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

Options

A. 3500 Hz
B. 4000 Hz
C. 5000 Hz
D. 3000 Hz

Explanation

To solve the problem, we will use the Doppler effect formula for sound. The scenario involves a moving source (the train) and a stationary observer (the object). The sound frequency emitted by the train is affected by the relative motion between the source and the observer.

### Step-by-Step Solution:

1. **Identify the given values:**
- Speed of the train (source), \( V_s = 220 \, \text{m/s} \)
- Frequency of the sound emitted, \( f = 1000 \, \text{Hz} \)
- Speed of sound in air, \( V = 330 \, \text{m/s} \)
- The observer (the object) is stationary, so \( V_o = 0 \).

2. **Determine the frequency heard by the stationary object:**
The frequency of the sound as heard by the stationary object can be calculated using the Doppler effect formula:
\[ f' = f \left( \frac{V + V_o}{V - V_s} \right) \]
Since the object is stationary, \( V_o = 0 \):
\[ f' = f \left( \frac{V}{V - V_s} \right) \]

3. **Substitute the values into the formula:**
\[ f' = 1000 \left( \frac{330}{330 - 220} \right) \]
\[ f' = 1000 \left( \frac{330}{110} \right) \]
\[ f' = 1000 \times 3 = 3000 \, \text{Hz} \]

4. **Calculate the frequency of the echo detected by the driver of the train:**
Now, the sound that reflects back to the train is now treated as if the train is the observer and the object is the source. The frequency of the echo can be calculated using the same Doppler effect formula, but this time the train is moving towards the stationary object:
\[ f'' = f' \left( \frac{V + V_s}{V} \right) \]
Substituting the values:
\[ f'' = 3000 \left( \frac{330 + 220}{330} \right) \]
\[ f'' = 3000 \left( \frac{550}{330} \right) \]
\[ f'' = 3000 \times \frac{55}{33} = 3000 \times \frac{5}{3} = 5000 \, \text{Hz} \]

5. **Final Answer:**
The frequency of the echo as detected by the driver of the train is \( 5000 \, \text{Hz} \).