Towards a stationary object, a train is moving with a speed of 300 m/s and emits a sound wave of frequency 2000 Hz. Small percentage of the sound is reflected back to train in the form of echo. The frequency of the echo as detected by a person sitting in the train will be (speed of sound is 340 m/s).

Options

A. 31000 Hz
B. 35000 Hz
C. 32000 Hz
D. 45000 Hz

Explanation

To solve the problem, we will apply the Doppler effect formula for sound waves. The scenario involves a train moving towards a stationary object, emitting sound waves, which are then reflected back as an echo. The frequency of the echo as detected by a person in the train needs to be calculated.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Speed of the train (observer) \( v_o = 300 \, \text{m/s} \)
- Frequency of the sound emitted by the train (source) \( f_0 = 2000 \, \text{Hz} \)
- Speed of sound in air \( v = 340 \, \text{m/s} \)

2. **Understand the Doppler Effect:**
The Doppler effect formula for sound when the source and observer are moving towards each other is given by:
\[ f' = f_0 \frac{v + v_o}{v - v_s} \]
where:
- \( f' \) is the observed frequency,
- \( f_0 \) is the emitted frequency,
- \( v \) is the speed of sound,
- \( v_o \) is the speed of the observer (train),
- \( v_s \) is the speed of the source (the train, in this case).

3. **Determine the Source and Observer Speeds:**
- The source (the sound wave reflected back) is effectively moving towards the observer (the person in the train) at the same speed as the train, which is \( v_s = 300 \, \text{m/s} \).
- The observer (the person in the train) is also moving towards the source of the sound.

4. **Substituting Values into the Formula:**
Since both the observer and the source are moving towards each other, we have:
\[ f' = 2000 \, \text{Hz} \times \frac{340 \, \text{m/s} + 300 \, \text{m/s}}{340 \, \text{m/s} - 300 \, \text{m/s}} \]

5. **Calculating the Values:**
- Calculate the numerator:
\[ 340 + 300 = 640 \, \text{m/s} \]
- Calculate the denominator:
\[ 340 - 300 = 40 \, \text{m/s} \]
- Now substitute these values back into the formula:
\[ f' = 2000 \, \text{Hz} \times \frac{640}{40} \]
- Simplifying further:
\[ f' = 2000 \, \text{Hz} \times 16 = 32000 \, \text{Hz} \]

6. **Final Answer:**
The frequency of the echo as detected by a person sitting in the train is \( 32000 \, \text{Hz} \).