A train moving at a speed of $220 ext{ ms}^{-1}$ towards a stationary object, emits a sound of frequency $10

Question

A train moving at a speed of $220 	ext{ ms}^{-1}$ towards a stationary object, emits a sound of frequency $1000 	ext{ Hz}$. Some of the sound reaching the object gets reflected back to the train as an echo. The frequency of the echo as detected by the driver of the train is (speed of sound in air is $330 	ext{ ms}^{-1}$)

Accepted Answer

1. **Frequency received by the stationary object:** The train acts as a source moving towards a stationary observer (the object). The apparent frequency $f'$ received by the object is given by the Doppler effect formula: $f' = f \left( \frac{v}{v - v_s} ight)$ .
   Substitute the given values ($f = 1000 	ext{ Hz}$, $v = 330 	ext{ m/s}$, $v_s = 220 	ext{ m/s}$):
   $$f' = 1000 \left( \frac{330}{330 - 220} ight) = 1000 \left( \frac{330}{110} ight) = 1000 	imes 3 = 3000 	ext{ Hz}$$
2. **Frequency of the echo heard by the driver:** The object now acts as a stationary source reflecting the sound of frequency $f' = 3000 	ext{ Hz}$. The driver acts as an observer moving towards this stationary source with a velocity $v_o = 220 	ext{ m/s}$. The frequency $f''$ detected by the driver is: $f'' = f' \left( \frac{v + v_o}{v} ight)$ .
   Substitute the values into the formula:
   $$f'' = 3000 \left( \frac{330 + 220}{330} ight) = 3000 \left( \frac{550}{330} ight) = 3000 	imes \frac{5}{3} = 5000 	ext{ Hz}$$
Therefore, the frequency of the echo detected by the driver is $5000 	ext{ Hz}$.

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