An analytic solution to the equations governing the motion of a point mass with quadratic resistance and generalizations

TL;DR

This paper derives an explicit analytical solution for the motion of a point mass under quadratic drag, extending to general drag laws and variable fluid velocities, providing a comprehensive mathematical framework for such dynamics.

Contribution

It presents the first explicit solution in elementary form for a general trajectory under quadratic drag, using a novel reduction of differential equations and linking to Hamiltonian mechanics.

Findings

Solution expressed as a ratio of series expansions

Extension to general power-law drag laws

Inclusion of variable fluid velocity scenarios

Abstract

The paper is devoted to the motion of a body in a fluid under the influence of gravity and drag. Depending on the regime considered, the drag force can exhibit a linear, quadratic or even more general dependence on the velocity of the body relative to the fluid. The case of quadratic drag is substantially more complex than the linear case, as it nonlinearly couples both components of the momentum equation. Careful screening of the literature on this classical topic showed that, unexpectedly, the solutions reported do not directly provide the particle velocity as a function of time but use auxiliary quantities or apply to special cases only. No explicit solution using elementary operations on analytical expressions is known for a general trajectory. After a detailed account of the literature, the paper provides such a solution in form of a ratio of two series expansions. This result is discussed in detail and related to previous approaches. In particular, it is shown to yield, as limiting cases, certain approximate solutions proposed in the literature. The solution technique employs a strategy to reduce systems of ordinary differential equations with a triangular dependence of the right-hand side on the vector of unknowns to a single equation in an auxiliary variable. For the particular case of quadratic drag, the auxiliary variable allows an interpretation in terms of canonical coordinates of motion within the framework of Hamiltonian mechanics. Another result of the paper is the extension of the solution technique to more general drag laws, such as general power laws or power laws with an additional linear contribution. Furthermore, generalization to variable velocity of the surrounding fluid is addressed by considering a linear velocity profile for which a solution is provided as well. Throughout, the results obtained are illustrated by numerical examples.