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 solve the problem, we need to find the apparent wavelengths of sound in front of and behind the locomotive. We will use the Doppler effect formulas for sound.

### Step 1: Identify the given values
- Speed of the train (VT) = 30 m/s
- Frequency of the whistle (f) = 500 Hz
- Speed of sound (V) = 345 m/s

### Step 2: Calculate the actual wavelength (λ)
The wavelength (λ) can be calculated using the formula:
\[ \lambda = \frac{V}{f} \]
Substituting the values:
\[ \lambda = \frac{345 \, \text{m/s}}{500 \, \text{Hz}} = 0.69 \, \text{m} \]

### Step 3: Calculate the apparent wavelength in front of the locomotive
Using the formula for the apparent wavelength in front of the source:
\[ \lambda' = \frac{V}{V + V_T} \cdot \lambda \]
Substituting the values:
\[ \lambda' = \frac{345}{345 + 30} \cdot 0.69 \]
\[ \lambda' = \frac{345}{375} \cdot 0.69 \]
\[ \lambda' = 0.92 \cdot 0.69 \approx 0.63 \, \text{m} \]

### Step 4: Calculate the apparent wavelength behind the locomotive
Using the formula for the apparent wavelength behind the source:
\[ \lambda' = \frac{V}{V - V_T} \cdot \lambda \]
Substituting the values:
\[ \lambda' = \frac{345}{345 - 30} \cdot 0.69 \]
\[ \lambda' = \frac{345}{315} \cdot 0.69 \]
\[ \lambda' = 1.095 \cdot 0.69 \approx 0.76 \, \text{m} \]

### Final Result
The apparent wavelengths of sound in front of and behind the locomotive are approximately:
- In front: \(0.63 \, \text{m}\)
- Behind: \(0.76 \, \text{m}\)