Question

A car travelling at a speed of \(54 \, \text{kmph}\) towards a wall sounds a horn of frequency \(400 \, \text{Hz}\). The difference in the frequencies of two sounds, one received directly from the car and the other reflected from the wall, noticed by a person standing between the car and the wall is: (Speed of sound in air is \(335 \, \text{m/s}\))

• A. \(359 \, \text{Hz}\)
• B. \(20 \, \text{Hz}\)
• C. \(70 \, \text{Hz}\)
• D. \(0\)

Hint:

The Doppler effect occurs due to relative motion between the sound source and the observer, affecting the perceived frequency.

Answer 1

The Correct Option is D

To solve this problem, we need to calculate the difference in frequencies between the sound received directly from the car and the sound reflected from the wall. This involves the Doppler effect for both the approaching car and the reflected sound. 1. Convert the car's speed to meters per second: \[ 54 \, \text{kmph} = 54 \times \frac{1000}{3600} = 15 \, \text{m/s} \] 2. Calculate the frequency received directly from the car: - The observer is stationary, and the car is moving towards the observer. - The formula for the Doppler effect when the source is moving towards the observer is: \[ f_{\text{direct}} = \frac{v}{v - v_s} f_0 \] where \( v = 335 \, \text{m/s} \) (speed of sound), \( v_s = 15 \, \text{m/s} \) (speed of the car), and \( f_0 = 400 \, \text{Hz} \) (original frequency). - Plugging in the values: \[ f_{\text{direct}} = \frac{335}{335 - 15} \times 400 = \frac{335}{320} \times 400 \approx 418.75 \, \text{Hz} \] 3. Calculate the frequency of the sound reflected from the wall: - The car is moving towards the wall, so the frequency of the sound hitting the wall is: \[ f_{\text{wall}} = \frac{v}{v - v_s} f_0 = \frac{335}{320} \times 400 \approx 418.75 \, \text{Hz} \] - The wall reflects this frequency, and the observer hears the reflected sound. Since the wall is stationary, the frequency received by the observer from the wall is the same as the frequency hitting the wall: \[ f_{\text{reflected}} = 418.75 \, \text{Hz} \] 4. Calculate the difference in frequencies: - The difference between the directly received frequency and the reflected frequency is: \[ \Delta f = f_{\text{reflected}} - f_{\text{direct}} = 418.75 - 418.75 = 0 \, \text{Hz} \] 5. Final Answer: - The difference in the frequencies noticed by the person is: \[ \boxed{0} \] This corresponds to option (4).