Landing Distance⚠ unverified

Understanding the Parameters

• The two-step model — first find how fast the aircraft is moving at touchdown ($V_{td}$, from the lift balance), then find how far it rolls stopping from that speed under braking friction ($s$). The touchdown speed feeds the stopping distance as $V_{td}^2$, so landing distance scales with the square of touchdown speed — the dominant sensitivity.

• Why landing distance $\propto W/(\rho S C_{L_{max}})$ — since $V_{td}^2 = 2W/(\rho S C_{L_{max}})$, the stopping distance is $s = W/(\mu g \rho S C_{L_{max}})$. Heavier or hot-and-high (low $\rho$) means faster touchdown and a longer roll; more wing area, more flap ($C_{L_{max}}$), or better braking ($\mu$) shortens it.

• Braking friction $\mu$ — the effective coefficient between tyres and runway under braking. Dry runway with good brakes/anti-skid gives $\mu \sim 0.3$–$0.5$; wet drops it to $\sim 0.2$; ice or hydroplaning can fall below $0.05$, lengthening the roll several-fold. This single number carries all the runway-condition risk — and the model's $\mu = 1$ default is unrealistically high (would be a very short stop).

• What the model leaves out — real landing distance is longer because: touchdown is above stall ($\sim1.15\,V_s$); there is a flare and free-roll before brakes bite; and, working the other way, aerodynamic drag, spoilers, and reverse thrust help decelerate (especially early, at high speed). The energy model here keeps only friction. Certified landing distances add large factors (typically $\times1.67$) on top of even a careful physical estimate.

Derivation (Approaching a Proof)

Step 1 — touchdown speed. At touchdown the aircraft is at the edge of flying, so lift still balances weight at maximum lift coefficient. This is the Stall Speed argument (see Lift Force):

$$\tfrac12\rho V_{td}^2 S C_{L_{max}} = W \quad\Longrightarrow\quad V_{td} = \sqrt{\frac{2W}{\rho S C_{L_{max}}}}.$$

Step 2 — stopping distance from energy. Once rolling, the brakes apply a friction force $F = \mu N$. In the simplest model the normal force is the full weight, $N = W = mg$, so the deceleration is constant:

$$F = \mu m g = m a \quad\Longrightarrow\quad a = \mu g.$$

Now equate the kinetic energy at touchdown to the work done by friction over the roll distance $s$ (work–energy theorem):

$$\tfrac12 m V_{td}^2 = F\,s = \mu m g\,s \quad\Longrightarrow\quad s = \frac{V_{td}^2}{2\,\mu\,g}. \qquad\blacksquare$$

Equivalently, from kinematics $V^2 = V_{td}^2 - 2 a s$ set to zero at $V = 0$ gives the same result. The mass cancels, so the braking distance depends only on touchdown speed and friction — but touchdown speed itself carries the weight, so heavier aircraft still stop longer.

Substituting Step 1 into Step 2 gives the combined form the calculator returns:

$$s = \frac{1}{2\mu g}\cdot\frac{2W}{\rho S C_{L_{max}}} = \frac{W}{\mu g \rho S C_{L_{max}}}.$$

Why real distance is longer. A more complete model integrates $m\,dV/dt = -\mu(W - L) - D$ through the roll, where the lift $L$ (still present until the aircraft slows) reduces the normal force and thus braking early on, while aerodynamic drag $D$ helps. The constant-$\mu g$ model ignores both, and assumes braking from stall speed with no flare — which is why certification applies large safety factors.

Dimensional check. $$\frac{V_{td}^2}{2\mu g} = \frac{(\text{m}/\text{s})^2}{(\text{–})(\text{m}/\text{s}^2)} = \frac{\text{m}^2/\text{s}^2}{\text{m}/\text{s}^2} = \text{m}. \checkmark$$ A distance, confirming the output is a length — not the speed $V_{td}$ the registry symbol suggests.

History and Development

• Energy methods for ground roll. Treating takeoff and landing as work–energy problems is classical aircraft-performance practice (Perkins & Hage, Anderson); the constant-deceleration braking model is the first approximation taught before the full integrated ground-roll equation.

• Braking technology and $\mu$. The whole history of landing safety is a fight to keep $\mu$ high and predictable: anti-skid/anti-lock braking (developed for aircraft from the 1940s–50s, e.g. Dunlop Maxaret), spoilers/lift-dumpers to plant weight on the wheels, thrust reversers, and drag chutes. Runway grooving and friction-measurement standards address the wet/contaminated-runway collapse of $\mu$.

• Certified vs physical distance. Regulators require large margins over any physics estimate: the unfactored landing distance is multiplied (commonly by $1.67$ for dry, more for wet/contaminated) to set the required landing distance, acknowledging pilot technique, approach-speed scatter, and degraded braking — exactly the effects this idealised model omits.

Related Concepts: Stall Speed, Lift Force, Dynamic Pressure, ISA Density, Lift-to-Drag Ratio, Drag Force

Notes: Registry latex/symbol show the intermediate $V_{td}$; the output is the stopping distance $s = V_{td}^2/(2\mu g)$ in metres. Crude model: constant braking friction only — no aero drag, reverse thrust, flare, or free roll, and touchdown taken exactly at stall (real $\sim1.15\,V_s$). $s \propto W/(\rho S C_{L_{max}}\mu)$. Wet/icy runways collapse $\mu$; certification adds $\sim\times1.67$. Defaults all $1.0$ ⇒ $s\approx0.102\ \text{m}$.