Space pirates: a pursuit curve problem involving retarded time

TL;DR

This paper extends the classical pursuit curve problem by incorporating finite information propagation speed, introducing retarded time concepts, deriving and numerically solving the modified differential equation, and analyzing pursuit success conditions.

Contribution

It presents a novel generalization of pursuit curves considering retarded time, providing a clear pedagogical example and a comprehensive analysis of pursuit success conditions.

Findings

Derived the differential equation for pursuit with retarded time

Numerically solved the generalized pursuit problem

Identified conditions for successful pursuit

Abstract

We revisit the classical pursuit curve problem solved by Pierre Bouguer in the 18th century, taking into account that information propagates at a finite speed. To a certain extent, this could be seen as a relativistic correction to that problem, though one does not need Einstein's theory of relativity in order to derive or understand its solution. The discussion of this generalized problem of pursuit constitutes an excellent opportunity to introduce the concept of retarded time without the complications inherent to the study of electromagnetic radiation (where it is usually seen for the first time), which endows the problem with a clear pedagogical motivation. We find the differential equation which describes the problem, solve it numerically, compare the solution to Bouguer's for different values of the parameters, and deduce a necessary and sufficient condition for the pursuer to catch the pursued, complementing previous work by Hoenselaers.