Grass Strip at 1,050 m Elevation, 31 °C and a 900 m Runway
You are planning a morning departure from a grass aerodrome at 1,050 m elevation in a Cessna 172, maximum take-off mass 1,100 kg, actual mass 1,090 kg. The runway is 900 m long with a 1% upslope in the take-off direction and no displaced threshold; the 15 m screen is at the far end, with trees 3 m beyond it. QNH is 1005 hPa, the outside air temperature is already 31 °C, and the wind is calm. The flight manual, for a paved, level, sea-level runway in ISA conditions, gives a take-off distance to 15 m of 460 m and a ground roll of 260 m. There is no other aerodrome within 150 km, and the next scheduled ATC opening is at 09:00 local. The aircraft has no oxygen and the planned route crosses a ridge at 1,600 m.
1. Compute the pressure altitude, the ISA deviation and the density altitude.
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Pressure altitude: QNH 1005 hPa is 8 hPa below 1013, roughly 8.3 m per hPa near the surface → 66 m. PA ≈ 1,050 + 66 = 1,116 m (about 3,660 ft). ISA temperature at that level = 15 − 1.98 × 1.116 ≈ 12.8 °C. With OAT 31 °C the ISA deviation is about +18 °C. Density altitude ≈ PA + 120 ft per °C of deviation = 1,116 m + (18 × 120 ft = 2,160 ft ≈ 658 m) ≈ 1,774 m, roughly 5,820 ft.
That is the figure to enter on the performance chart: the wing and propeller “see” an aerodrome more than 700 m higher than the one on the map. The engine loses about 3% of its power per 1,000 ft of density altitude - here around 17% - which also affects the climb gradient over the trees, not just the ground roll.
2. Using the flight-manual figures and reasonable corrections, decide whether the take-off is acceptable.
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The 460 m to 15 m applies to a paved, level, sea-level, ISA runway. Density altitude 5,820 ft is about 5.8 thousand feet above sea level; at roughly 10% per 1,000 ft the distance becomes 460 × 1.58 ≈ 727 m. Grass adds roughly 15–20% → about 850 m. A 1% upslope adds roughly 5–10% → around 900 m, against a 900 m strip with trees 3 m beyond the screen and no stopway. Even allowing for rules of thumb over-predicting slightly, the margin is zero or negative, so the take-off is not acceptable at 1,090 kg.
The answer is not to “be careful” but to change the variables: wait for cooler air, offload to two occupants and less fuel, use a longer paved runway if one is in range, or read the actual POH chart for the type instead of a rule of thumb. If the only option is to go as loaded, the correct decision is to stay on the ground.
3. The wind shifts to 210°/10 kt, giving a 10 kt tailwind component on the upslope runway. Reassess.
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A tailwind is the most expensive correction of all: roughly 10% more take-off distance per 2 kt, so 10 kt adds about 50% to the ground roll and to the distance to 15 m. Applied to the already-corrected 900 m, that is around 1,350 m, which no longer fits the strip under any interpretation. It also erodes the initial climb gradient over the trees, because both the airspeed at the screen and the distance covered on the ground deteriorate.
The action is to change to a runway with a headwind or crosswind component, or to cancel. A 10 kt tailwind on a 900 m grass strip at 5,800 ft density altitude is one of the classic combinations behind take-off accidents; the aircraft is not at fault, and neither is the wind - the decision is. Take the tailwind component from the reported surface wind, not from winds aloft, and re-check it during taxi, because it can change between the hold point and the roll.