A car travelling towards a hill at `10 m//s` sound its horn which a frequency `500 H_(Z)`. This is heard in a second car travelling behind the first car in the same direction with speed `20 m//s`. The sound can also be heard in the second car by reflections of sound the hill. The beat frequency heard by the driver of the sound car will be (speed of sound in air `= 340 m//s)`

Options

A. `31 H_(Z)`
B. `24 H_(Z)`
C. `21 H_(Z)`
D. `34 H_(Z)`

Explanation

To solve the problem, we need to calculate the beat frequency heard by the driver of the second car. The beat frequency is the difference between the frequencies of the original sound and the reflected sound. We will use the Doppler effect to find the frequencies.

### Step-by-step Solution:

1. **Identify the Given Values:**
- Frequency of the horn, \( f = 500 \, \text{Hz} \)
- Speed of sound in air, \( v = 340 \, \text{m/s} \)
- Speed of the first car (source of sound), \( v_s = 10 \, \text{m/s} \)
- Speed of the second car (observer), \( v_o = 20 \, \text{m/s} \)

2. **Calculate the Frequency Heard by the Second Car (Original Sound):**
The frequency heard by the second car due to the moving source (first car) can be calculated using the Doppler effect formula:
\[ f' = f \left( \frac{v + v_o}{v - v_s} \right) \]
Substituting the values:
\[ f' = 500 \left( \frac{340 + 20}{340 - 10} \right) = 500 \left( \frac{360}{330} \right) \]
\[ f' = 500 \times 1.0909 \approx 545.45 \, \text{Hz} \]

3. **Calculate the Frequency of the Reflected Sound:**
The reflected sound will act as a new source for the second car. The frequency heard by the second car from the reflection can be calculated using the same formula, but now the source is stationary (the hill), and the observer (second car) is moving towards the reflection:
\[ f'' = f' \left( \frac{v + v_o}{v} \right) \]
Here, \( f' \) is the frequency calculated from the first car, and since the hill is stationary, \( v_s = 0 \):
\[ f'' = 545.45 \left( \frac{340 + 20}{340} \right) = 545.45 \left( \frac{360}{340} \right) \]
\[ f'' = 545.45 \times 1.0588 \approx 578.79 \, \text{Hz} \]

4. **Calculate the Beat Frequency:**
The beat frequency is the absolute difference between the two frequencies:
\[ f_{beat} = |f'' - f'| = |578.79 - 545.45| \approx 33.34 \, \text{Hz} \]

5. **Final Result:**
The beat frequency heard by the driver of the second car is approximately \( 33.34 \, \text{Hz} \).