Question

The frequency of the sound of a car horn as received by an observer towards whom the car is moving differs from the frequency of the horn by 2.0%. Assuming that the velocity of sound of air is $350m{{\operatorname{s}}^{-1}},$ the velocity of car is
A. $6.0m{{\operatorname{s}}^{-1}}$
B. $7.5m{{\operatorname{s}}^{-1}}$
C. $7.0m{{\operatorname{s}}^{-1}}$
D. $8.5m{{\operatorname{s}}^{-1}}$

Answer 1

Question is based on Doppler Effect, first calculate the ratio of frequency and equate the relative velocity.
1. Frequency of car horn $={{n}_{{}^\circ }}$
2. Frequency that received by observer $=n'$

Complete step by step solution:
Ratio of eq (1) to eq (2) given by the question = $\dfrac{n'}{{{n}_{{}^\circ }}}$ = 1.02 ….(1)
Use concept of relativity
By Doppler effect= $\dfrac{n'}{{{n}_{{}^\circ }}}$ = $\dfrac{V}{V{{V}_{5}}}$ …..(2)
V5 = velocity of sound
V = velocity of car
Now put the value of eqn (1) to the eq(2)
1.02 = $\dfrac{V}{V{{V}_{5}}}$
V = 350 (Given in question)
1.02 = $\dfrac{350}{350{{V}_{5}}}$
350 – V5 = $\dfrac{350}{1.02}$
350 – V5 = 343.13
V5 = 350 – 343.13
V5 = 7 m/s (approx.)

Finally velocity of car in air is 7 m/s

Note: (1) Carefully calculate the ratio of frequency, it is $\dfrac{n'}{{{n}_{{}^\circ }}}$ not $\dfrac{{{n}_{{}^\circ }}}{n'}$
(2) Carefully calculate the relative velocity of sound and car.