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(C) An octave higher means the frequency is doubled,so $f' = 2f$.The frequency of the sound heard by the driver after reflection from the cliff is giv

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$\frac{v}{3}$

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(C) An octave higher means the frequency is doubled,so $f' = 2f$.The frequency of the sound heard by the driver after reflection from the cliff is given by the Doppler effect formula for a moving source and observer.First,the cliff receives the sound with frequency $f_1 = f \left( \frac{v}{v - v_c} \right)$,where $v_c$ is the velocity of the car.Then,the cliff acts as a stationary source reflecting this sound back to the moving driver (observer). The frequency $f'$ heard by the driver is $f' = f_1 \left( \frac{v + v_c}{v} \right)$.Substituting $f_1$,we get $f' = f \left( \frac{v}{v - v_c} \right) \left( \frac{v + v_c}{v} \right) = f \left( \frac{v + v_c}{v - v_c} \right)$.Given $f' = 2f$,we have $2f = f \left( \frac{v + v_c}{v - v_c} \right)$.$2(v - v_c) = v + v_c$.$2v - 2v_c = v + v_c$.$v = 3v_c$.Therefore,$v_c = \frac{v}{3}$.

$A$ racing car moving towards a cliff sounds its horn. The driver observes that the sound reflected from the cliff has a pitch one octave higher than the actual sound of the horn. If $v$ is the velocity of sound,the velocity of the car will be

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