An airplane in flight is subject to an air resistance force proportional to the square of its speed [itex]v[/itex]. 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. 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 [itex]v^2[/itex], so that the total air resistance force can be expressed by [itex]F_{air}=\alpha v^2 + \beta/v^2[/itex], where [itex]\alpha[/itex] and [itex]\beta[/itex] 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 [itex]\alpha[/itex] and [itex]\beta[/itex]. In steady flight, the engine must provide a forward force that exactly balances the air resistance force.(adsbygoogle = window.adsbygoogle || []).push({});

Calculate the speed (in km/h) for which the airplane will have the maximum endurance (that is, will remain in the air the longest time).

Since the plane is in steady flight, the thrust of the engines equals the drag force from the air. The power of the engines is thus

[itex]P = F_{air} v[/itex].

I assume that the power is directly proportional to the fuel flow rate, which is inversely proportional to the endurance. Letting the derivative wrt v equal zero gives

[itex]\frac{\mathrm{d}P}{\mathrm{d}v} = 0[/itex]

[itex]v = \sqrt[4]{\frac{\beta}{3\alpha}}[/itex].

Which is not correct. What am I doing wrong?

Thanks

James

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# Homework Help: Maximum range/endurance of an aeroplane

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