An eaeroplane is to go along straight line from A to B, and back again. The relative speed with respect to wind is V. The wind blows perpendicular to line AB with speed v. The distance between A and B is `l`. The total time for the round trip is:

Options

A. `(2l)/(sqrt(V^(2)-nu^(2)))`
B. `(2vl)/(V^(2)-nu^(2))`
C. `(2vl)/(V^(2)-nu^(2))`
D. `(2l)/(sqrt(V^(2)+nu^(2)))`

Explanation

To solve the problem, we need to determine the total time taken for an airplane to travel from point A to point B and back again, considering the effects of wind. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We have an airplane flying from point A to point B and back. The distance between A and B is \( l \), the speed of the airplane relative to the wind is \( V \), and the wind blows perpendicular to the line AB with speed \( v \).

### Step 2: Analyze the Motion
When the airplane is flying, its velocity can be broken down into two components:
- A horizontal component (along the line AB): \( V \cos \theta \)
- A vertical component (against or with the wind): \( V \sin \theta \)

### Step 3: Relate the Vertical Component to Wind Speed
Since the wind blows perpendicular to the path of the airplane, we can set the vertical component of the airplane's speed equal to the wind speed:
\[ V \sin \theta = v \]
From this, we can express \( \sin \theta \):
\[ \sin \theta = \frac{v}{V} \]

### Step 4: Use the Pythagorean Identity
We know from trigonometry that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Substituting for \( \sin \theta \):
\[ \left(\frac{v}{V}\right)^2 + \cos^2 \theta = 1 \]
This leads us to find \( \cos \theta \):
\[ \cos^2 \theta = 1 - \left(\frac{v}{V}\right)^2 \]
\[ \cos \theta = \sqrt{1 - \frac{v^2}{V^2}} = \frac{\sqrt{V^2 - v^2}}{V} \]

### Step 5: Calculate the Effective Speed
The effective speed of the airplane in the horizontal direction (along AB) is:
\[ V \cos \theta = V \cdot \frac{\sqrt{V^2 - v^2}}{V} = \sqrt{V^2 - v^2} \]

### Step 6: Calculate the Total Distance
The total distance for the round trip is:
\[ \text{Total Distance} = l + l = 2l \]

### Step 7: Calculate the Total Time
The total time \( T \) for the round trip can be calculated using the formula:
\[ T = \frac{\text{Total Distance}}{\text{Effective Speed}} = \frac{2l}{\sqrt{V^2 - v^2}} \]

### Final Answer
Thus, the total time for the round trip is:
\[ T = \frac{2l}{\sqrt{V^2 - v^2}} \]

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