Let me walk you through the example in the text — Page 77, Lesson 86

Let’s pick up right where the lift formula left off, because now we’re going to use it to look at the relationship between speed and angle of attack at a constant altitude — meaning constant air density. Here’s the core idea: as speed changes, angle of attack must be adjusted to keep lift constant. Let me walk you through the example in the text. If your indicated airspeed, IAS, is doubled, then at constant altitude your true airspeed, TAS, will also double. Now, because lift depends on the square of speed, doubling the speed increases the dynamic pressure — and hence the lift — by a factor of four. So to keep the lift force constant, you have to decrease the angle of attack so that the lift coefficient, CL, drops to one quarter of its previous value. Let me make sure you’ve got the terminology straight. Dynamic pressure is the pressure due to the motion of the air, and it’s what the square of speed acts on in the lift formula. The lift coefficient, CL, is a dimensionless number that represents how much lift a wing shape produces at a given angle of attack. So when you double speed, dynamic pressure quadruples, and you compensate by cutting CL to a quarter. Now, there’s a useful relationship stated here: IAS varies approximately as the square root of dynamic pressure. The proportionality is written as IAS ∝ √Q. That means if dynamic pressure goes up by a factor of four, IAS only doubles — because the square root of four is two. And for simplicity, at constant altitude, TAS changes in proportion to IAS — double one, double the other. But the text is careful to note this is only true up to about Mach 0.4; above that, compressibility effects mean it’s no longer accurate. Now let’s look at how the lift formula can be transposed to calculate things a professional pilot actually cares about. The formula is L = ½ ρ V² CL S. If we transpose it to solve for CL, we get CL = L / (½ ρ V² S). Since in level flight lift, density, and wing area are constant, we can write that CL is proportional to 1 / V². Here’s the worked example: if speed is increased in level flight by 30% above the minimum level flight speed, we write that as 1.3V. The proportional change in CL is then 1 divided by (1.3)², which is 1 divided by 1.69, which equals 0.59, or 59%. So while maintaining level flight at a speed 30% above minimum level flight speed, the CL would be 59% of CLMAX — the maximum lift coefficient. Let me now give you the review points, because they tie everything together. In straight and level flight, lift must balance weight, so at any moment, weight and the lift required are constant. First, to maintain constant lift if density varies because of altitude change, you must change TAS. If altitude increases, density decreases, so TAS must be increased. If altitude decreases, density increases, so TAS must be decreased. Maintaining a constant IAS will compensate for density changes. Second, to maintain constant lift if speed is changed at constant altitude — constant density — you must adjust angle of attack. If speed is increased, angle of attack must be decreased. If speed is doubled, angle of attack must be decreased to make CL one quarter of its previous value. If speed is decreased, angle of attack must be increased — if speed is halved, angle of attack must be increased to make CL four times its previous value. And finally, generally a cruise speed is chosen so the aircraft operates at its optimum angle of attack, which is L/D MAX — the maximum lift-to-drag ratio — at approximately 4 degrees. So the whole picture is: lift must equal weight, and you have two levers to keep that balance — speed and angle of attack. Change one, and you adjust the other according to the square relationship. That’s the heart of this section.