A man is watching two trains, one leaving and the other coming in with equal speeds of `4"m s"^(-1)`. If they sound their whistles, each of frequency 240 Hz, the number of beats per second heard by the man is (velocity of sound in air `=330"m s"^(-1)` )

Options

A. 6
B. 3
C. 0
D. 12

Explanation

To solve the problem, we need to determine the number of beats per second heard by a man watching two trains, one approaching and the other receding, both sounding whistles of frequency 240 Hz. The speed of sound in air is given as 330 m/s, and the speed of the trains is 4 m/s.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Frequency of the whistle (f) = 240 Hz
- Speed of sound in air (v_s) = 330 m/s
- Speed of the trains (v_t) = 4 m/s

2. **Calculate the Frequency Heard from the Approaching Train:**
- When the train is approaching the observer, the frequency heard (f_a) can be calculated using the formula:
\[ f_a = f \times \frac{v_s}{v_s - v_t} \]
- Substituting the values:
\[ f_a = 240 \times \frac{330}{330 - 4} = 240 \times \frac{330}{326} \]

3. **Calculate the Frequency Heard from the Receding Train:**
- When the train is receding from the observer, the frequency heard (f_r) can be calculated using the formula:
\[ f_r = f \times \frac{v_s}{v_s + v_t} \]
- Substituting the values:
\[ f_r = 240 \times \frac{330}{330 + 4} = 240 \times \frac{330}{334} \]

4. **Calculate the Beat Frequency:**
- The beat frequency (f_b) is the difference between the frequencies heard from the two trains:
\[ f_b = |f_a - f_r| \]
- Substituting the expressions for f_a and f_r:
\[ f_b = \left| 240 \times \frac{330}{326} - 240 \times \frac{330}{334} \right| \]
- Factor out the common terms:
\[ f_b = 240 \times 330 \times \left| \frac{1}{326} - \frac{1}{334} \right| \]

5. **Simplify the Expression:**
- To simplify \(\frac{1}{326} - \frac{1}{334}\):
\[ \frac{1}{326} - \frac{1}{334} = \frac{334 - 326}{326 \times 334} = \frac{8}{326 \times 334} \]
- Substitute this back into the beat frequency equation:
\[ f_b = 240 \times 330 \times \frac{8}{326 \times 334} \]

6. **Calculate the Numerical Value:**
- Now, calculate the numerical value:
\[ f_b \approx 240 \times 330 \times \frac{8}{108356} \approx 5.8 \text{ Hz} \]
- Rounding to the nearest integer gives us approximately 6 beats per second.

7. **Final Answer:**
- The number of beats per second heard by the man is **6 beats per second**.