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 (Figure 1) . 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 *v*2, so that the total
air resistance force can be expressed by
*F**a**i**r*=*α**v*2+*β*/*v*2,
where *α* and *β* are positive constants that depend
on the shape and size of the airplane and the density of the air.
To simulate a Cessna 150, a small single-engine airplane, use
*α* = 0.250 N⋅s2/m2 and *β* = 3.60×10^{5}
N⋅m2/s2 . In steady flight, the engine must provide a forward force
that exactly balances the air resistance force.

part a) Calculate the speed at which this airplane will have the
maximum *range* (that is, travel the greatest distance) for
a given quantity of fuel.

part b) Calculate the speed for which the airplane will have the
maximum *endurance* (that is, will remain in the air the
longest time).

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