Question

A stationary source emits a sound wave of frequency 500Hz. Two observers moving along a line passing through the source detects sound to be of frequencies 480 Hz and 530 Hz. Their respective sounds are in $m{{s}^{-1}}$ (Given speed of sound is = $300m{{s}^{-1}}$ )
a)16,14
b)12,18
c)12,16
d)8,18

Answer 1

In the above question there are two observers who perceive the frequency of the sound differently of the same source. This is because there is a relative motion between the observer and the source. It is given to us the source is stationary and the two observers cross the source. Hence using the equation of apparent frequency that comes from Doppler’s effect, we will obtain the velocities of the two observers respectively.

Formula used:
${{v}_{1}}=\dfrac{V+{{v}_{A}}}{V}v$
${{v}_{2}}=\dfrac{V-{{v}_{B}}}{V}v$

Complete step-by-step answer:
Let us say the two observers i.e. A hears the frequency of the source as ${{v}_{1}}=530Hz$ and B hears the apparent frequency of the source as ${{v}_{2}}=480Hz$. Let us say the observer A moves with velocity ${{v}_{A}}$ and observer B moves with velocity ${{v}_{B}}$. IN the above data we can see that the frequency of the source heard by observer A is greater than that of B. Hence from this we can imply that A is moving towards the source and B is moving away from the source. If V is speed of sound in air and v is the actual frequency of the source, than the apparent frequency heard by A is,
$\begin{aligned} & {{v}_{1}}=\dfrac{V+{{v}_{A}}}{V}v \\\ & 530Hz=\dfrac{300+{{v}_{A}}}{300}500 \\\ & \Rightarrow 300+{{v}_{A}}=\dfrac{530\times 3}{5}=318 \\\ & \Rightarrow {{v}_{A}}=318-300=18m{{s}^{-1}} \\\ \end{aligned}$

Similarly, the apparent frequency heard by the observer B is equal to,
$\begin{aligned} & {{v}_{2}}=\dfrac{V-{{v}_{B}}}{V}v \\\ & 480Hz=\dfrac{300-{{v}_{B}}}{300}500 \\\ & \Rightarrow 300-{{v}_{B}}=\dfrac{480\times 3}{5}=288 \\\ & \Rightarrow -{{v}_{B}}=288-300=-12 \\\ & \Rightarrow {{v}_{B}}=12m/s \\\ \end{aligned}$
Hence the velocity of observer B is 12m/s and that of observer A is 18m/s.
Therefore the correct answer of the above question is option c.

So, the correct answer is “Option c”.

Note: The important point to be kept in mind is that if the frequency heard by the observer depends on the relative motion of source as well as the observer. Further the direction determines whether the frequency heard by the observer will be greater or smaller than the actual frequency. Its relative frequency heard also depends on the medium.