An airplane flies northward from town `A` and `B` and then back again. There is a steady wind blowing towards the north so that for the first state of the trip, the airplane is flying in the same direction as the wind and for the return trip of the journey, the airplane is flying opposite of the wind. The total trip time `T_(omega)` as compared to the total trip time in the absence of any winds `T_(0)` is:

Options

A. `T_(w)=T_(0)`
B. `T_(w)gtT_(0)`
C. `T_(w)ltT_(0)`
D. Data is insufficient

Answer

To solve the problem, we need to analyze the airplane's journey in the presence of wind and compare it to the journey in the absence of wind. Let's break it down step by step.

### Step 1: Define Variables
Let:
- \( V_A \) = speed of the airplane in still air
- \( V_W \) = speed of the wind
- \( d \) = distance between towns A and B
- \( T_0 \) = total trip time in the absence of wind
- \( T_{\omega} \) = total trip time in the presence of wind

### Step 2: Calculate Time from A to B
When the airplane flies from A to B, it is flying in the same direction as the wind. Therefore, the effective speed of the airplane is:
\[ V_{AB} = V_A + V_W \]
The time taken to travel from A to B, denoted as \( t_1 \), is given by:
\[ t_1 = \frac{d}{V_{AB}} = \frac{d}{V_A + V_W} \]

### Step 3: Calculate Time from B to A
When the airplane returns from B to A, it is flying against the wind. Therefore, the effective speed of the airplane is:
\[ V_{BA} = V_A - V_W \]
The time taken to travel from B to A, denoted as \( t_2 \), is given by:
\[ t_2 = \frac{d}{V_{BA}} = \frac{d}{V_A - V_W} \]

### Step 4: Calculate Total Time in Presence of Wind
The total time \( T_{\omega} \) for the round trip in the presence of wind is:
\[ T_{\omega} = t_1 + t_2 = \frac{d}{V_A + V_W} + \frac{d}{V_A - V_W} \]
Factoring out \( d \):
\[ T_{\omega} = d \left( \frac{1}{V_A + V_W} + \frac{1}{V_A - V_W} \right) \]

### Step 5: Calculate Total Time in Absence of Wind
In the absence of wind, the time \( T_0 \) for the round trip is simply:
\[ T_0 = \frac{d}{V_A} + \frac{d}{V_A} = \frac{2d}{V_A} \]

### Step 6: Compare \( T_{\omega} \) and \( T_0 \)
To compare \( T_{\omega} \) with \( T_0 \), we can express \( T_{\omega} \) in terms of \( T_0 \):
\[ T_{\omega} = d \left( \frac{1}{V_A + V_W} + \frac{1}{V_A - V_W} \right) \]
This can be simplified further:
\[ T_{\omega} = d \left( \frac{(V_A - V_W) + (V_A + V_W)}{(V_A + V_W)(V_A - V_W)} \right) = d \left( \frac{2V_A}{V_A^2 - V_W^2} \right) \]
Thus,
\[ T_{\omega} = \frac{2d}{V_A} \cdot \frac{V_A}{V_A^2 - V_W^2} = T_0 \cdot \frac{1}{1 - \left(\frac{V_W}{V_A}\right)^2} \]

### Conclusion
Since \( V_W \) is positive, \( T_{\omega} > T_0 \). Therefore, the total trip time \( T_{\omega} \) in the presence of wind is greater than the total trip time \( T_0 \) in the absence of wind.

### Final Answer
The total trip time \( T_{\omega} \) is greater than \( T_0 \).

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