Two friends A and B are standing a distance x apartin an open field and wind is blowing from A to B.A beats a drum and B hears the sound `t_1` timed after he sees the event. A and B intechange their positions and te experiment is repeated. This time B hears teh drum `t_2` time after he sees the event. Calculte teh velocity of sound in still air v and the velocity of wind u. Neglect the tiem light takes in travelling between the friends.

Explanation

To solve the problem, we will analyze the two scenarios described and derive equations based on the information given.

### Step-by-Step Solution:

1. **Understanding the Setup:**
- Two friends, A and B, are standing a distance \( x \) apart.
- Wind is blowing from A to B.
- When A beats the drum, B hears it after a time \( t_1 \).
- When A and B switch places, B hears the drum after a time \( t_2 \).

2. **Case 1: A beats the drum, B hears it.**
- The sound travels from A to B with the wind.
- The effective speed of sound with respect to the ground is \( v + u \) (where \( v \) is the speed of sound in still air and \( u \) is the speed of the wind).
- The time taken for sound to travel distance \( x \) is \( t_1 \).
- Therefore, we can write the equation:
\[ x = (v + u) t_1 \]
- Rearranging gives:
\[ v + u = \frac{x}{t_1} \quad \text{(Equation 1)} \]

3. **Case 2: B beats the drum, A hears it.**
- Now, the sound travels from A to B against the wind.
- The effective speed of sound with respect to the ground is \( v - u \).
- The time taken for sound to travel distance \( x \) is \( t_2 \).
- Therefore, we can write the equation:
\[ x = (v - u) t_2 \]
- Rearranging gives:
\[ v - u = \frac{x}{t_2} \quad \text{(Equation 2)} \]

4. **Adding the Two Equations:**
- From Equation 1: \( v + u = \frac{x}{t_1} \)
- From Equation 2: \( v - u = \frac{x}{t_2} \)
- Adding these two equations:
\[ (v + u) + (v - u) = \frac{x}{t_1} + \frac{x}{t_2} \]
\[ 2v = \frac{x}{t_1} + \frac{x}{t_2} \]
- Thus, we find:
\[ v = \frac{x}{2} \left( \frac{1}{t_1} + \frac{1}{t_2} \right) \quad \text{(Equation 3)} \]

5. **Subtracting the Two Equations:**
- Now, subtract Equation 2 from Equation 1:
\[ (v + u) - (v - u) = \frac{x}{t_1} - \frac{x}{t_2} \]
\[ 2u = \frac{x}{t_1} - \frac{x}{t_2} \]
- Thus, we find:
\[ u = \frac{x}{2} \left( \frac{1}{t_1} - \frac{1}{t_2} \right) \quad \text{(Equation 4)} \]

### Final Results:
- The velocity of sound in still air \( v \) is given by:
\[ v = \frac{x}{2} \left( \frac{1}{t_1} + \frac{1}{t_2} \right) \]
- The velocity of wind \( u \) is given by:
\[ u = \frac{x}{2} \left( \frac{1}{t_1} - \frac{1}{t_2} \right) \]