Abstract
The displacement of a sprint kayak can be described by a one-dimensional mathematical model, which, in its simplest case, is analogous to the free-fall problem with quadratic drag and constant propulsion. To describe realistic cases, it is necessary to introduce a propulsion capable of reproducing the characteristics of the kayak stroke, including periodicity, average force and effects of stroke frequency, among others. Addressing the problem in terms of a Fourier series allows us to separate the equation into two parts, one of which is equivalent to the constant propulsion case and results in an asymptotic expression, while the second accounts for the periodic contributions. This approach allows us to solve several cases of interest: to propose a quadrature rule for the asymptotic part that allows fast estimations; to compare results with the literature; and finally to propose a general mathematical method for this problem which could help to understand some key strategies in the kayak race.
Highlights
Professional sports have seen important transformations due to the use of new data collection techniques to create models and optimize performance
The value of this new analytical model that we introduce is that: (i) it is built from first principles, and the reduced dimensionality allows us to model the main drivers of the system. (ii) The analytical model allows us to write quadrature relations that are useful to describe the race in simple terms, using relevant physical constants. (iii) The analytical expressions are general and allow us to use and compare different paddle models to analyze the race and understand the relevant drivers of performance; and (iv) it can be used to analyze real data from races and relate the performance to simple physical constants that are relevant and easy to understand
We introduced a simple analytical model for the fluid–hull–paddlers system, which allows to understand the basic physics of the problem and discuss many relevant aspects of the race in simple and useful terms
Summary
Professional sports have seen important transformations due to the use of new data collection techniques to create models and optimize performance. Analytic models provide an alternative approach, which allows to study the sports from basic principles and provides insights on the main drivers of performance and strategies to improve it Their results are expressed in terms of simple physical constants and can be generalized to different situations. The results here presented could have implications in this sport, and in others characterized by periodic propulsion and time-independent drag The value of this new analytical model that we introduce is that: (i) it is built from first principles, and the reduced dimensionality allows us to model the main drivers of the system. All these characteristics are in contrast to numerical models that can be used to integrate numerically the differential equation, which can be more accurate but cannot be generalized, and do not provide simple physical pictures to understand and improve the performance
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