# Newton: Flight is Impossible!

• If I have seen further than others, it is by standing upon the shoulders of giants…A man may imagine things that are false, but he can only understand things that are true, for if the things be false, the apprehension of them is not understanding…Errors are not in the art but in the artificers…I can calculate the motion of heavenly bodies, but not the madness of people. (Isaac Newton)

Newton computed the lift $L$ of a thin flat rectangular wing of (one-sided) surface area $S$traveling with velocity $V$ through still air at an angle of attack $\alpha$ (in radians), to be

• $L\approx \alpha^2V^2S$

by considering the lower part of the wing to be deviating incoming air downwards in the direction of the wing.

In other words, Newton argued that the lift coefficient $c_L\equiv \frac{2L}{V^2S}\approx\alpha^2$, so that for $\alpha =0.1$ as a common angle of attack, $c_L=0.02$.

For a human being + wing of weight 1000 Newton (100 kp) traveling on a wing of surface area $S=10$ square meter at velocity $V=100$ meter/second (360 km/hour) would be required. Or with $V=10$, a wing area of $S=1000$ square meters would be needed. Impossible!

The flight of birds must have been totally inexplicable to Newton. And of course: No hope for Icarus!

• How did Newton argue to come up with the above formula?

To understand what is correct, one has to also understand what is not correct.

# Kutta and Zhukovsky: Flight is Posssible!

Newton’s computation ruled aerodynamics for more than 200 years until the two brothers Wilbur and Orwille Wright in 1903 showed that powered human flight was possible. Newton’s lift coefficient was then quicky increased to

• $c_L=2\pi \alpha$

by the two mathematicians Kutta and Zhukovsky, based on a different mathematical argument, where the angle of attack $\alpha$ appears to the first power, much bigger than the second power in Newton’s formula. For $\alpha =\frac{10}{180}$ this gave $c_L=0.3$ and theory and observation was no longer in glaring contradiction.

In reality $c_L$ can be bigger for well designed wings $c_L\approx 18 \alpha$ that is with $\alpha$ in degrees $c_L\approx 0.1\alpha$ so that $c_L\approx 1$ for $\alpha =10$ degrees (50 times bigger than Newton’s!!).

# Flight in a Nutshell

The basics of flight can be summarized in the following two formulas

• $W=L=\frac{1}{2}c_LV^2S, \quad P=DV=\frac{L}{F}V$,

where $W$ is weight (in Newton), $L$ is lift (in Newton), $V$ velocity (meter/sec), $S$ wing area (square meter), $P$ power (Watt), $D$ is drag (Newton), and $c_L\approx 0.1\alpha$ with $\alpha$ angle of attack (degrees) is lift coefficient and $F = \frac{L}{D}=10-70$ is the finesse coefficient.

With $c_L=2.0$ we obtain the wing loading $\frac{W}{S}=V^2$ ranging from 10 Newton/square meter for a Gossamer Condor, 25 for a common tern, 100 for a wandering albatross, and up to 10 000 for an Airbus 340 at take-off. The velocity ranges from 5 meter per second for a Gossamer Condor, 10 for a starling, over 30 for a Canada goose up to 250 for an Airbus 380.

The quantity $\frac{P}{WV}=\frac{D}{L}=\frac{1}{F}$ measures energy consumption per meter traveled distance and ranges from 0.15 for pigeons, 0.05 for albatrosses and Boeing 747, 0.035 for Lance Armstrong and French TGV, 0.025 for sailplanes.

The power $P$ ranges from 1 Watt for a starling at 10 meter/sec, 450 kWatts for a Spitfire and 200.000 kWatts for a Boeing 747. A human being is capable of 200 Watt.