The take-off of a departing aircraft is subjected to varying forces, the largest of which are function of the speed of the aircraft itself: the thrust of a jet engine, the aerodynamic drag of the airframe are essentially polynomial function of the airspeed.
First principle field performance determination has relied for the last half century on stepwise integration, a brute force approach that requires properly tuned integration steps, which has the benefit of being fairly reliable but with relatively low computational efficiency.
An alternate, mathematically more formal approach would be to algebraically integrate an accelerative function combining all the forces in presence. While comparatively complex, the derivation of the solution equations permits fast and accurate integration between boundary conditions, which could be orders of magnitude more efficient than stepwise integration, even with relatively high degree of polynomial force functions, while being essentially free of any round-off error which may accumulate at each step of a linearized stepwise integration as the result would instead be derived strictly from the value at the integration limits; the speed advantage remains for most practical models, up to a 6th degree acceleration to speed polynomial function. Moreover, the integrated equations could be used for several airspeed evaluations, leveraging the computational efficiency higher still.