Strategies for human-powered water ferries

A classic trajectory optimization problem involves a watercraft crossing a river with a constant current flow. As long as the current speed is not too fast, the craft can make it directly across by pointing its nose upstream at the proper angle. If the current velocity depends on position, the problem is solved by using the calculus of variation. However, these optimal trajectories do not take into account the relevant constraints for a human-powered watercraft, which should include fatigue, and limited power. In this article, we consider two situations with analytical solutions. First, the current is constant but the watercraft speed is a decaying function of time specified by two parameters. Under certain conditions, we find the crossing conditions analytically, and compare them to perturbative and numerical solutions. Second, we study the possibility of crossing a zone where the current speed is larger than the watercraft maximum speed. In addition to their practical importance in coastal sailing or sea kayaking, the mathematical methods used in this article are very general and provide students the opportunity to develop skills in modeling real-world physical systems.