Question

Two cars are moving towards each other at the speed of \(50\,\text{m s}^{-1}\). If one of the cars blows a horn at a frequency of \(250\,\text{Hz}\), the wavelength of the sound perceived by the driver of the other car is
\[ \text{(Speed of sound in air }=350\,\text{m s}^{-1}\text{)} \]

• A. \(18.7\ \text{cm}\)
• B. \(105\ \text{cm}\)
• C. \(75\ \text{cm}\)
• D. \(10.5\ \text{cm}\)

Hint:

For Doppler effect: \[ f'=f\left(\frac{v\pm v_o}{v\mp v_s}\right) \] Use \(+\) sign when observer moves towards source and \(-\) sign when source moves towards observer.

Answer 1

The Correct Option is B

Step 1: Identify the Doppler effect scenario.
Two cars approach each other, both moving at $v_s = v_o = 50$ m/s. One car (source) blows a horn at frequency $f = 250$ Hz. Speed of sound $v = 350$ m/s. We need to find the wavelength of sound perceived by the other driver (observer).
Step 2: Find the apparent frequency using the Doppler formula.
Since the source moves toward the observer and the observer moves toward the source: \[ f' = f \cdot \frac{v + v_o}{v - v_s} = 250 \times \frac{350 + 50}{350 - 50} \] \[ f' = 250 \times \frac{400}{300} = 250 \times \frac{4}{3} = \frac{1000}{3} \approx 333.33 \text{ Hz} \]
Step 3: Find the perceived wavelength.
The wavelength of sound perceived by the observer is the speed of sound divided by the apparent frequency: \[ \lambda' = \frac{v}{f'} = \frac{350}{1000/3} = \frac{350 \times 3}{1000} = \frac{1050}{1000} = 1.05 \text{ m} \]
Step 4: Convert to centimetres.
\[ \lambda' = 1.05 \text{ m} = 105 \text{ cm} \]
Step 5: Understand the physics of perceived wavelength.
When a moving observer approaches a source, they encounter more wavefronts per second (higher frequency), but the wavelength they measure is based on the speed of sound in the medium divided by the Doppler-shifted frequency. The factor $\frac{v}{f'}$ gives this perceived wavelength correctly.
Step 6: State the final answer.
The wavelength of sound perceived by the driver of the other car is 105 cm. \[ \boxed{105 \text{ cm}} \]