Question 1210546: two planes take off from the origin o,at the same time.Plane A travel to point P which is 100 miles north of O,and then returns .Plane B travels to Q which is 100 miles east of O and then returns.Each plane can travel through the air at 100 mi/hr.There is a 40 mi/hr westerly wind(a wind from the west).Determine which plane returns to O first
Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website! To determine which plane returns first, we need to calculate the total travel time for each plane, accounting for how the wind affects their ground speed.
### Plane B: Traveling East and West (The "Wind-Aligned" Case)
Plane B travels to point Q (100 miles East) and back. The 40 mi/hr wind is blowing from the West (moving East).
* **Outbound (Eastward):** The wind is a tailwind.
* Ground Speed =
* Time =
* **Return (Westward):** The wind is a headwind.
* Ground Speed =
* Time =
* **Total Time for B:**
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### Plane A: Traveling North and South (The "Crosswind" Case)
Plane A travels to point P (100 miles North) and back. To maintain a straight North-South path, the plane must angle itself into the 40 mi/hr westerly wind. This creates a right triangle where the airspeed is the hypotenuse and the wind is one side.
* **Calculating Ground Speed ():**
Using the Pythagorean theorem:
* **Total Time for A:** Since the ground speed is the same for both the Northbound and Southbound legs (the wind is a constant crosswind):
* Total Time =
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### Conclusion
Comparing the results:
* **Plane A:** ~2.18 hours
* **Plane B:** ~2.38 hours
**Plane A (traveling North/South) returns to O first.** This is a classic physics result: a crosswind slows a plane down less than a headwind/tailwind combination of the same magnitude over a round trip.
Would you like me to show you the mathematical proof for why the crosswind is always faster, regardless of the specific wind speed?
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