Question

A man moving towards a vertical cliff at a constant velocity of $u\ m{{s}^{-1}}$,fires a gun and hears the echo after $t$ seconds. If he was at a distance of $d$ meters from the cliff when he fired the gun, then the velocity of sound in air in $m{{s}^{-1}}$ is
$\begin{aligned} & \text{A}\text{. }\dfrac{2d}{t} \\\ & \text{B}\text{. }\dfrac{2d}{t}+u \\\ & \text{C}\text{. }\dfrac{2d}{t}-u \\\ & \text{D}\text{.}\ \dfrac{d}{t} \\\ \end{aligned}$

Answer 1

The speed of the sound will be the distance travelled by the sound divided by the time taken. The man is moving towards the cliff with a constant velocity. The total distance travelled by the sound will be the distance travelled by the sound in $t$ seconds. Which is equal to twice the distance from the cliff minus the distance travelled by the man in $t$ seconds. After finding the distance divide by $t$ to get the speed of sound.

Complete answer:
The man is moving towards a vertical cliff at a constant velocity of $u\ m{{s}^{-1}}$ and he fires a gun and hears the echo after $t$ seconds. The man was at a distance of $d$ meters from the cliff when he fired the gun.
Let the velocity of the sound in air be ${{v}_{s}}$
If the man was standing still at distance $d$ from the cliff and hears echo after $t$ seconds then the distance travelled by the sound is ${{v}_{s}}t=2d$.
According to the question the man is moving towards the cliff with velocity $u\ m{{s}^{-1}}$. When the man fires the gun and hears the echo after $t$ seconds, then the man would have travelled the distance $ut$.
So the total distance travelled by the sound is $\left( 2d-ut \right)$
So the velocity of the sound is
$\begin{aligned} & {{v}_{s}}t=2d-ut \\\ & \Rightarrow {{v}_{s}}=\dfrac{2d}{t}-u \\\ \end{aligned}$

So the correct option is C.

Note:
Velocity of sound depends upon the medium through which it is travelling. For a denser medium the velocity of sound will be more. Due to this we can hear the sound of a train coming from far away by just touching the train tracks. Sound waves also have a peculiar property called the Doppler effect. Due to Doppler effect If we approach a sound emitting source or vice versa the pitch of the sound will be higher and if we move away from the sound source or the source is moving away from us the pitch of the sound will be reduced.