A train is moving at 30 m/s in still air. The frequency of the locomotive whistle is 500 Hz and the speed of sound is 345 m/s. The apparent wavelengths of sound in front of and behind the locomotive are respectively :-

Options

A. 0.63 m, 0.80 m
B. 0.63 m, 0.75 m
C. 0.60 m, 0.85 m
D. 0.60 m, 0.75 m

Explanation

To find the apparent wavelengths of sound in front of and behind the locomotive, we will use the Doppler effect formula. Here are the steps to solve the problem:

### Step 1: Identify the given values
- Speed of the train (source), \( v_s = 30 \, \text{m/s} \)
- Frequency of the whistle, \( f = 500 \, \text{Hz} \)
- Speed of sound, \( v = 345 \, \text{m/s} \)

### Step 2: Calculate the wavelength in still air
Using the formula for wavelength:
\[ \lambda_0 = \frac{v}{f} \]
Substituting the values:
\[ \lambda_0 = \frac{345 \, \text{m/s}}{500 \, \text{Hz}} = 0.69 \, \text{m} \]

### Step 3: Calculate the apparent wavelength in front of the locomotive
For the observer in front of the source, the formula for the apparent frequency is:
\[ f' = f \left(\frac{v}{v - v_s}\right) \]
Substituting the values:
\[ f' = 500 \left(\frac{345}{345 - 30}\right) = 500 \left(\frac{345}{315}\right) \approx 500 \times 1.0952 \approx 547.6 \, \text{Hz} \]
Now, we can find the apparent wavelength using:
\[ \lambda_1 = \frac{v}{f'} \]
Substituting the values:
\[ \lambda_1 = \frac{345}{547.6} \approx 0.630 \, \text{m} \]

### Step 4: Calculate the apparent wavelength behind the locomotive
For the observer behind the source, the formula for the apparent frequency is:
\[ f' = f \left(\frac{v}{v + v_s}\right) \]
Substituting the values:
\[ f' = 500 \left(\frac{345}{345 + 30}\right) = 500 \left(\frac{345}{375}\right) \approx 500 \times 0.92 \approx 460 \, \text{Hz} \]
Now, we can find the apparent wavelength using:
\[ \lambda_2 = \frac{v}{f'} \]
Substituting the values:
\[ \lambda_2 = \frac{345}{460} \approx 0.750 \, \text{m} \]

### Step 5: Conclusion
The apparent wavelengths of sound in front of and behind the locomotive are approximately:
- In front: \( \lambda_1 \approx 0.630 \, \text{m} \)
- Behind: \( \lambda_2 \approx 0.750 \, \text{m} \)

### Final Answer:
The apparent wavelengths of sound in front of and behind the locomotive are \( 0.630 \, \text{m} \) and \( 0.750 \, \text{m} \), respectively.

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