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 H_(Z)` . What frequency is heard by a passenger on a train moving in the opposite direction to the first at `18.0 m//s` and (a) approaching the first? (b) receding from the first? Speed of sound in air = `340 m//s` .

Explanation

To solve the problem, we will use the Doppler effect formula for sound waves. The formula for the frequency heard by an observer when both the source and observer are moving is given by:

1. **When the observer is approaching the source**:
\[ f' = f \times \frac{v_s + v_o}{v_s - v_s'} \]

2. **When the observer is receding from the source**:
\[ f'' = f \times \frac{v_s - v_o}{v_s + v_s'} \]

Where:
- \( f' \) is the frequency heard by the observer.
- \( f \) is the frequency emitted by the source (262 Hz).
- \( v_s \) is the speed of sound in air (340 m/s).
- \( v_o \) is the speed of the observer (train 2, 18 m/s).
- \( v_s' \) is the speed of the source (train 1, 30 m/s).

### Step-by-Step Solution

#### (a) Frequency heard by the passenger approaching the first train:

1. **Identify the values**:
- Frequency of the whistle, \( f = 262 \, \text{Hz} \)
- Speed of sound, \( v_s = 340 \, \text{m/s} \)
- Speed of observer (train 2), \( v_o = 18 \, \text{m/s} \)
- Speed of source (train 1), \( v_s' = 30 \, \text{m/s} \)

2. **Substitute the values into the formula for approaching**:
\[ f' = 262 \times \frac{340 + 18}{340 - 30} \]

3. **Calculate the values in the fraction**:
\[ f' = 262 \times \frac{358}{310} \]

4. **Calculate the frequency**:
\[ f' = 262 \times 1.15484 \approx 302 \, \text{Hz} \]

#### (b) Frequency heard by the passenger receding from the first train:

1. **Substitute the values into the formula for receding**:
\[ f'' = 262 \times \frac{340 - 18}{340 + 30} \]

2. **Calculate the values in the fraction**:
\[ f'' = 262 \times \frac{322}{370} \]

3. **Calculate the frequency**:
\[ f'' = 262 \times 0.87027 \approx 228 \, \text{Hz} \]

### Final Answers:
- (a) The frequency heard by the passenger approaching the first train is approximately **302 Hz**.
- (b) The frequency heard by the passenger receding from the first train is approximately **228 Hz**.