Stall Speed⚠ unverified

Understanding the Parameters

• Why a wing stalls — lift rises with angle of attack only up to a point. Beyond the critical angle (typically $\sim15^\circ$), the airflow separates from the upper surface, lift collapses, and drag spikes. $C_{L_{max}}$ is the lift coefficient at that critical angle — the most lift the wing can make. Below $V_s$, even pulling to the critical angle does not produce enough lift for level flight.

• Weight $W$ — heavier aircraft stall faster ($V_s \propto \sqrt{W}$). A fully loaded aircraft has a higher stall speed than an empty one, which is why landing speeds depend on weight and why aircraft dump or burn fuel before an emergency landing.

• Air density $\rho$ — thinner air (high altitude, hot day) raises the true stall speed ($V_s \propto 1/\sqrt{\rho}$). Crucially, though, the indicated stall speed (what the airspeed indicator shows, which senses Dynamic Pressure $\tfrac12\rho V^2$) is independent of altitude — the wing stalls at the same dynamic pressure everywhere, so a pilot sees the same number. This is why airspeed indicators, not true-airspeed, guard the stall.

• Wing area $S$ and $C_{L_{max}}$ — both in the denominator, so a bigger wing or a higher max lift coefficient lowers stall speed. This is the entire purpose of flaps and slats: deploying them raises $C_{L_{max}}$ (and $S$ on Fowler flaps), dropping stall speed so the aircraft can approach and land slower.

• The $1.3\,V_s$ approach speed — the standard final-approach reference speed ($V_{ref}$) is $1.3\,V_s$, a 30 % margin above the stall. Rotation and lift-off on takeoff sit near $1.1$–$1.2\,V_s$. Stall speed is thus the yardstick for the whole low-speed regime.

Derivation (Approaching a Proof)

In steady, level flight lift exactly balances weight. The lift is given by the standard lift equation (see Lift Force):

$$L = \tfrac12\,\rho\,V^2\,S\,C_L = W.$$

At any given speed the aircraft flies at whatever $C_L$ this balance demands. As the aircraft slows, $V^2$ falls, so the required $C_L$ must rise to keep $L = W$. But $C_L$ cannot exceed $C_{L_{max}}$ — the wing's physical limit. The stall speed is the speed at which the required $C_L$ has climbed all the way to $C_{L_{max}}$; any slower and level flight is impossible. Set $C_L = C_{L_{max}}$ and solve for $V$:

$$\tfrac12\,\rho\,V_s^2\,S\,C_{L_{max}} = W \quad\Longrightarrow\quad V_s^2 = \frac{2W}{\rho\,S\,C_{L_{max}}} \quad\Longrightarrow\quad V_s = \sqrt{\frac{2W}{\rho\,S\,C_{L_{max}}}}. \qquad\blacksquare$$

The accelerated (manoeuvring) stall. In a level turn or pull-up the wing must support not $W$ but $nW$, where $n$ is the load factor (Turn Radius). Replacing $W$ with $nW$ gives

$$V_{s,\,n} = \sqrt{\frac{2nW}{\rho S C_{L_{max}}}} = V_s\sqrt{n}.$$

So a 2-g turn raises the stall speed by $\sqrt2 \approx 1.41$; a 4-g turn doubles it. This is why aggressive manoeuvring can stall a wing at speeds far above the placarded 1-g stall — the origin of many loss-of-control accidents.

Dimensional check. $$\frac{2W}{\rho\,S\,C_{L_{max}}} = \frac{\text{N}}{(\text{kg}/\text{m}^3)(\text{m}^2)(\text{–})} = \frac{\text{kg}\cdot\text{m}/\text{s}^2}{\text{kg}/\text{m}} = \frac{\text{m}^2}{\text{s}^2},$$ and the square root gives $\text{m}/\text{s}$ ✓.

History and Development

• The lift equation's lineage. The $\tfrac12\rho V^2 S C_L$ form was nailed down by the Wright brothers' 1901 wind-tunnel work (correcting the era's faulty Smeaton coefficient) and formalised through Prandtl-era aerodynamics. Stall speed is its most direct practical consequence.

• High-lift devices. The drive to lower stall (and thus landing) speed produced a century of high-lift engineering: Handley Page and Gustav Lachmann's leading-edge slat (patented independently ~1918–1920), the Fowler flap (1927), and the complex multi-element flap systems of modern airliners, which can more than double $C_{L_{max}}$. Every one exists to move $V_s$ down.

• A regulatory cornerstone. Certification rules (FAR/CS-23, -25) define reference speeds — $V_{ref}$, $V_{2}$, manoeuvring speed $V_A$ — as fixed multiples of stall speed, and the manoeuvring speed is specifically $V_A = V_s\sqrt{n_{limit}}$ from the accelerated-stall relation above, the speed below which full control deflection cannot overstress the airframe because the wing stalls first.

Related Concepts: Lift Force, Dynamic Pressure, Landing Distance, Best Climb Speed, Turn Radius, ISA Density, Lift-to-Drag Ratio, Lift Curve Slope

Notes: 1-g level-flight stall; accelerated stall is $V_s\sqrt{n}$ in a manoeuvre. Indicated stall speed is altitude-independent (wing stalls at fixed dynamic pressure) even though true $V_s\propto1/\sqrt\rho$. $C_{L_{max}}$ depends on flap/slat configuration — the reason high-lift devices exist. Approach speed $V_{ref}=1.3\,V_s$; manoeuvring speed $V_A=V_s\sqrt{n_{limit}}$. Defaults ⇒ $V_s\approx46.7\ \text{m/s}$.