Mathematical Modeling of Stroke Rate and Stroke Length for Swimming Speed and Energy Efficiency

Abstract

This study develops a theoretical framework for identifying stroke rate and stroke length combinations that balance swimming speed and energy expenditure. Cycle averaged speed is expressed as their product, while a simplified hydrodynamic model links metabolic power to drag and effective efficiency. Energy expenditure is evaluated per unit distance and over a fixed event distance. The analysis shows that constant efficiency determines an energy efficient speed but cannot identify a unique stroke combination. A dimensionless calculation demonstrates why swimming more slowly does not necessarily minimize total energy expenditure. An assumed efficiency function with quadratic penalties in logarithmic stroke variables then yields an explicit optimal rate and length allocation at a given speed. Physiological and technical limits restrict these solutions, and an event specific speed requirement or energy budget selects among feasible tradeoffs. A calibration and validation protocol is provided for practical application. The results are conditional analytical deductions; individualized parameter estimates and empirical performance improvements remain to be established.