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The driver of a car travelling at a speed of 30 m/s30 \text{ m/s} towards a hill sounds a horn of frequency 600 Hz600 \text{ Hz}. If the velocity of sound in air is 330 m/s330 \text{ m/s}, the frequency of reflected sound as heard by the driver is:

A

550 Hz550 \text{ Hz}

B

555.5 Hz555.5 \text{ Hz}

C

720 Hz720 \text{ Hz}

D

500 Hz500 \text{ Hz}

Step-by-Step Solution

Let the velocity of sound be v=330 m/sv = 330 \text{ m/s}, the velocity of the car be vc=30 m/sv_c = 30 \text{ m/s}, and the actual frequency of the horn be f=600 Hzf = 600 \text{ Hz}. When the sound travels towards the hill, the car acts as a moving source approaching a stationary reflector (the hill). The apparent frequency f1f_1 received by the hill is: f1=f(vvvc)f_1 = f \left( \frac{v}{v - v_c} \right) When the sound is reflected from the hill, the hill acts as a stationary source emitting frequency f1f_1, and the car acts as an observer moving towards the source. The frequency heard by the driver is: f=f1(v+vcv)f' = f_1 \left( \frac{v + v_c}{v} \right) Substituting f1f_1 into the second equation gives the formula for the apparent frequency of an echo heard by a moving source: f=f(v+vcvvc)f' = f \left( \frac{v + v_c}{v - v_c} \right) f=600(330+3033030)f' = 600 \left( \frac{330 + 30}{330 - 30} \right) f=600(360300)f' = 600 \left( \frac{360}{300} \right) f=600×1.2=720 Hzf' = 600 \times 1.2 = 720 \text{ Hz} Therefore, the driver hears a frequency of 720 Hz720 \text{ Hz}.

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