Question

Two cars A and B are moving away from each other in opposite directions. Both the cars are moving with a speed of $20\text{ }m{{s}^{-1}}$with respect to the ground. If an observer in car A detects a frequency 2000 Hz of the sound coming from car B, what is the natural frequency of the sound source in car B? (Speed of sound in air =$340m{{s}^{-1}}$)
$\begin{aligned} & \text{A}\text{. 2250Hz} \\\ & \text{B}\text{. 2060Hz} \\\ & \text{C}\text{. 2150Hz} \\\ & \text{D}\text{. 2300Hz} \\\ \end{aligned}$

Answer 1

Hint: First, understand that whenever there is relative motion between a listener and the source, an apparent change in frequency is observed. List down the information given in the question. Using the Doppler formula for an apparent frequency change, calculate the natural frequency of the sound source.

Complete step by step solution:

Let ${{v}_{o}}$be the speed of an observer sitting in car A i.e. it is the speed of car A.
${{v}_{s}}$be the speed of car B.
Thus, ${{v}_{o}}={{v}_{s}}=20m{{s}^{-1}}$
Let $v$be the speed of sound in air. Therefore, $v=340m{{s}^{-1}}$
Given that, an observer in car A detects a frequency 2000 Hz of the sound coming from car B, therefore, apparent frequency = $f$ = 2000 Hz
Let ${{f}_{0}}$ be the natural frequency which we have to find.
According to the Doppler Effect, the apparent frequency is given as
$f=\left( \dfrac{v\pm {{v}_{o}}}{v\mp {{v}_{s}}} \right){{f}_{0}}$
Since both cars are moving away from each other, we have,
$f=\left( \dfrac{v-{{v}_{o}}}{v+{{v}_{s}}} \right){{f}_{0}}$
Substituting the given values in this equation, we get,
$\begin{aligned} & 2000=\left( \dfrac{340-20}{340+20} \right){{f}_{0}} \\\ & \therefore {{f}_{0}}=\left( \dfrac{340+20}{340-20} \right)\times 2000 \\\ & \therefore {{f}_{0}}=2250Hz \\\ \end{aligned}$
Hence, the natural frequency of the sound source in car B is 2250 Hz.

Additional Information:
The Doppler Effect is well observed when you stand by a passing railway on the railway platform. You will observe that frequency of sound waves emitted by the horn on the railway will increase when it approaches you and decreases when it moves away from you.
The Doppler Effect is also observed in light. But in this case, the speed of the observer doesn’t matter much as it is very very less than the speed of light.

Note: Note that Doppler Effect is the change in observed frequency by the listener due to relative motion between the listener and the source. Remember that, original frequency of the source is not changed, only the different frequency is observed because the waves get stretched or squeezed due to the relative motion of the listener and source.