An Analytical Solution of the Extended Lifeguard Problem with Water Current

Abstract

This paper presents an analytical solution to an extended version of the classical lifeguard problem, incorporating the effects of water current. The problem involves determining the optimal path for a lifeguard to reach a drowning swimmer who is being carried by a current. Unlike the classical problem, which has been well-studied in the context of Snell’s law and Fermat’s principle, this extended version introduces additional complexity through the vector addition of velocities. This paper derives the complete mathematical solution, presents numerical methods for practical applications, and analyzes the behavior of the system under various parameter regimes. The results demonstrate how the optimal rescue path deviates from the classical solution due to the presence of current, providing practical insights for real-world rescue scenarios.