A piolt files due East from A to B. The velocity of the aeroplane in still air is v and the velocity of air relative to ground is u. The distance between A and B is l. Calculate the time for a round trip if air velocity is due to East. In still air time for round trip is `t_(0)`

Options

A. `(t_(0))/(1-(u^(2))/(v^(2)))`
B. `(t_(0))/(1+(u^(2))/(v^(2)))`
C. `(2t_(0))/(1+(u^(2))/(v^(2)))`
D. `(2t_(0))/(1-(u^(2))/(v^(2)))`

Answer

To solve the problem, we need to calculate the time taken for a round trip of an airplane flying from point A to point B and back to point A, considering the effect of wind. Here’s a step-by-step solution:

### Step 1: Understand the Given Information
- The airplane flies due east from A to B.
- Velocity of the airplane in still air = \( V \)
- Velocity of air (wind) relative to the ground = \( U \)
- Distance between A and B = \( L \)
- Time for a round trip in still air = \( T_0 \)

### Step 2: Calculate Time for the Trip from A to B
When the airplane is flying from A to B, the effective velocity of the airplane with respect to the ground is the sum of its own velocity and the wind velocity:
\[ \text{Effective velocity from A to B} = V + U \]
The time taken to travel from A to B (denoted as \( T_1 \)) can be calculated using the formula:
\[ T_1 = \frac{L}{V + U} \]

### Step 3: Calculate Time for the Trip from B to A
When the airplane is flying back from B to A, the wind opposes its motion. Thus, the effective velocity of the airplane with respect to the ground is:
\[ \text{Effective velocity from B to A} = V - U \]
The time taken to travel from B to A (denoted as \( T_2 \)) is given by:
\[ T_2 = \frac{L}{V - U} \]

### Step 4: Calculate Total Time for the Round Trip
The total time for the round trip (denoted as \( T \)) is the sum of the times for both legs of the trip:
\[ T = T_1 + T_2 = \frac{L}{V + U} + \frac{L}{V - U} \]
To combine these fractions, we find a common denominator:
\[ T = L \left( \frac{(V - U) + (V + U)}{(V + U)(V - U)} \right) \]
This simplifies to:
\[ T = L \left( \frac{2V}{V^2 - U^2} \right) = \frac{2LV}{V^2 - U^2} \]

### Step 5: Relate Total Time to \( T_0 \)
We know that the time for a round trip in still air is:
\[ T_0 = \frac{2L}{V} \]
Now we can express \( T \) in terms of \( T_0 \):
\[ T = \frac{2LV}{V^2 - U^2} = T_0 \cdot \frac{V}{V^2 - U^2} \]
Thus, we can rewrite \( T \) as:
\[ T = T_0 \cdot \frac{1}{1 - \frac{U^2}{V^2}} \]

### Final Answer
The time for the round trip considering the wind is:
\[ T = \frac{T_0}{1 - \frac{U^2}{V^2}} \]