Two persons `A` and `B` , each carrying a source of frequency `300 H_(Z)` , are standing a few metre apart. A starts moving towards `B` with velocity `30 m//s` . If speed of sound is `300 m//s` , which of the following is true?

Options

A. (a) Number of beats heard by `A` is heigher than that heard by `B`
B. (b) The number of beats heard by `B` are `30 H_(Z)`
C. Both (a) and (b) are correct
D. Both (a) and (b) are wrong

Explanation

To solve the problem, we need to analyze the situation using the Doppler effect for sound waves. We have two observers, A and B, with A moving towards B.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Frequency of the sound source (f) = 300 Hz
- Speed of sound (v) = 300 m/s
- Speed of observer A (v_o) = 30 m/s
- Observer B is stationary (v_s = 0).

2. **Calculate the Apparent Frequency for Observer A:**
- Since A is moving towards B, we can use the Doppler effect formula:
\[ f_A = f \times \frac{v + v_o}{v - v_s} \]
- Here, \(v_s = 0\) (B is stationary), so the formula simplifies to:
\[ f_A = f \times \frac{v + v_o}{v} \]
- Substituting the values:
\[ f_A = 300 \times \frac{300 + 30}{300} = 300 \times \frac{330}{300} = 330 \, \text{Hz} \]

3. **Calculate the Apparent Frequency for Observer B:**
- Observer B hears the sound from A, who is moving towards him. Therefore, we use the formula:
\[ f_B = f \times \frac{v + v_s}{v - v_o} \]
- Here, \(v_s = 0\) (A is the source), so the formula simplifies to:
\[ f_B = f \times \frac{v}{v - v_o} \]
- Substituting the values:
\[ f_B = 300 \times \frac{300}{300 - 30} = 300 \times \frac{300}{270} = 333.33 \, \text{Hz} \]

4. **Calculate the Beat Frequency:**
- The beat frequency is the difference between the frequencies heard by A and B.
- For A, the frequency heard is \(f_A = 330 \, \text{Hz}\) and for B, it is \(f_B = 333.33 \, \text{Hz}\).
- Therefore, the number of beats heard by A:
\[ \text{Beats}_{A} = f_A - f = 330 - 300 = 30 \, \text{Hz} \]
- And the number of beats heard by B:
\[ \text{Beats}_{B} = f_B - f = 333.33 - 300 = 33.33 \, \text{Hz} \]

5. **Conclusion:**
- A hears 30 beats per second and B hears 33.33 beats per second.
- Thus, the statement "Number of beats heard by A is higher than that of B" is incorrect.
- The statement "Number of beats heard by B are 30 Hz" is also incorrect.
- Therefore, both A and B are correct in their respective observations.