An airplane flies from a town A to a town B when there is no wind and takes a total time `T_(0)` for a return trip. When there is a wind blowing in a direction from town A to town B, the plane's time for a similar return trip, `T_(w)`, would satisfy

Options

A. `T_(0)lt T_(w)`
B. `T_(0) gt T_(w)`
C. `T_(0) = T_(w)`
D. the result depends on the wind velocity between the towns

Explanation

To solve the problem, we need to analyze the airplane's travel time under two different conditions: without wind and with wind. Let's break it down step by step.

### Step 1: Define the Variables
- Let the distance from town A to town B be \( x \).
- Let the speed of the airplane be \( v \).
- Let the speed of the wind be \( v_w \).
- The total time taken for the return trip without wind is \( T_0 \).

### Step 2: Calculate the Time Without Wind
When there is no wind, the airplane travels from A to B and back to A. The total distance for the round trip is \( 2x \). The time taken for the return trip without wind can be expressed as:
\[ T_0 = \frac{2x}{v} \]
This is our **Equation 1**.

### Step 3: Calculate the Time With Wind
When there is wind blowing from A to B, the airplane's effective speed while flying from A to B becomes \( v + v_w \) (the speed of the airplane plus the speed of the wind). Conversely, while returning from B to A, the effective speed becomes \( v - v_w \) (the speed of the airplane minus the speed of the wind).

#### Time to Fly from A to B:
\[ T_{AB} = \frac{x}{v + v_w} \]

#### Time to Fly from B to A:
\[ T_{BA} = \frac{x}{v - v_w} \]

### Step 4: Total Time With Wind
The total time for the round trip with wind \( T_w \) is the sum of the time taken to fly from A to B and from B to A:
\[ T_w = T_{AB} + T_{BA} = \frac{x}{v + v_w} + \frac{x}{v - v_w} \]

### Step 5: Simplify the Expression for \( T_w \)
To combine the two fractions, we find a common denominator:
\[ T_w = x \left( \frac{(v - v_w) + (v + v_w)}{(v + v_w)(v - v_w)} \right) = x \left( \frac{2v}{v^2 - v_w^2} \right) \]

Thus, we can write:
\[ T_w = \frac{2x}{v^2 - v_w^2} \cdot v \]

### Step 6: Relate \( T_w \) to \( T_0 \)
From Equation 1, we know that:
\[ T_0 = \frac{2x}{v} \]

Now, substituting \( T_0 \) into the expression for \( T_w \):
\[ T_w = T_0 \cdot \frac{1}{1 - \frac{v_w^2}{v^2}} \]

### Conclusion
From our analysis, we can conclude that:
\[ T_w = T_0 \cdot \frac{1}{1 - \frac{v_w^2}{v^2}} \]
This indicates that the time taken with wind \( T_w \) is greater than the time taken without wind \( T_0 \) since \( \frac{v_w^2}{v^2} \) is a positive fraction less than 1.

### Final Relation
Thus, the final relation we derive is:
\[ T_w > T_0 \]