You are flying to New York. You've been reading the in-flight magazine, which has an article about the physics of flying. You learned that the airflow over the wings creates a lift force that is always perpendicular to the wings. In level flight, the upward lift force exactly balances the downward weight force. The pilot comes on to say that, because of heavy traffic, the plane is going to circle the airport for a while. She says that you'll maintain a speed of 300 mph at an altitude of 20,000 ft. You start to wonder what the diameter of the plane's circle around the airport is. You notice that the pilot has banked the plane so that the wings are 10∘ from horizontal. The safety card in the seatback pocket informs you that the plane's wing span is 250 ft.
What is the diameter of the airplane's path around the airport?
The centrifugal force on the circling plane is m.v²/R The weight
force = m.g
[m.v²/R]/(m.g) = v²/(g.R) = tan(10˚) giving R = v²/[9.81×tan(10˚)]
= v²/1.73
Now we need v in m/s (which is 134.112 for 300 mph) and we'll get R
in meters:
R = (134.112)²/1.73 = 28,900 m = 10.396 km = 6.45977491 miles. So
the diameter = 12.9195 miles.
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