Landing performance

Why it matters

An aircraft that can take off from a runway must also be able to land on it at its landing weight, on a wet day, with a tailwind component, and still stop with margin. Landing overruns are among the most common serious accidents, and certified landing distances carry large safety factors for exactly that reason. Landing analysis decides flap and spoiler design, brake and reverser sizing, and the runway an operator can use.

Key ideas

Phases of landing.

1. Approach from the screen height (15 m / 50 ft) along a steady glide path, typically 3°, at the approach speed V_a ≈ 1.3·V_stall (V_REF for transports).
2. Flare: a curved path that reduces the descent rate before touchdown; the mean flare speed is about 1.23·V_stall and the load factor during the flare is about n = 1.2.
3. Touchdown at V_TD ≈ 1.15·V_stall.
4. Free roll for 1–3 s while spoilers deploy, the nose comes down and brakes are applied.
5. Braking roll to rest. Landing distance = approach + flare + free roll + braking roll.

Stall speed in landing configuration. V_stall = √(2W/(ρ·S·C_L,max)) with full landing flaps (C_L,max typically 2.2–3.0 for transports). Landing weight is less than takeoff weight because fuel has been burnt.

Approach and flare geometry. In the flare the aircraft follows a circular arc of radius R = V_f²/(g·(n − 1)). For an approach angle θ_a, the flare starts at height h_f = R·(1 − cos θ_a) and covers horizontal distance s_f = R·sin θ_a. The straight approach from the screen height covers s_a = (h_screen − h_f)/tan θ_a.

Braking roll equation. After touchdown thrust is at idle or in reverse: (W/g)·dV/dt = −[T_rev + D + μ_r·(W − L)]. μ_r is the effective braking coefficient — about 0.4–0.5 on a dry runway with good anti-skid, 0.2–0.3 wet, and below 0.1 on ice (take values from your data book). Spoilers destroy lift so that L ≈ 0 and the full weight presses on the braked wheels; they also add drag. Writing the deceleration as A + B·V², with

• A = g·(T_rev/W + μ_r)
• B = (g/W)·½ρS·(C_D,g − μ_r·C_L,g) the braking distance from V_TD to rest is s_b = (1/(2B))·ln(1 + B·V_TD²/A). If aerodynamic forces are neglected, B → 0 and s_b = V_TD²/(2·A), i.e. V_TD²/(2·μ_r·g) without reverse thrust.

What matters most. Since V_TD² ∝ W/(ρ·C_L,max), the braking roll grows with weight, falls with density and falls with better flaps. Reverse thrust and spoilers matter most at high speed, where they supplement brakes; wheel brakes dominate at low speed. A headwind reduces the ground speed at touchdown (V_TD − V_w), and with roughly V² dependence even a modest headwind cuts the roll noticeably. Downhill slope, wet or contaminated runways and tailwind all increase the distance.

Approach speed sensitivity. Because distance scales with V², a 10 % excess approach speed lengthens the braking roll by about 21 %. Pilots are trained to fly V_REF accurately for this reason.

Regulatory margins. Certified (demonstrated) landing distances are factored — for transports the required field length is the demonstrated distance divided by 0.6 on a dry runway, with a further increase for wet runways (see the applicable regulations).

Formulas

V_stall = √(2W / (ρ·S·C_L,max)), V_a ≈ 1.3·V_stall, V_f ≈ 1.23·V_stall, V_TD ≈ 1.15·V_stall

R = V_f² / (g·(n − 1)), h_f = R·(1 − cos θ_a), s_f = R·sin θ_a

s_a = (h_screen − h_f) / tan θ_a

s_fr = N·V_TD (free roll for N seconds)

(W/g)·dV/dt = −[T_rev + D + μ_r·(W − L)]

s_b = (1/(2B))·ln(1 + B·V_TD²/A), A = g·(T_rev/W + μ_r), B = (g/W)·½·ρ·S·(C_D,g − μ_r·C_L,g)

s_b = V_TD² / (2·μ_r·g) (aerodynamic forces and reverse thrust neglected)

Symbols: W landing weight (N); ρ density (kg/m³); S wing area (m²); C_L,max landing-configuration maximum lift coefficient; V speeds (m/s); n flare load factor; R flare radius (m); θ_a approach angle; h_screen screen height (15 m); h_f flare height (m); s_a, s_f, s_fr, s_b approach, flare, free-roll and braking distances (m); N free-roll time (s); T_rev reverse thrust (N); μ_r effective braking coefficient; C_L,g, C_D,g lift and drag coefficients during the roll; g = 9.81 m/s².

Worked examples

Example 1 (standard): braking roll with brakes only. Given: landing weight W = 500 kN, S = 120 m², C_L,max = 2.4, sea level (ρ = 1.225 kg/m³), V_TD = 1.15·V_stall, μ_r = 0.4. Neglect aerodynamic forces and reverse thrust; free roll 2 s.

1. V_stall = √(2W/(ρSC_L,max)) = √(1 000 000/352.8) = 53.24 m/s; V_TD = 1.15 × 53.24 = 61.23 m/s.
2. s_b = V_TD²/(2μ_r g) = 3748.6/(2 × 0.4 × 9.81) = 477.6 m.
3. Free roll = 2 × 61.23 = 122.5 m, so ground distance = 600 m.

Example 2 (GATE level): full landing distance with spoilers and reversers. Given: same aircraft; approach angle 3°, flare speed 1.23·V_stall with n = 1.2, screen height 15 m. In the braking roll: C_L,g = 0.1 (spoilers out), C_D,g = 0.12, μ_r = 0.3, reverse thrust 50 kN constant; free roll 2 s.

1. Flare: V_f = 1.23 × 53.24 = 65.48 m/s; R = V_f²/(g(n − 1)) = 4288.3/(9.81 × 0.2) = 2186 m.
2. h_f = R(1 − cos 3°) = 2186 × 0.00137 = 2.99 m; s_f = R·sin 3° = 114.4 m.
3. Approach: s_a = (15 − 2.99)/tan 3° = 12.01/0.05241 = 229.1 m.
4. Braking: A = g(T_rev/W + μ_r) = 9.81 × (0.1 + 0.3) = 3.924 m/s²; B = (g/W)·½ρS(C_D,g − μ_rC_L,g) = (9.81/500 000) × 73.5 × 0.09 = 1.298 × 10⁻⁴ m⁻¹.
5. s_b = (1/(2B))·ln(1 + B·V_TD²/A) = 3852.6 × ln(1 + 0.12398) = 3852.6 × 0.11688 = 450.3 m.
6. Total = 229.1 + 114.4 + 122.5 + 450.3 = 916 m from the screen to rest.

Common mistakes

• Using takeoff weight or cruise-configuration C_L,max for landing.
• Forgetting the free-roll distance, which can be 100–150 m for a transport.
• Leaving L = W in the friction term after spoilers deploy, or forgetting that lift reduces braking effectiveness when spoilers are not used.
• Applying the dry-runway braking coefficient to a wet runway.
• Treating a 10 % speed excess as a 10 % distance increase; it is about 21 %.

For GATE AE

Expect braking-roll numericals with s = V²/(2a) and the A + BV² integration, approach and flare geometry with the circular-arc model, effects of weight, density, headwind and approach-speed error on landing distance, and conceptual questions on spoilers, reverse thrust and braking coefficients. Practise splitting the landing distance into its segments and checking which one dominates.

Quick check

1. How does braking roll scale with touchdown speed?
2. Why are spoilers deployed immediately after touchdown?
3. What is the flare radius for V_f = 60 m/s and n = 1.2?
4. Touchdown speed 60 m/s, headwind 6 m/s: by what factor does the simple braking roll fall?

Answers: 1. As V_TD². 2. To dump lift so the wheels carry the weight and brakes work, and to add drag. 3. 3600/(9.81 × 0.2) = 1835 m. 4. (54/60)² = 0.81.