A person is listening to two trains one approaching him while the other moving away from him. The speed of both the trains is 5 m/s. If both trains give off whistle of their nature frequency of 280 Hz then the observer will hear ... no of beats/s. (Velocity of sound = 350 m/s)

Options

A. 6
B. 7
C. 5
D. 8

Answer

To solve the problem, we need to determine the frequency of the sound waves heard by the observer from both trains and then calculate the number of beats per second produced by the two frequencies.

### Step-by-Step Solution:

1. **Identify Given Values**:
- Natural frequency of the trains, \( n = 280 \, \text{Hz} \)
- Speed of sound, \( V = 350 \, \text{m/s} \)
- Speed of the trains (sources), \( V_s = 5 \, \text{m/s} \)

2. **Calculate the Frequency of the Train Approaching the Observer**:
- When the train is approaching, the frequency \( n_1 \) can be calculated using the formula:
\[ n_1 = n \cdot \frac{V}{V - V_s} \]
- Substituting the values:
\[ n_1 = 280 \cdot \frac{350}{350 - 5} = 280 \cdot \frac{350}{345} \]
- Calculating \( n_1 \):
\[ n_1 = 280 \cdot 1.0145 \approx 284 \, \text{Hz} \]

3. **Calculate the Frequency of the Train Moving Away from the Observer**:
- When the train is moving away, the frequency \( n_2 \) can be calculated using the formula:
\[ n_2 = n \cdot \frac{V}{V + V_s} \]
- Substituting the values:
\[ n_2 = 280 \cdot \frac{350}{350 + 5} = 280 \cdot \frac{350}{355} \]
- Calculating \( n_2 \):
\[ n_2 = 280 \cdot 0.9859 \approx 276 \, \text{Hz} \]

4. **Calculate the Number of Beats**:
- The number of beats per second is given by the absolute difference between the two frequencies:
\[ \text{Number of beats} = |n_1 - n_2| = |284 - 276| = 8 \, \text{beats/s} \]

### Final Answer:
The observer will hear **8 beats per second**.