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Aerospace Engineering DISCUSSION
I am studying the lift equation L = ½ × ρ × V² × S × CL and trying to apply it to a light aircraft: mass 1000 kg, wing area 16 m², maximum lift coefficient 1.5 without flaps.
If lift goes with the square of airspeed, why does an aircraft not climb away every time it speeds up? And how do I get the stall speed from these numbers, including the effect of altitude and of a banked turn?
In steady level flight lift equals weight, so lift is fixed and the equation tells you what CL the wing must produce at each speed: CL = 2W / (ρ × V² × S). Fly twice as fast and the required CL falls to a quarter, which the pilot gets by reducing the angle of attack. The V² relationship describes lift at a fixed angle of attack; in flight the angle of attack is the variable that is adjusted to keep lift equal to weight.
If speed increases and the angle of attack is held, lift does exceed weight and the aircraft accelerates upward.
Stall speed is the speed at which the required CL reaches CLmax: Vs = sqrt(2W / (ρ × S × CLmax)). With W = 1000 × 9.81 = 9810 N, sea-level density 1.225 kg/m³, S = 16 m² and CLmax = 1.5: Vs = sqrt(19620 / 29.4) = sqrt(667) = 25.8 m/s, about 50 knots.
Notice that weight and wing area appear only as W/S, the wing loading (613 N/m² here). Stall speed goes with the square root of wing loading, which is why that one ratio says so much about how an aircraft handles at low speed.
Altitude: at 3000 m the standard density is about 0.909 kg/m³, so the true airspeed at the stall rises by sqrt(1.225 / 0.909) = 1.16, to about 30 m/s. The indicated airspeed at the stall hardly changes, because the airspeed indicator measures dynamic pressure, ½ρV², and the stall happens at the same dynamic pressure. That is why pilots use one set of indicated stall speeds at all altitudes.
Turns and pull-ups: the wing has to produce n times the weight, where n is the load factor, so stall speed rises by sqrt(n). In a level turn n = 1 / cos(bank angle). At 60° of bank n = 2 and the stall speed becomes 25.8 × 1.41 = 36.5 m/s.
The more general statement is that a wing stalls at a critical angle of attack, not at a speed. The speed quoted in a handbook is only the speed at which that angle is reached in 1 g flight at a stated weight.
The other levers in the formula are CLmax and weight. Flaps raise CLmax; if extending them takes it from 1.5 to 2.0, the stall speed drops by sqrt(1.5 / 2.0) = 0.87, to about 22.4 m/s. A lighter aircraft stalls slower in proportion to the square root of its weight: at 800 kg the clean stall speed is 25.8 × sqrt(0.8) = 23.1 m/s.
CLmax itself is not a constant of the aerofoil alone. It depends on Reynolds number, surface contamination such as ice or insects, and wing planform, so handbook values come from flight test rather than from the equation.
