A car is approaching towards a cliff at a speed of 20\ m/s . The driver sounds a whistle of frequency 800\ Hz . What will be the frequency of the echo as heard by the car driver? Velocity of sound…

Solution

The frequency of the echo heard by the driver involves two Doppler shifts:

1. The sound travels from the moving car to the stationary cliff (no shift at the cliff).
2. The echo travels from the stationary cliff back to the moving car (shift occurs because the observer is moving toward the source).

Step 1: Frequency of the echo as emitted by the cliff

Since the cliff is stationary, the frequency of the sound reaching the cliff is the same as the emitted frequency: $$f_1=f_0=800\mathrm{Hz}.$$

Step 2: Frequency of the echo heard by the driver

The car is moving toward the cliff at $u=20m/s$, so the driver hears the echo at a higher frequency due to the Doppler effect for a moving observer: $$f^{\prime }=f_1\left(\frac{v+u}{v}\right),$$ where $v=350m/s$ is the speed of sound in air.

Substitute the values: $$f^{\prime }=800\left(\frac{350+20}{350}\right)=800\left(\frac{370}{350}\right).$$

Step 3: Calculate the final frequency

$$f^{\prime }=800\times \frac{370}{350}=800\times 1.05714\approx 845.71\mathrm{Hz}.$$

Final Answer: The frequency of the echo as heard by the driver is 846 Hz (rounded to the nearest whole number).