Rate of Climb from Excess Power

Climb is a power problem, not a thrust problem, and this equation is why. An aircraft in a steady climb is doing work against gravity at a rate of weight times climb rate. The only place that work can come from is power the engine is producing beyond what drag is already consuming. Divide the excess by the weight and you have the vertical speed: $RoC = (P_a - P_r)/W$.

Notice what is absent. There is no flight path angle, no airspeed, no aspect ratio. All of them are hidden inside the two power terms, and understanding those terms is understanding climb performance. POWER AVAILABLE is shaft power times propeller efficiency for a piston or turboprop, and thrust times true airspeed for a jet. POWER REQUIRED is drag times true airspeed, which means it inherits the U-shape of the drag curve but with its minimum at a LOWER speed than minimum drag — because multiplying by $V$ shifts the minimum left.

That shift is the reason best-rate-of-climb speed and best-glide speed are different numbers on the same placard. Best glide sits at minimum DRAG. Best rate of climb sits near minimum POWER REQUIRED, which is slower — around 76% of the minimum-drag speed for a parabolic polar. Best ANGLE of climb, the one that clears the trees, is different again and slower still, because it maximises excess THRUST rather than excess power.

The engine type changes the whole character of the climb. A propeller aircraft makes roughly constant power across its speed range, so its excess power is largest at low speed and it climbs best there. A jet makes roughly constant thrust, so its power available rises with speed, and its excess power peaks much faster — which is why a jet's best climb speed is a large fraction of its cruise speed and a light aircraft's is barely above its approach speed.

Both terms change with altitude and not at the same rate, which gives the ceiling. Power available falls with air density, roughly in proportion for a normally aspirated engine. Power required rises, because holding the same lift in thinner air means flying faster in true terms. Where the curves meet, the climb rate is zero: that is the absolute ceiling, and because approaching it takes an infinitely long time, the SERVICE ceiling is defined at a small residual climb rate instead — conventionally 100 ft/min for piston aircraft and 500 ft/min for jets. A handy imperial check on the whole equation: excess horsepower times 33,000, divided by weight in pounds, gives the climb rate in feet per minute.