Question

Two cars $\mathrm{A}$ and $\mathrm{B}$ are moving away from each other in opposite directions. Both the cars are moving at a speed of $20\mathrm{m}/\mathrm{s}$ with respect to the ground. If an observer in car $\mathrm{A}$ detects a frequency of $2000\mathrm{Hz}$ of the sound coming from car $\mathrm{B}$, what is the natural frequency of the sound source in car $\mathrm{B}$? (speed of sound in air $=340\mathrm{m}/\mathrm{s}$)

• A. $2250\mathrm{Hz}$
• B. $200\mathrm{Hz}$
• C. $2300\mathrm{Hz}$
• D. $2150\mathrm{Hz}$

Answer 1

The correct option is A

$2250\mathrm{Hz}$

Step 1. Given data

Two cars $\mathrm{A}$ and $\mathrm{B}$ are moving away in opposite direction at speed of $20\mathrm{m}/\mathrm{s}$ and speed of sound in are is $340\mathrm{m}/\mathrm{s}$.

We have to find the natural frequency of the sound source in car $\mathrm{B}$.

Step 2. Formula to be used

The Doppler effect is the apparent change in the frequency of a wave motion when there is relative motion between the source of the waves and the observer.

So,

$\mathrm{f}=\left(\frac{\mathrm{v}\pm {\mathrm{v}}_{\mathrm{o}}}{\mathrm{v}\mp {\mathrm{v}}_{\mathrm{s}}}\right){\mathrm{f}}_0$

Here, ${\mathrm{f}}_0$ is the actual frequency of the sound waves, $\mathrm{f}$ is observed frequency, $\mathrm{v}$ is the speed of the sound waves, ${\mathrm{v}}_0$ is the velocity of the observer and ${\mathrm{v}}_{\mathrm{s}}$ is the velocity of the source.

Step 3. Set the values according to the formula.

Let ${\mathrm{v}}_{\mathrm{o}}$ be the speed of an observer sitting in car $\mathrm{A}$ i.e. it is the speed of car $\mathrm{A}$ seen in the above figure.

and
${\mathrm{v}}_{\mathrm{s}}$ be the speed of car $\mathrm{B}$.
So,

${\mathrm{v}}_0={\mathrm{v}}_{\mathrm{s}}$

$=20\mathrm{m}/\mathrm{s}$
Consider $\mathrm{v}$ is the speed of sound in air.

Therefore,

$\mathrm{v}=340\mathrm{m}/\mathrm{s}$
From the given, an observer in car $\mathrm{A}$ detects a frequency of $2000\mathrm{Hz}$ of the sound coming from car $\mathrm{B}$,

Therefore, the apparent frequency $\mathrm{f}=2000\mathrm{Hz}$
Let us assume that ${\mathrm{f}}_0$ is the natural frequency.
According to the Doppler Effect, the apparent frequency is given as

$\mathrm{f}=\left(\frac{\mathrm{v}\pm {\mathrm{v}}_{\mathrm{o}}}{\mathrm{v}\mp {\mathrm{v}}_{\mathrm{s}}}\right){\mathrm{f}}_0$
Since both cars are moving away from each other, we have,

$\mathrm{f}=\left(\frac{\mathrm{v}-{\mathrm{v}}_{\mathrm{o}}}{\mathrm{v}+{\mathrm{v}}_{\mathrm{s}}}\right){\mathrm{f}}_0$

Step 4. Find the natural frequency of the sound source in car $\mathrm{B}$.
Substitute the given values in the above equation, we get,

$2000=\left(\frac{340-20}{340+20}\right){\mathrm{f}}_0$

$=\left(\frac{340+20}{340-20}\right)\times 2000$

$=2250\mathrm{Hz}$

Thus, the natural frequency of the sound source in car $\mathrm{B}$ is $2250\mathrm{Hz}$.

Hence, option (A) is the correct answer.