Two vehicles `A` and `B` are moving towards each other with same speed `u = 25 m//s`. They blow horns of the same frequency `f = 550 Hz`. Wind is blowing at speed `w = 20 m//s` in the direction of motion of `A`. The driver of vehicle `A` hears the sound of horn blown by vehicle `B` and the sound of horn of his own vehicle after reflection from the vehicle `B`. If difference of wavelength of both sounds received by `A` is `(5)/(P)`. Find `P`. Velocity of sound is `= 320 m//s`.

Explanation

To solve the problem step by step, we need to analyze the situation using the principles of sound waves and the Doppler effect.

### Step 1: Understand the setup
Two vehicles A and B are moving towards each other with the same speed \( u = 25 \, \text{m/s} \). They blow horns of the same frequency \( f = 550 \, \text{Hz} \). Wind is blowing at speed \( w = 20 \, \text{m/s} \) in the direction of motion of vehicle A. The speed of sound is given as \( v = 320 \, \text{m/s} \).

### Step 2: Calculate the wavelength of the sound received by A from B
The wavelength \( \lambda \) of the sound received by A from B can be calculated using the formula:
\[ \lambda = \frac{v - w - u}{f} \]
Here, \( v \) is the speed of sound, \( w \) is the wind speed (which opposes the sound), and \( u \) is the speed of vehicle B (which is also moving towards A).

Substituting the values:
\[ \lambda_1 = \frac{320 - 20 - 25}{550} = \frac{275}{550} = 0.5 \, \text{m} \]

### Step 3: Calculate the frequency of the sound received by B
The frequency \( f' \) of the sound received by B from A can be calculated using the Doppler effect formula:
\[ f' = \frac{v + w}{v - u} f \]
Substituting the values:
\[ f' = \frac{320 + 20}{320 - 25} \times 550 = \frac{340}{295} \times 550 \]
Calculating \( f' \):
\[ f' \approx 0.576 \times 550 \approx 316.8 \, \text{Hz} \]

### Step 4: Calculate the wavelength of the reflected sound
The wavelength \( \lambda' \) of the sound reflected from B back to A can be calculated as:
\[ \lambda' = \frac{v - w - u}{f'} \]
Substituting the values:
\[ \lambda' = \frac{320 - 20 - 25}{316.8} \approx \frac{275}{316.8} \approx 0.868 \, \text{m} \]

### Step 5: Calculate the difference in wavelengths
The difference in wavelengths \( \Delta \lambda \) is given by:
\[ \Delta \lambda = \lambda' - \lambda_1 \]
Substituting the values:
\[ \Delta \lambda = 0.868 - 0.5 = 0.368 \, \text{m} \]

### Step 6: Relate the difference in wavelengths to the given equation
According to the problem, the difference in wavelengths is given as:
\[ \Delta \lambda = \frac{5}{P} \]
Setting the two expressions for \( \Delta \lambda \) equal to each other:
\[ 0.368 = \frac{5}{P} \]
Solving for \( P \):
\[ P = \frac{5}{0.368} \approx 13.6 \]

### Step 7: Final calculation
Since \( P \) should be an integer, we round \( P \) to the nearest whole number, which gives us:
\[ P \approx 14 \]

### Conclusion
The value of \( P \) is approximately \( 14 \).