Hot-day take-off from a highland strip
On a February afternoon you plan to depart from a 1,300 m tarred runway at 5,577 ft elevation in a Cessna 172N. The QNH is 1024 hPa, OAT 32 °C, wind calm, and the runway slopes up by 1.5% in the take-off direction. The aircraft's basic empty mass is 660 kg, MTOW 1,043 kg. On board are three adults at 85 kg each, 110 L of fuel (density 0.72 kg/L), and 20 kg of baggage. The POH gives take-off distance over a 50 ft obstacle as 2,350 ft at 8,000 ft density altitude and 2,900 ft at 9,000 ft density altitude, ISA. The strip has a 15 m high tree line 400 m beyond the departure end. The pilot has not used a density-altitude chart for this aircraft before and proposes to use the '1,300 m is long enough' rule.
1. Calculate pressure altitude and density altitude, showing each step.
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Pressure altitude = aerodrome elevation + 30 ft per hPa by which QNH is above 1013. With QNH 1024: PA = 5,577 + (1013 - 1024) x 30 = 5,577 - 330 = 5,247 ft. ISA temperature at that pressure altitude is about 15 - 1.98 x 5.247 = 4.6 °C. Density altitude = PA + 120 x (OAT - ISA) = 5,247 + 120 x (32 - 4.6) = 5,247 + 3,288 = about 8,535 ft. The aerodrome is therefore performing at roughly 8,500 ft density altitude, not 5,577 ft. The calculation must use the actual QNH and temperature, not the elevation alone.
2. Estimate the take-off distance over 50 ft and decide whether the runway and tree line are acceptable. Show the reasoning.
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Interpolate between the two POH values: at 8,535 ft, distance = 2,350 + 0.535 x (2,900 - 2,350) = 2,350 + 294 = 2,644 ft, or about 806 m. The runway is 1,300 m, so the nominal runway margin is about 494 m. That is not the whole problem. The POH figure is for a level, dry, hard runway, zero wind and a 50 ft obstacle at the end of the take-off distance. The 15 m trees are only 400 m from the departure end, inside the 806 m distance, so the aircraft will be well below 50 ft when it reaches them. The 1.5% upslope further reduces acceleration and climb. At 8,500 ft density altitude the climb gradient is also reduced. The '1,300 m is long enough' rule is unsafe. Reduce mass by carrying less fuel or fewer passengers/bags, wait for cooler air, use a runway without upslope or obstacles, or cancel. Clear the trees using the POH climb-gradient data, not runway length alone.
3. Calculate the take-off mass and state what else must be checked before accepting the POH figures.
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Mass = 660 kg empty + 3 x 85 kg adults + 110 L x 0.72 kg/L fuel + 20 kg baggage = 660 + 255 + 79.2 + 20 = 1,014.2 kg. MTOW is 1,043 kg, so the mass margin is only 28.8 kg. That margin is small: extra baggage, a heavier passenger or more fuel would exceed MTOW. The centre of gravity must also be checked in the envelope; an aft CG increases rotation and stall risk. Before using the POH figures, confirm the actual mass, CG, runway surface and slope, wind, obstacle height and distance, and use the correct POH chart for the aircraft. The performance limitation here is density altitude and obstacle clearance, not mass alone.