An airplane in flight is subject to an air resistance
force proportional to the square of its speed v. But there is an additional
resistive force because the airplane has wings. Air flowing over the
wings is pushed down and slightly forward, so from Newton’s third law
the air exerts a force on the wings and airplane that is up and slightly
backward (Fig.). The upward force is the lift force that keeps the
airplane aloft, and the backward force is called induced drag. At flying
speeds, induced drag is inversely proportional to v2, so the total air
resistance force can be expressed by Fair = av2 + b/v2, where a and
b are positive constants that depend on the shape and size of the airplane
and the density of the air. For a Cessna 150, a small single-engine
airplane, a = 0.30 N . s2/m2 and b = 3.5 * 105 N . m2/s2. In steady
flight, the engine must provide a forward force that exactly balances the
air resistance force. (a) Calculate the speed 1in km>h2 at which this airplane
will have the maximum range (that is, travel the greatest distance)
for a given quantity of fuel. (b) Calculate the speed (in km>h) for which
the airplane will have the maximum endurance (that is, remain in the air
the longest time).
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