Two vehicles, each moving with speed `u` on the same horizontal straight road, are approaching each other. Wind blows along the road with velocity `w`. One of these vehicles blows a whistle of frequency `f_(1)`. An observer in the other vehicle hears the frequency of the whistle to be `f_(2)`. the speed of sound in still air is `V_C`. The correct statement `(s)` is (are)

Options

A. If the wind blows from the source to the observer, `f_2 gt f_1`
B. If the wind blows from the observer to the source, `f_2 gt f_1`
C. If the wind blows from observer to the source, `f_2 lt f_1`
D. If the wind blows from the source to the observer, `f_2 lt f_1`

Explanation

To solve the problem, we need to analyze the situation using the Doppler effect for sound. Here are the steps to derive the relationship between the frequencies heard by the observer and emitted by the source.

### Step-by-Step Solution:

1. **Identify the Variables:**
- Let the frequency of the whistle emitted by the source vehicle be \( f_1 \).
- Let the frequency heard by the observer in the other vehicle be \( f_2 \).
- The speed of sound in still air is \( V_C \).
- The speed of both vehicles is \( u \).
- The wind speed is \( w \).

2. **Determine the Effective Speed of Sound:**
- When wind blows, it affects the speed of sound.
- If the wind blows in the direction of the sound, the effective speed of sound becomes \( V_C + w \).
- If the wind blows against the direction of sound, the effective speed of sound becomes \( V_C - w \).

3. **Use the Doppler Effect Formula:**
- The general formula for the frequency heard by the observer when both source and observer are moving towards each other is:
\[ f_2 = f_1 \times \frac{V + V_O}{V - V_S} \]
- Here, \( V \) is the effective speed of sound, \( V_O \) is the speed of the observer, and \( V_S \) is the speed of the source.

4. **Substitute the Values:**
- Since both vehicles are moving towards each other with speed \( u \):
\[ f_2 = f_1 \times \frac{V_C + w + u}{V_C + w - u} \]
- This formula holds true if the wind is blowing in the same direction as the sound. If the wind is blowing against the sound, we would use \( V_C - w \) instead.

5. **Analyze the Ratio:**
- In both cases (wind blowing with or against), the numerator will always be greater than the denominator because:
- The numerator is \( V + u \) and the denominator is \( V - u \).
- Thus, \( V + u > V - u \).
- Therefore, we can conclude that:
\[ f_2 > f_1 \]

6. **Conclusion:**
- The frequency heard by the observer \( f_2 \) is always greater than the frequency emitted by the source \( f_1 \), regardless of the direction of the wind.