The engine of a train moving with speed `10ms^(-1) towards a platform sounds a whistle at frequency `400 Hz`.The frequency heard by a passerger inside the train is: (neglect air speed . Speed of sound in air = `330 ms^(-1)`)

Options

A. `400 Hz`
B. `200 Hz`
C. `412 Hz`
D. `388 Hz`

Answer

To solve the problem, we need to determine the frequency heard by a passenger inside the train when the train is moving towards a platform and sounding a whistle.

### Step-by-Step Solution:

1. **Identify the given values:**
- Speed of the train (v_s) = 10 m/s (towards the platform)
- Frequency of the whistle (f) = 400 Hz
- Speed of sound in air (v) = 330 m/s

2. **Understand the scenario:**
- The passenger is inside the train, which is moving towards the source of the sound (the whistle). Since the passenger is moving with the train, they will hear the frequency of the whistle without any Doppler effect due to the train's motion.

3. **Use the Doppler Effect formula:**
The formula for the frequency heard by an observer when the source is moving towards the observer is given by:
\[ f' = f \left( \frac{v + v_o}{v - v_s} \right) \]
where:
- \( f' \) = frequency heard by the observer
- \( f \) = frequency of the source
- \( v \) = speed of sound in air
- \( v_o \) = speed of the observer (0 m/s since the observer is inside the train)
- \( v_s \) = speed of the source (the train) = 10 m/s

4. **Substituting the values:**
Since the passenger is inside the train, the speed of the observer \( v_o \) is 0. Therefore, the formula simplifies to:
\[ f' = f \left( \frac{v}{v - v_s} \right) \]
Substituting the values:
\[ f' = 400 \left( \frac{330}{330 - 10} \right) \]
\[ f' = 400 \left( \frac{330}{320} \right) \]

5. **Calculating the frequency:**
\[ f' = 400 \times 1.03125 = 412.5 \text{ Hz} \]

6. **Conclusion:**
The frequency heard by the passenger inside the train is approximately **412.5 Hz**.