Two factories are sounding their sirens at 800 Hz. A man goes from one factory to the other at a speed of 2 m/s. The velocity of sound is 320 m/s. The number of beats heard by the person is 1 s will be

Options

A. 2
B. 4
C. 8
D. 10

Explanation

To solve the problem of how many beats are heard by the person moving between the two factories, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Frequency of sirens (f₀) = 800 Hz
- Speed of sound (v) = 320 m/s
- Speed of the observer (man) (v₀) = 2 m/s

2. **Calculate the Apparent Frequency from Factory A (F_a):**
- Since the man is moving away from Factory A, the apparent frequency can be calculated using the Doppler effect formula:
\[ F_a = f₀ \cdot \frac{v - v₀}{v} \]
- Substituting the values:
\[ F_a = 800 \cdot \frac{320 - 2}{320} = 800 \cdot \frac{318}{320} \]
- Simplifying this:
\[ F_a = 800 \cdot 0.99375 = 795 Hz \]

3. **Calculate the Apparent Frequency from Factory B (F_b):**
- Since the man is moving towards Factory B, the apparent frequency can be calculated as:
\[ F_b = f₀ \cdot \frac{v + v₀}{v} \]
- Substituting the values:
\[ F_b = 800 \cdot \frac{320 + 2}{320} = 800 \cdot \frac{322}{320} \]
- Simplifying this:
\[ F_b = 800 \cdot 1.00625 = 805 Hz \]

4. **Calculate the Beat Frequency (F_beats):**
- The beat frequency is the absolute difference between the two apparent frequencies:
\[ F_{beats} = |F_b - F_a| = |805 - 795| = 10 Hz \]

5. **Determine the Number of Beats Heard in 1 Second:**
- Since the beat frequency is 10 Hz, this means that the person hears 10 beats in one second.

### Final Answer:
The number of beats heard by the person in one second is **10 beats**.

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