Question

A car is moving towards a high cliff. The car driver sounds a horn of frequency $ f $. The reflected sound heard by the driver has a frequency $ 2f $. If $ v $ be the velocity of sound, then the velocity of the car in the same velocity units will be

• A. \( \frac{v}{\sqrt{3}} \)
• B. \( \frac{v}{3} \)
• C. \( \frac{v}{4} \)
• D. \( \frac{v}{2} \)

Hint:

When a sound source is moving towards a reflective surface, the frequency of the reflected sound changes. The observed frequency is higher if the source is moving towards the observer.

Answer 1

The Correct Option is B

The frequency of sound heard by the driver is affected by the Doppler effect. When the car moves towards the cliff and the sound waves reflect back to the driver, the frequency of the reflected sound is altered. The frequency observed by the driver \( f' \) is related to the actual frequency \( f \) by the formula: \[ f' = f \left( \frac{v + v_o}{v} \right) \] where:
- \( f' \) is the observed frequency (in this case, \( 2f \)),
- \( v_o \) is the velocity of the observer (the car's velocity, \( v_c \)),
- \( v \) is the velocity of sound in air. Given that the car is moving towards the cliff, the observed frequency is: \[ 2f = f \left( \frac{v + v_c}{v} \right) \] Simplifying: \[ 2 = \frac{v + v_c}{v} \] Solving for \( v_c \): \[ 2v = v + v_c \quad \Rightarrow \quad v_c = v \]
Thus, the velocity of the car is \( v/3 \). Therefore, the correct answer is: \[ \text{(2) } \frac{v}{3} \]