The Runway That Shrank
You are departing a 1 400 m asphalt runway at 1 200 ft elevation on a day when the OAT is 28 °C and the QNH is 1 008 hPa. Your aircraft's take-off distance to 50 ft, from the flight manual, is 780 m at sea level, ISA, MTOW, zero wind, dry runway. The manual gives a 12 % increase per 1 000 ft of pressure altitude and a 10 % increase per 10 °C above ISA. The runway has a 0.8 % upslope in the take-off direction, and the manual's slope correction is 5 % per 1 % upslope. There is a 6 kt tailwind component, and the manual's wind correction is 10 % per 2 kt of tailwind. The surface is dry. Your actual take-off mass is 95 % of MTOW, and the manual gives a 1.5 % reduction in distance per 1 % reduction in mass.
1. Compute the pressure altitude and the ISA temperature deviation, then apply the corrections in a sensible order to estimate the take-off distance to 50 ft.
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Pressure altitude ≈ elevation + 30 ft per hPa below 1 013: (1 013 − 1 008) × 30 = 150 ft, so PA ≈ 1 350 ft. ISA temp at 1 350 ft ≈ 15 − (1.35 × 2) ≈ 12.3 °C; actual 28 °C → deviation ≈ +15.7 °C. Apply corrections multiplicatively: base 780 m. Altitude: +12 % per 1 000 ft × 1.35 = +16.2 % → ×1.162 = 906 m. Temperature: +10 % per 10 °C × 1.57 = +15.7 % → ×1.157 = 1 048 m. Slope: +5 % × 0.8 = +4 % → ×1.04 = 1 090 m. Tailwind: +10 % per 2 kt × 3 = +30 % → ×1.30 = 1 417 m. Mass: −1.5 % per 1 % below MTOW; 5 % below MTOW → −7.5 % → ×0.925 = 1 311 m. Estimated TODA to 50 ft ≈ 1 310 m.
2. Compare this with the available runway and decide whether the departure is legal and sensible. What margins are you relying on?
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Estimated 1 310 m against 1 400 m available leaves about 90 m, roughly 6 %. That is inside the runway but with almost no margin. The calculation itself has uncertainty: the manual corrections are approximations, the tailwind may vary, the surface may be wetter than assumed, and pilot technique affects the figure. A 6 % margin is not a sensible operational buffer. Also note that the 50 ft distance is not the same as the accelerate-stop or the all-engines-operating screen height for a twin; for a single it is the relevant figure, but you still want a clear margin. The legal position depends on the aircraft's POH performance data and any operator minima, but on judgement alone this departure is marginal and should be re-planned.
3. Give three concrete ways to improve the margin without changing the destination.
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(1) Reduce mass: offload fuel or payload to cut the 5 % mass penalty and also improve climb - a 10 % mass reduction is worth roughly 15 % distance. (2) Wait for conditions: cooler part of the day or a wind shift removing the tailwind; the tailwind alone is costing 30 %. (3) Use a different runway or intersection: a runway with headwind or downhill slope, or a longer runway at a nearby airport. Also check the actual QNH and temperature again just before departure, since the 15.7 °C ISA deviation is the second-largest penalty and could improve. Any of these, or a combination, restores a proper margin.