On a platform a man is watching two trains one leaving the station the other approaching the station, both with the same velocity 2 m/s and blowing their horn with a frequency 480 Hz. The number of beats heard by the man will be (Velocity of sound in air is 320 m/s)

Options

A. 6
B. 3
C. 2
D. 4

Answer

To solve the problem, we will use the Doppler effect to find the apparent frequencies of the horns of the two trains as heard by the man on the platform.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Actual frequency of the horn, \( f_0 = 480 \, \text{Hz} \)
- Velocity of sound in air, \( v = 320 \, \text{m/s} \)
- Velocity of the trains (sources), \( v_s = 2 \, \text{m/s} \)
- Velocity of the observer (man), \( v_o = 0 \, \text{m/s} \)

2. **Calculate the Apparent Frequency from the Approaching Train (Train A):**
- For the train approaching the observer, the formula for the apparent frequency \( f_1 \) is:
\[ f_1 = f_0 \left( \frac{v + v_o}{v - v_s} \right) \]
- Substituting the values:
\[ f_1 = 480 \left( \frac{320 + 0}{320 - 2} \right) = 480 \left( \frac{320}{318} \right) \]

3. **Calculate the Apparent Frequency from the Leaving Train (Train B):**
- For the train leaving the observer, the formula for the apparent frequency \( f_2 \) is:
\[ f_2 = f_0 \left( \frac{v + v_o}{v + v_s} \right) \]
- Substituting the values:
\[ f_2 = 480 \left( \frac{320 + 0}{320 + 2} \right) = 480 \left( \frac{320}{322} \right) \]

4. **Calculate the Difference in Frequencies:**
- The number of beats heard by the man is given by the difference in the two frequencies:
\[ \Delta f = f_1 - f_2 \]
- Substituting the expressions for \( f_1 \) and \( f_2 \):
\[ \Delta f = 480 \left( \frac{320}{318} \right) - 480 \left( \frac{320}{322} \right) \]
- Factor out \( 480 \times 320 \):
\[ \Delta f = 480 \times 320 \left( \frac{1}{318} - \frac{1}{322} \right) \]

5. **Simplify the Expression:**
- To simplify \( \frac{1}{318} - \frac{1}{322} \):
\[ \frac{1}{318} - \frac{1}{322} = \frac{322 - 318}{318 \times 322} = \frac{4}{318 \times 322} \]
- Substitute back into the expression for \( \Delta f \):
\[ \Delta f = 480 \times 320 \times \frac{4}{318 \times 322} \]

6. **Calculate the Numerical Value:**
- Performing the calculations gives:
\[ \Delta f \approx 6 \, \text{Hz} \]

### Final Answer:
The number of beats heard by the man will be **6 Hz**.

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