Mathematical Relations to Analyze Aircraft Performance for Trajectory Planning


  1. Labonté, G.
Corporate Authors
Defence R&D Canada - Ottawa, Ottawa ONT (CAN);Royal Military Coll of Canada, Kingston ONT (CAN)
We revise the derivation of the Bréguet-Coffin endurance and range formulas in the context of Newton’s Second Law of Motion. We believe that the formulas we derived constitute valuable tools for analyzing aircraft performance and for analysing aircraft trajectory planning. We point out that, in principle, the momentum of the mass of air required to burn the fuel that is taken in at rest and ejected at the airplane speed, should be taken into account since this mass is about 14.7 times that of the fuel burned. We give the exact solutions to the equations for the fuel consumption, obtained with this consideration, when the aircraft is in level flight. We consider two modes of flight: one with a constant angle of attack (as in the Bréguet-Coffin case), and one at constant speed. Comparison of the endurances and the ranges obtained with and without the correction terms show that some of the added terms can actually be neglected. We then solve exactly the fuel consumption equations in which these terms are neglected, for climbing airplanes in three different modes of motion – flight at a constant angle of attack, flights at a constant speed, and flights at a constant Mach number. With the help of these solutions, we derive formulas for the speed, the altitude and the power required as a function of time, and as a function of the altitude. The behaviour of the solutions obtained is exhibited through many sample calculations that involve the model CP-1 airplane described in An

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Report Number
DRDC-OTTAWA-CR-2010-249 — Contractor Report
Date of publication
01 Dec 2010
Number of Pages
Electronic Document(PDF)

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