Two cars A and B are moving away from each other in opposite directions. Both the cars are moving with speed of 20 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 = 340 m s$$^{-1}$$)

We are told that car A and car B are moving in opposite directions, each with a speed of $$20 \,\text{m s}^{-1}$$ relative to the ground. An observer sitting in car A hears a sound of frequency $$2000 \,\text{Hz}$$ coming from the horn of car B. The speed of sound in still air is given as $$340 \,\text{m s}^{-1}$$. Our goal is to determine the actual (natural) frequency $$f$$ of the horn in car B.

For sound waves, the Doppler-effect formula that links the heard (apparent) frequency $$f'$$ and the natural frequency $$f$$ is

$$ f' = f \, \dfrac{v \pm v_o}{\,v \mp v_s\,}, $$

where

• $$v$$ is the velocity of sound in the medium, here $$340 \,\text{m s}^{-1}$$,
• $$v_o$$ is the speed of the observer relative to the medium (positive when the observer moves towards the source),
• $$v_s$$ is the speed of the source relative to the medium (positive when the source moves away from the observer),
• the upper sign in the numerator is used when the observer approaches the source; the upper sign in the denominator is used when the source recedes from the observer.

In our situation both vehicles are receding from each other:

• The observer in car A is moving away from the source in car B, so $$v_o = 20 \,\text{m s}^{-1}$$ is taken with a minus sign in the numerator.
• The source in car B is moving away from the observer, so $$v_s = 20 \,\text{m s}^{-1}$$ is taken with a plus sign in the denominator.

Hence the formula becomes

$$ f' \;=\; f \,\dfrac{v - v_o}{v + v_s}. $$

We know $$f' = 2000 \,\text{Hz}$$, $$v = 340 \,\text{m s}^{-1}$$, $$v_o = 20 \,\text{m s}^{-1}$$, and $$v_s = 20 \,\text{m s}^{-1}$$. Substituting these values gives

$$ 2000 \;=\; f \,\dfrac{340 - 20}{340 + 20}. $$

Simplifying the fractions step by step, we first work out the numerator and denominator:

$$ 340 - 20 = 320, \qquad 340 + 20 = 360. $$

So we now have

$$ 2000 \;=\; f \,\dfrac{320}{360}. $$

To isolate $$f$$, multiply both sides of the equation by the reciprocal of the fraction $$\dfrac{360}{320}$$:

$$ f \;=\; 2000 \,\times \dfrac{360}{320}. $$

Carrying out the multiplication in the numerator, we get

$$ 2000 \times 360 \;=\; 720\,000. $$

Dividing by the denominator:

$$ f \;=\; \dfrac{720\,000}{320}. $$

Performing the division:

$$ f \;=\; 2250 \,\text{Hz}. $$

Thus the natural frequency of the horn in car B is $$2250 \,\text{Hz}$$.

Hence, the correct answer is Option C.