# One-Dimensional Mathematical Model for Kayak Propulsion
## 1 Introduction  
Professional sports have seen important transformations due to the use of new data collection techniques to create models and optimize performance. These data-driven models provide important insight into the sports but they may be hard to extend to different situations. 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.  
In the case of competition canoeing, many different aspects have been studied in recent years, which reflects the complexity of this sport. The hydrodynamic aspects of the propulsion have been investigated extensively; the hydrodynamic drag or its components was analyzed both computationally and experimentally [1; 2; 3; 4; 5; 6], even under uniform displacement conditions. The paddle [7] or stroke [8] hydrodynamics were also analyzed; correlations between the stroke frequency and hull speed [9; 10; 11], the stroke force versus time functional relation [9; 12] or the hull-stroke interaction [13; 14] were found. There are also studies on the forces applied to the paddle blade and the power, accelerations and velocities that it produce in the hulls [9; 12; 15; 16]. Additionally there are kinematics kayak models: the one of six degree of freedom proposed by [PERSON] [17] or the one introduced by [PERSON] [18] to calibrate a kayak ergometer in the most precise way.  
These papers show the importance of understanding correctly hydrodynamic drag coefficients, the effect of the forces on the paddler-hull system, and the functional relationships of the stroke on the kayak to model the races and provide optimization routes for the materials of the equipment, the techniques or the strategies to increase performance.  
In the present work, we propose a solvable one-dimensional mathematical model for kayak displacement as a function of the applied force. Our model allows to connect thephysics of the system to quantities relevant to the race and the performance of the athlete. It also provides a tool to analyze and develop strategies in different race scenarios.  
The model is characterized by periodic propulsion and constant hydrodynamic drag and mass, which are the only relevant parameters of the kayak in our model, and are useful to analyze different aspects of the race. We introduce different functional relations in order to model the stroke and with it, the propulsion. The analytical and numerical resolution of such a model will allow us to find the good parameters to characterize the kayak advance, and to relate the analytical model with previous experimental studies. Another consequence is the description of a general mathematical technique to solve both analytically or numerically the problem of a boat propelled by a periodic force. The results here presented could have implications not only in this sport, but also 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. (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 finally, (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.  
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.