Balanced Field Length Lab (2D)

The problem the decision speed solves

If an engine fails during the take-off roll the crew has two options. Stop: reject the take-off and brake to rest within the runway. Go: continue on the remaining engine, lift off, and clear a 35 ft (10.7 m) screen height before the runway ends. The speed at which the decision is made is V1. A low V1 is easy to stop from but leaves a long, slow one-engine acceleration; a high V1 means a short go but a stop that may not fit. The balanced field length (BFL) is the runway needed at the V1 where the two distances are equal.

Ground-run equation

m dV/dt = n T(V, rho) - D - mu_r (W - L) - W sin(slope)

with thrust T = T0 (rho/rho0)^0.75 (1 - 0.0016 Va) per engine (n = 2 with all engines, 1 after the failure), drag D = 1/2 rho Va^2 S (CD0 + k CL^2), rolling friction mu_r = 0.02, Va = V + headwind. The density rho follows the standard atmosphere for the elevation and temperature deviation, which is why hot and high airports need much longer runways. The stall speed is Vs = sqrt(2 W / (rho S CLmax)); rotation is at VR = 1.12 Vs, the screen height must be reached at about V2 = 1.2 Vs.

Accelerate-go distance

After V1 the aircraft accelerates on one engine (with extra drag from the windmilling engine and rudder) to VR, rotates (lift coefficient increasing at a fixed rate) until the lift equals the weight, then climbs with the flight-path angle relaxing towards (T - D)/W. The distance to the 35 ft screen height is the accelerate-go distance (ATD).

Accelerate-stop distance

All-engine acceleration to V1, then a transition (default 2 s: the first second with one engine's thrust, then idle, brakes and spoilers) and full braking with an effective braking coefficient mu_b (0.45 dry, about 0.25 wet, 0.12 slush):

m dV/dt = -D - mu_b (W - 0.4 L) - W sin(slope)

The sum of the acceleration, transition and braking distances is the accelerate-stop distance (ASD). With constant braking and no drag the stop is simply V^2 / (2 mu_b g).

Numerical checks on this page

(1) The ground run to 60 m/s with drag and rolling friction switched off is compared with the closed form for thrust that falls linearly with speed. (2) A braking run with constant coefficient and no drag is compared with V^2/(2 mu g). (3) The balanced V1 found by bisection on (ATD - ASD) is compared with a brute-force scan for the minimum of max(ATD, ASD).

Honest simplifications

This is a teaching model, not an aircraft flight manual. Fixed coefficients stand in for a real aircraft database: constant CD0 and CLmax for one flap setting, thrust lapse as a simple linear function of speed, no ground effect, no pitch dynamics (the lift coefficient is ramped), a fixed rolling coefficient, no wheel-brake energy limit, reverse thrust not credited, V1 limited to VR, no minimum-control-speed limit (VMCG), no obstacle after the screen height and no climb-gradient requirements. The effective braking coefficient hides anti-skid efficiency, runway texture and contamination. Never use these numbers for real flight planning; use the performance data of the actual aircraft.

• Load Typical twin-jet, sea level and press Set V1 = balanced: the go and stop distances become equal on graph 1.
• Drag V1 down (V1 set too low) and see the go distance grow because the one-engine acceleration lasts longer. Raise it (V1 set too high) and the stop distance runs past the runway end.
• Compare Typical with Heavy, hot and high: the balanced field length grows and graph 3 shows the mass limit for a given runway.
• Switch the decision to GO or STOP: the main aircraft follows your decision and the translucent one flies the other at the same time.
• Set a tailwind or a slippery runway (wet, slush) and see which distance is affected most.