# A Simple Solution for Maximum Range Flight

## TL;DR

This paper derives a simple, optimal flight speed and altitude profile for maximum range using variational calculus within standard flight models, applicable to different speed restrictions and illustrated with jet examples.

## Contribution

It introduces a straightforward variational approach to determine maximum range flight trajectories without complex optimization methods.

## Key findings

- Optimal flight plans consist of climbing, cruising at optimal speed, and descending phases.
- The method applies to both unrestricted and speed-restricted flights.
- Numerical examples demonstrate the approach's effectiveness with a business jet.

## Abstract

Within the standard framework of quasi-steady flight, this paper derives a speed that realizes the maximal obtainable range per unit of fuel. If this speed is chosen at each instant of a flight plan $h(x)$ giving altitude $h$ as a function of distance $x$, a variational problem for finding an optimal $h(x)$ can be formulated and solved. It yields flight plans with maximal range, and these turn out to consist of mainly three phases using the optimal speed: starting with a climb at maximal continuous admissible thrust, ending with a continuous descent at idle thrust, and in between with a transition based on a solution of the Euler-Lagrange equation for the variational problem. A similar variational problem is derived and solved for speed-restricted flights, e.g. at 250 KIAS below 10000 ft. In contrast to the literature, the approach of this paper does not need more than standard ordinary differential equations solving variational problems to derive range-optimal trajectories. Various numerical examplesbased on a Standard Business Jet are added for illustration.