Density altitude on the plateau: a ratio is not a performance calculation
You are briefing a dual navigation leg out of a plateau aerodrome: elevation 4,300 ft, QNH 1010, OAT 34 °C, runway 2,600 m with a 1.2% upslope, wind calm, no obstacles in the first 2 km, no fuel on the field. Aircraft is a C172S, MTOW 1,157 kg, ramp mass 1,120 kg. The POH gives a sea-level ISA ground roll of 290 m and a 50 ft distance of 505 m, and states that the chart must not be extrapolated above 8,000 ft density altitude. The student computes pressure altitude 4,390 ft and density altitude about 7,700 ft, then says: "The runway is 2,600 m and we need about 500 m, so we have a five-times margin - we're fine." The engine is a carburetted O-360 with a fixed-pitch propeller, and the student's leaning technique so far has been "full rich until cruise".
1. Critique the student's reasoning. Which performance factors has the calculation ignored, and why is "the runway is long" not an adequate answer?
Read the worked answer
The student has compared two incompatible numbers and applied no correction.
First, the figures used are sea-level ISA figures. At 7,700 ft density altitude, ground roll and the 50 ft distance both increase substantially - a working rule of thumb is roughly a 10% increase in ground roll per 1,000 ft of density altitude, and the airframe POH chart is the authority, not the rule. Second, the 1.2% upslope adds to ground roll. Third, and most important for safety, ground roll is not the limiting quantity: the 50 ft distance, the climb gradient after that, and the obstacle and turn clearance are. At high density altitude a normally aspirated engine produces significantly less power and the propeller less thrust, so the aircraft may clear the 50 ft screen and then climb at 200–300 ft/min with a poor gradient - a terrain or circuit conflict problem the runway length says nothing about.
Fourth, the student is operating right at the edge of the POH chart. At 7,700 ft, an error in QNH, temperature or mass pushes the calculation off the chart, and the chart is explicitly not to be extrapolated. Fifth, the aircraft may not achieve book figures: age, rigging, propeller condition, tyre and brake condition, and pilot technique all cost distance. Sixth, the mass is 1,120 kg against MTOW 1,157 kg - check whether the chart is for MTOW, which would be conservative, or for a lower mass.
Finally, no abort point has been nominated. On a marginal takeoff the decision speed and the stopping distance matter as much as the climb - and stopping downhill on a contaminated or hot runway is a separate calculation.
2. What is your decision for the flight, and how do you brief it so that the student owns the calculation rather than being handed the answer?
Read the worked answer
The decision is to go only after a proper calculation, and to be willing to move the departure to a cooler part of the day. The instructional aim is the method, because the student will face this again without me.
Brief sequence:
Make the student compute it. Pressure altitude from elevation and QNH (about 30 ft per hPa), ISA temperature at that level, ISA deviation, then density altitude. Then read the POH chart honestly and, if the result sits outside the chart, say so and use the conservative end of the data rather than extrapolating.
Apply the corrections in order: density altitude, then upslope, then any surface condition, then a personal safety margin (for training flights I would want at least 1.5× the calculated 50 ft distance, and a clear nominated abort point).
Brief the abort point by name and by distance, and have the student call it on the roll.
Brief the engine failure after takeoff: below a stated height, land ahead within the remaining runway or the cleared area; above it, the nominated field or turn-back altitude - decided on the ground, not at 200 ft with the engine quiet.
Ask the counterfactual: "What would change your answer?" A 5 kt tailwind, 3 °C warmer, an extra passenger, a wet runway. This is the question that tests understanding rather than arithmetic.
If the student cannot defend the numbers, we do not depart - and that itself is the lesson.
3. Explain how you would teach leaning and the after-takeoff engine management for this departure, and what errors you would expect to see in the first two attempts.
Read the worked answer
At 7,700 ft density altitude the mixture is not a detail - it is a power setting. Teach it as a procedure with a purpose, in the aircraft, at the run-up bay.
Teaching sequence. First, let the student see the effect: at run-up, lean for maximum RPM at the target setting and note the rise; then enrich and note the fall. Second, give the rule: at high density altitude the takeoff mixture is normally lean for best power, not full rich, because full rich costs power and fouls plugs; follow the POH, which is the authority for this type. Third, the climb: monitor CHT and EGT, enrich if the engine runs rough or temperatures rise above limits, and brief the cruise climb speed to keep cooling airflow. Fourth, the transition: after takeoff, gear and flaps as applicable, power set, mixture adjusted for the climb, and a positive call on engine parameters.
Expected errors. Full rich on the roll with rough running and reduced power; leaning at the wrong moment so the engine stumbles just after rotation; chasing the mixture instead of setting it and monitoring; leaning at high power without reference to CHT; forgetting carburettor heat in the descent into humid air; and a fixation on the mixture that costs airspeed and heading discipline in the initial climb.
Correct by making the procedure a checklist item with a callout, not an improvisation, and by having the student verbalise the reason each time until the reason, not the action, is memorised. Debrief with the engine parameters recorded so the student sees the evidence, and rehearse the rough-running response on the ground before it happens in the air.