A railroad train is travelling at 30.0 m/s in still air. The frequency of the note emitted by the train whistle is 262 Hz. Speed of sound in air is 340 m/s.

Options

A. Frequency heard by a passenger on another train moving in the opposite direction to the first at 18.0 m/s and approaching the first is 302 Hz
B. Frequency heard by a passenger on another train moving in the opposite direction to the first at 18.0 m/s and receding from the first is 228 Hz
C. Frequency heard by a passenger on another train moving in the same direction of the first at 18.0 m/s and approaching the first is 272 Hz
D. Frequency heard by a passenger on another train moving in the same direction of the first at 18.0 m/s and receding the first is 253 Hz

Answer

To solve the problem, we need to find the apparent frequency of the sound heard by a passenger moving at a speed of 18 km/h (which we will convert to m/s) when the train is both approaching and receding.

### Step-by-Step Solution:

1. **Convert the speed of the passenger from km/h to m/s:**
\[ \text{Speed of passenger} = 18 \text{ km/h} = \frac{18 \times 1000}{3600} \text{ m/s} = 5 \text{ m/s} \]

2. **Identify the given values:**
- Speed of the train (source) \( v_s = 30.0 \text{ m/s} \)
- Frequency of the train whistle \( f = 262 \text{ Hz} \)
- Speed of sound in air \( v = 340 \text{ m/s} \)
- Speed of the passenger (observer) \( v_o = 5 \text{ m/s} \)

3. **Calculate the apparent frequency when the train is approaching the passenger:**
The formula for the apparent frequency when the source is moving towards the observer is:
\[ f' = f \cdot \frac{v + v_o}{v - v_s} \]
Substituting the values:
\[ f' = 262 \cdot \frac{340 + 5}{340 - 30} = 262 \cdot \frac{345}{310} \]
Now, calculate the value:
\[ f' = 262 \cdot 1.113 = 292.836 \text{ Hz} \approx 293 \text{ Hz} \]

4. **Calculate the apparent frequency when the train is receding from the passenger:**
The formula for the apparent frequency when the source is moving away from the observer is:
\[ f' = f \cdot \frac{v - v_o}{v + v_s} \]
Substituting the values:
\[ f' = 262 \cdot \frac{340 - 5}{340 + 30} = 262 \cdot \frac{335}{370} \]
Now, calculate the value:
\[ f' = 262 \cdot 0.905 = 237.71 \text{ Hz} \approx 238 \text{ Hz} \]

### Final Answers:
- Apparent frequency when approaching: **293 Hz**
- Apparent frequency when receding: **238 Hz**