Question

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

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

Answer 1

The Correct Option is A

To solve this problem, we need to understand the concept of the Doppler Effect, which explains the change in frequency or wavelength of a wave in relation to an observer who is moving relative to the wave source. Here, a car is approaching a cliff and the sound of the horn is reflected back to the driver.

Given:

• Frequency of horn = \(f\)
• Reflected frequency heard by the driver = \(2f\)
• Velocity of sound = \(v\)

The situation involves two Doppler Effect events: first when the sound waves reach the cliff and second when the reflected waves return to the driver.

Step-by-step Solution:

1. Frequency heard by the cliff (\(f'\)):

The frequency heard by the cliff, \(f'\), when the car is moving towards it is given by the Doppler Effect formula:

f' = \frac{v + v_c}{v} \cdot f

Where \(v_c\) is the velocity of the car. Since the cliff is stationary, the velocity of the observer is zero.

2. Frequency reflected back to the driver (\(f''\)):

The cliff reflects the frequency \(f'\) back towards the moving car:

f'' = \frac{v}{v - v_c} \cdot f'

Substitute the expression for \(f'\):

f'' = \frac{v}{v - v_c} \cdot \left(\frac{v + v_c}{v}\right) \cdot f

3. Given Condition:

It is given that the frequency heard by the driver is \(2f\):

\frac{v}{v - v_c} \cdot \frac{v + v_c}{v} \cdot f = 2f

Cancel out \(f\) from both sides and solve:

\frac{v + v_c}{v - v_c} = 2

Cross multiply to solve for \(v_c\):

v + v_c = 2(v - v_c)

v + v_c = 2v - 2v_c

3v_c = 2v - v

3v_c = v

Therefore, v_c = \frac{v}{3}.

Conclusion:

The velocity of the car, in terms of the velocity of sound, is \(\frac{v}{3}\). Thus, the correct answer is:

Answer: \(\frac{v}{3}\)