Unit 5: Aircraft Performance in Accelerated Flight

I. Performance Framework

Accelerated-flight performance follows Newton’s second law and the work–energy principle: an aircraft accelerates whenever thrust, aerodynamic forces, weight, and ground reactions do not balance. Take-off and landing calculations divide the motion into ground and airborne phases, while climb calculations allocate excess power between altitude gain and acceleration.

• Governing equations: Motion along the flight path is determined by the resultant tangential force, while normal force determines path curvature.
  
  TEXT

    ΣF_t = m(dV/dt)
    ΣF_n = mV²/R

  
  Here, ΣF_t and ΣF_n are tangential and normal resultant forces, m is aircraft mass, V is true speed, t is time, and R is path radius.
• Standard assumptions: Introductory calculations commonly assume constant mass, no wind, a level runway, fixed aerodynamic configuration, steady atmospheric conditions, and representative average thrust or acceleration.
• Reference speeds: Stall, lift-off, decision, safety, approach, and touchdown speeds must correspond to the applicable aircraft mass, flap setting, atmospheric condition, and regulatory procedure.
• Environmental corrections: High density altitude, high temperature, tailwind, uphill slope, wet surfaces, and contamination generally increase required field length.
• Safety convention: Actual approved performance includes prescribed margins, engine-failure cases, runway slope, wind limits, and manufacturer data; simplified equations reveal physical trends but do not replace flight-manual charts.

II. Take-Off — Ground Acceleration, Transition, and Obstacle Clearance

A. Take-off performance

Take-off performance measures the distance and time needed to accelerate from rest, lift off, and reach a specified obstacle height safely.

• Phases: The take-off comprises ground roll, rotation, lift-off, transition to climb, and obstacle clearance—commonly evaluated at a screen height such as 35 ft for relevant certification cases.
• Forces during ground roll: The accelerating force is thrust minus drag and rolling resistance.
  
  TEXT

    F_a = T - D - μ_r(W - L)

  
  Here, F_a is accelerating force, T is thrust, D is drag, μ_r is rolling-friction coefficient, W is weight, and L is lift.
• Principal variables: Greater weight raises stall and lift-off speeds; lower air density reduces aerodynamic lift and engine performance; headwind reduces ground speed for a given airspeed.
• Configuration trade-off: Take-off flap increases lift and may shorten ground roll, but excessive flap increases drag and can reduce climb gradient after lift-off.

B. Calculation of ground roll

Ground roll is calculated by integrating the aircraft’s acceleration from rest to lift-off speed.

• Variable-acceleration method: Using dV/dt = V(dV/ds) gives:
  
  TEXT

    s_g = ∫[0 to V_LOF] mV dV / [T - D - μ_r(W - L)]

  
  Here, s_g is ground-roll distance, s is distance, and V_LOF is lift-off speed; the other symbols retain their earlier meanings.
• Aerodynamic forces: Lift and drag vary approximately with the square of airspeed.
  
  TEXT

    L = ½ρV²SC_L
    D = ½ρV²SC_D

  
  Here, ρ is air density, S is wing area, and C_L and C_D are lift and drag coefficients.
• Constant-acceleration estimate: If an average acceleration a_avg is available:
  
  TEXT

    s_g ≈ V_LOF² / (2a_avg)

  
  This approximation is useful for preliminary estimates but masks changes in thrust, drag, lift, and rolling resistance.
• Wind distinction: Aerodynamic forces depend on airspeed, whereas runway distance depends on ground speed; a headwind therefore reduces ground roll without changing the required lift-off airspeed substantially.

C. Calculation of airborne distance to clear an obstacle

Airborne distance extends from lift-off to the point at which the flight path reaches the specified obstacle height.

• Transition model: A simplified circular pull-up from a horizontal path to climb angle γ has approximate radius:
  
  TEXT

    R ≈ V² / [g(n - 1)]

  
  Here, g is gravitational acceleration and n = L/W is load factor near the horizontal portion of the transition.
• Transition geometry:
  
  TEXT

    x_tr = R sinγ
    h_tr = R(1 - cosγ)

  
  Here, x_tr is horizontal transition distance and h_tr is height gained during transition.
• Straight-climb portion:
  
  TEXT

    x_cl = (h_obs - h_tr) / tanγ
    s_air = x_tr + x_cl

  
  Here, h_obs is obstacle height, x_cl is horizontal climb distance, and s_air is total airborne horizontal distance.
• Limitation: Real calculations account for speed changes, rotation dynamics, landing-gear retraction, engine condition, and variation of climb angle rather than assuming constant V and γ.

D. Balanced field length

Balanced field length is the runway length for which accelerate-stop distance equals accelerate-go distance at the selected decision speed.

• Accelerate-stop case: The aircraft accelerates toward V₁, rejects the take-off after the critical failure, and stops using aerodynamic drag, wheel braking, spoilers, and permitted reverse thrust.
• Accelerate-go case: After the critical engine failure, the aircraft continues through V₁, lifts off, and reaches the prescribed screen height with reduced thrust.
• Balance condition:
  
  TEXT

    ASD(V₁) = AGD(V₁) = BFL

  
  Here, ASD is accelerate-stop distance, AGD is accelerate-go distance, and BFL is balanced field length.
• Effect of decision speed:
  1. Higher V₁: Increases accelerate-stop distance because more kinetic energy must be dissipated.
  2. Lower V₁: Increases accelerate-go distance because the aircraft spends longer accelerating with one engine inoperative.
• Operational qualification: Stopways, clearways, runway condition, brake-energy limits, and regulatory definitions can produce declared-distance requirements that differ from the idealized balanced length.

III. Landing — Descent, Flare, and Deceleration

A. Landing performance

Landing performance determines the horizontal distance from a specified screen height to a complete stop.

• Phases: Total landing distance contains approach, flare, touchdown or free-roll transition, and braking ground roll.
• Reference speed: Approach speed is normally scheduled as a certified multiple of stall speed with corrections for wind, gusts, mass, and configuration.
• Energy requirement: Touchdown kinetic energy is:
  
  TEXT

    KE = ½mV_TD²

  
  Here, KE is kinetic energy and V_TD is touchdown ground speed; because energy varies with speed squared, excess touchdown speed significantly increases stopping distance.
• Influencing conditions: Tailwind, downhill slope, delayed braking, worn brakes, standing water, and poor tire–runway friction increase landing distance.

B. Calculation of approach distance

Approach distance is the horizontal distance travelled along the final descent from screen height to flare initiation.

• Geometric model: For a straight approach at constant descent angle:
  
  TEXT

    s_app = (h_s - h_f) / tan|γ_a|

  
  Here, s_app is approach horizontal distance, h_s is screen height, h_f is flare-initiation height, and γ_a is the negative approach angle.
• Example: With h_s = 15 m, h_f = 5 m, and |γ_a| = 3°, the distance is approximately 10/tan3° = 191 m.
• Wind effect: The geometric distance is unchanged for a fixed path, but headwind reduces elapsed time and ground speed; operational landing-distance treatment follows approved wind-credit rules.

C. Calculation of flare distance

Flare distance is the horizontal distance required to curve from the descending approach path toward a nearly horizontal touchdown path.

• Circular-arc approximation: Near horizontal flight, the flare radius may be estimated by:
  
  TEXT

    R_f ≈ V_f² / [g(n_f - 1)]

  
  Here, R_f is flare radius, V_f is flare speed, and n_f is average flare load factor.
• Flare geometry:
  
  TEXT

    s_f = R_f sin|γ_a|
    Δh_f = R_f(1 - cos|γ_a|)

  
  Here, s_f is flare horizontal distance and Δh_f is height lost during flare.
• Technique effects: A gentle flare uses a large radius and more distance; an abrupt flare uses a smaller radius but may cause excessive load factor, ballooning, or unstable touchdown.
• Speed effect: Since R_f varies with V_f², excessive approach speed lengthens the flare and promotes floating.

D. Calculation of landing ground roll

Landing ground roll is the runway distance from touchdown until the aircraft stops or reaches a specified taxi speed.

• Retarding forces: Braking friction, aerodynamic drag, reverse thrust, and runway slope determine deceleration; spoilers reduce lift and place more weight on the wheels.
• Integration method:
  
  TEXT

    s_LG = ∫[0 to V_TD] mV dV / [D + μ_b(W - L) - T]

  
  Here, s_LG is landing ground roll and μ_b is effective braking-friction coefficient; T is positive forward thrust and becomes negative when reverse thrust is represented.
• Constant-deceleration estimate:
  
  TEXT

    s_LG ≈ V_TD² / (2a_d)

  
  Here, a_d is average deceleration magnitude.
• Surface limitation: On wet or contaminated runways, available tire friction falls and hydroplaning may occur, so dry-runway friction assumptions are unsafe.

IV. Near-Surface Aerodynamics

A. Ground effects

Ground effect is the alteration of wing aerodynamics caused by proximity to the runway, principally through reduced downwash and induced drag.

• Physical mechanism: The surface restricts trailing-vortex development, reducing induced angle of attack and induced drag for a given lift.
• Height parameter: Strength depends mainly on wing height relative to span, h/b, where h is wing height above the surface and b is wingspan; the effect becomes strongest close to the ground.
• Take-off consequence: An aircraft may lift off at insufficient climb speed and remain temporarily supported in ground effect, yet be unable to climb away when induced drag increases outside it.
• Landing consequence: Reduced induced drag decreases deceleration during flare, causing float and a longer touchdown distance, especially after a fast approach.
• Model limitation: Ground effect depends on wing geometry, flap setting, aircraft attitude, and surface proximity, so aircraft-specific data are preferable to a universal correction factor.

V. Accelerated Climbing Flight

A. Acceleration in climb

Acceleration in climb occurs when excess thrust or power is divided between increasing altitude and increasing speed.

• Tangential force equation:
  
  TEXT

    T - D - W sinγ = m(dV/dt)

  
  Here, γ is climb angle and dV/dt is true-speed acceleration; positive acceleration requires thrust to exceed drag plus the component of weight along the path.
• Specific excess power:
  
  TEXT

    P_s = (TV - DV)/W
    P_s = dh/dt + (V/g)(dV/dt)

  
  Here, P_s is specific excess power, h is altitude, and dh/dt is rate of climb.
• Energy allocation:
  1. Steady-speed climb: If dV/dt = 0, all specific excess power produces altitude gain, so P_s = dh/dt.
  2. Accelerating climb: If speed increases, part of P_s raises kinetic energy, reducing the climb rate available at the same thrust.
• Operational interpretation: Lowering the nose generally trades climb rate for acceleration; raising it trades acceleration for altitude until stall or thrust limits intervene.
• Atmospheric distinction: True airspeed may increase during a constant indicated-airspeed climb as density falls, so energy calculations must use true speed rather than indicated speed.