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 (\(\textbf{Fig. P6.94}\)). 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 the total air resistance force can be expressed by \(F_air = \alpha v^{2} +
\beta /v{^2}\), where \(\alpha\) and \(\beta\) 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, \(\alpha = 0.30 \, \mathrm{N} \cdot
\mathrm{s^{2}/m^{2}}\) and \(\beta = 3.5 \times 10^5 \, \mathrm{N} \cdot
\mathrm{m^2/s^2}\). In steady flight, the engine must provide a forward force
that exactly balances the air resistance force. (a) Calculate the speed (in
km/h) 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).