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Chapter 9: Problem 76

A racing car moving towards a cliff sounds its horn. The driver observes that the sound reflected from the cliff has a frequency one octave higher than the actual frequency of the horn. If \(v\) is the velocity of sound, then the velocity of the car is (A) \(\frac{v}{2}\) (B) \(\frac{v}{\sqrt{2}}\) (C) \(\frac{v}{4}\) (D) \(\frac{v}{3}\)

Short Answer

Expert verified
The velocity of the car is \(v_s = \frac{v}{2}\). The correct answer is (A) \(\frac{v}{2}\).

Step by step solution

01

Identify the Doppler effect formula for moving source and stationary observer

The Doppler effect formula for a moving source and a stationary observer is given by: \(f' = f \frac{v}{v - v_s}\) where: - \(f'\) is the frequency observed by the stationary observer, - \(f\) is the frequency emitted by the moving source, - \(v\) is the velocity of sound in the medium, - \(v_s\) is the velocity of the source (positive if moving towards the observer, negative if moving away). In our case, the racing car is the moving source, and the cliff is the stationary observer.
02

Express the condition of one octave higher frequency

When the frequency is one octave higher, it means the frequency has doubled. Therefore, we can write the condition as follows: \(2f = f \frac{v}{v - v_s}\)
03

Solve for the velocity of the car \(v_s\)

Now, we need to solve the equation for the velocity of the car, \(v_s\). We can do this by rearranging the equation and then isolating \(v_s\): \(\frac{2f}{f} = \frac{v}{v - v_s}\) \(2 = \frac{v}{v - v_s}\) \(2(v - v_s) = v\) \(2v - 2v_s = v\) \(-v = -2v_s\) \(v_s = \frac{v}{2}\)
04

Conclusion

The velocity of the car is: \(v_s = \frac{v}{2}\) Thus, the answer is (A) \(\frac{v}{2}\).

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Most popular questions from this chapter

A bus is moving towards a huge wall with a velocity of \(5 \mathrm{~m} / \mathrm{s}\). The driver sounds a horn of frequency \(200 \mathrm{~Hz}\). What is the frequency of beats heard by a passenger of the bus, if the speed of sound in air is \(330 \mathrm{~m} / \mathrm{s}\).

Change in temperature of the medium changes (A) frequency of sound waves. (B) amplitude of sound waves. (C) wavelength of sound waves. (D) loudness of sound waves.

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In Fig. \(9.40\), a block of mass \(M=1 \mathrm{~kg}\) is attached to one end of massless spring of spring constant \(k=100 \mathrm{~N} / \mathrm{m}\) and other end of spring is fixed. Initially, spring in its natural length. A horizontal force \(F=10 \mathrm{~N}\) at \(t=0\) is applied on the block. | Column-I | | :--- | | (A) Amplitude of SHM in cm is | | (B) Maximum speed of block in | | cm//s is | | (C) The time in second after which | | compression in the spring | | is half of amplitude if block | | starts from extreme position | | (D) Maximum compression in the | | spring in cm is |
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