The lift equation

The Lift Equation

The lift equation is the central formula for the quantitative calculation of aerodynamic lift.

The equation

L = ½ · ρ · V² · S · CL

with:

• L = lift in newtons (N)
• ρ = air density (kg/m³)
• V = true freestream velocity / TAS (m/s)
• S = wing area / reference area (m²)
• CL = dimensionless lift coefficient

Source: standard form in every aerodynamics textbook (Anderson Ch. 1; FAA-H-8083-25B Ch. 5).

Dynamic pressure q

The term ½ρV² = q (dynamic pressure) can be factored:

L = q · S · CL

So lift = dynamic pressure × area × coefficient. q links the flow energy to the mechanical effect on the wing.

CL — the lift coefficient

CL is a dimensionless number between 0 and ~2.0 (typically ~1.0 in cruise, ~1.8 with full flaps).

CL depends on:

• Angle of attack α (see next lesson "Lift coefficient vs angle of attack"): linearly rising up to near stall.
• Airfoil shape (camber, thickness): more camber → higher CL at the same α.
• Reynolds number (small influence at PPL Re).
• Mach number (relevant only > M 0.5; negligible at PPL).
• Flap setting (see lesson "Flaps").

Application in steady flight

Vertical equilibrium: L = W (lift = weight).

Solving for stall speed Vs:

Vs = √(2·W / (ρ·S·CL_max))

at given CL_max (max attainable CL). This is the minimum speed at which L = W can be maintained.

Worked example

Cessna 172, data:

• W = 1100 kg × 9.81 = 10,791 N (weight at MTOM)
• S = 16.2 m² (wing area)
• ρ = 1.225 kg/m³ (ISA MSL)
• CL_max = 1.6 (with full flaps, source: Cessna POH)

Vs (stall speed with full flaps):

Vs = √(2 × 10,791 / (1.225 × 16.2 × 1.6)) = √(21,582 / 31.75) = √679.7 ≈ 26 m/s ≈ 51 KIAS

Matches roughly the POH value for Vs0 (stall in landing configuration).

How does lift change with altitude?

At same TAS and higher altitude, ρ decreases → L decreases. To maintain L = W, either:

• Increase V (TAS): at altitude one has higher TAS at same IAS.
• Increase α: CL rises.
• Set flaps: CL rises (see lesson "Flaps").

How does lift change with weight?

At higher weight (e.g. fully loaded C172) at the same speed CL must be higher → more α → closer to stall. Hence Vs grows with √W: doubling weight → Vs about √2 ≈ 1.4× higher.

Limits of the formula

• Inviscid: ignores drag. Sufficient for L.
• Incompressible: valid for M < 0.3.
• Steady: no accelerations.
• Valid only below stall (α < α_stall).