\dfrac{2y}{4} \\
 & y=\dfrac{1320}{2}=660m \\
\end{align}$
Hence substituting x and y in equation 1 we get,
$\begin{align}
  & D=x+y \\
 & D=495+660 \\
 & D=1155m \\
\end{align}$
Hence the distance between the two cliffs is 1155 meters.

Note:
Echo’s are a result of reflection of sound. In caves and closed surfaces echoes are more prominently heard as the reflection of sound takes place frequently. In the above question a man is standing somewhere in between the two cliffs, if the man had to stand at the midpoints between the two cliffs then a single echo would have been heard by the man.