An aeroplane 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-v^2)`
B. `(2vl)/(V^2-v^2)`
C. `(2Vl)/(V^2-v^2)`
D. `(2l)/(sqrt(V^2+v^2))`

Explanation

To solve the problem, we need to determine the total time taken for the round trip of the airplane from point A to point B and back to point A, considering the effects of wind.

### Step-by-Step Solution:

1. **Identify the Given Variables:**
- Let the distance between points A and B be \( l \).
- The relative speed of the airplane with respect to the wind is \( V \).
- The speed of the wind blowing perpendicular to the line AB is \( v \).

2. **Determine the Effective Velocity Components:**
- The airplane's effective velocity in the direction of the trip (along line AB) is affected by the wind.
- The velocity of the airplane can be resolved into two components:
- The component along the direction of travel (horizontal): \( V \cos \theta \)
- The component perpendicular to the direction of travel (vertical): \( V \sin \theta \)
- Since the wind is blowing perpendicular to the line AB, we can relate the components using the Pythagorean theorem:
\[ V^2 = (V \cos \theta)^2 + (V \sin \theta)^2 \]
- The vertical component of the airplane's velocity must balance the wind speed:
\[ V \sin \theta = v \]
- Thus, we can express \( \sin \theta \):
\[ \sin \theta = \frac{v}{V} \]

3. **Calculate the Horizontal Component:**
- Using \( \cos^2 \theta + \sin^2 \theta = 1 \):
\[ \cos^2 \theta = 1 - \left(\frac{v}{V}\right)^2 \]
- Therefore, the horizontal component becomes:
\[ V \cos \theta = V \sqrt{1 - \left(\frac{v}{V}\right)^2} = \sqrt{V^2 - v^2} \]

4. **Calculate the Time for One Leg of the Trip (A to B):**
- The time taken to travel from A to B is given by:
\[ T_{AB} = \frac{l}{V \cos \theta} = \frac{l}{\sqrt{V^2 - v^2}} \]

5. **Calculate the Time for the Return Trip (B to A):**
- The time taken to return from B to A is the same as the time taken from A to B, since the conditions are symmetric:
\[ T_{BA} = T_{AB} = \frac{l}{\sqrt{V^2 - v^2}} \]

6. **Calculate the Total Time for the Round Trip:**
- The total time for the round trip is:
\[ T_{total} = T_{AB} + T_{BA} = 2 \cdot \frac{l}{\sqrt{V^2 - v^2}} = \frac{2l}{\sqrt{V^2 - v^2}} \]

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