The frequency of echo will be __________ Hz if the train blowing a whistle of frequency 320 Hz is moving with a velocity of 36 km/h towards a hill from which an echo is heard by the train driver. Velocity of sound in air is 330 m/s.

Solution

Physics sir · 4 steps

The echo goes through two Doppler shifts: the hill hears a moving source, then the driver moves into the echo.

1. Convert the speed. 36 km/h = 36 × 5/18 = 10 m s⁻¹. So the train moves at vₛ = 10 m s⁻¹ and sound moves at v = 330 m s⁻¹.
2. Hill as listener. The source moves towards the stationary hill, so the hill receives f₁ = 320 × 330/(330 - 10) = 320 × 330/320 = 330 Hz. The hill sends this frequency back as the echo.
3. Driver as listener. The hill is now a stationary source and the driver moves towards it at 10 m s⁻¹. So f₂ = 330 × (330 + 10)/330 = 340 Hz.

   f_(echo) = f (v + vₛ)/(v - vₛ) = 320 × 340/320 = 340 Hz

4. So. The driver hears the echo at 340 Hz.

The common slip: 330. Applying the Doppler shift only once, for the train approaching the hill, gives 330 Hz. That is what the hill hears, not the driver.

The idea to keep. For an echo off a fixed wall, treat the wall as a listener first and then as a source. A moving source and listener together give f (v + vₛ)/(v - vₛ).

Answer from the NTA final answer key. Solution by PrepWiser.