# One-Dimensional Mathematical Model for Kayak Propulsion
## 4 Estimations of the Model
### Asymptotic Curve Fitting  
It is also possible to adjust speed-distance curves published by other authors. For example, [PERSON] et al. ([PERSON], 2018) present speed-distance curves over 200 m with elite paddlers. These curves show the acceleration interval, the passage through a speed maximum and a third phase of slight deceleration associated with fatigue. The data are obtained with an interval of 10 m and are not precise enough to observe the dynamics of each stroke, but they allow estimations of the asymptotic term. As our model does not consider fatigue, we assume that the maximum velocity corresponds to 99% of the asymptotic velocity and that, thereafter, no fatigue appears, i.e., the velocity is maintained or increases very slightly toward its asymptotic velocity. As the data were measured with elite paddlers, we also assume that their average force is of the order of those already published by [PERSON] et al. (2018) or [PERSON] et al. (2018), and therefore are of the order of \(f_{ave}=130\) N. From the expression obtained in Equation (6), and with the assumptions made, we can clear the resistance coefficient.  
\[d=\frac{f_{ave}}{v_{a}/a}=\frac{130}{5.78/0.99}=3.81\text{ kg/m}.\]  
and then we obtain the mass from the integrated expression for the distance, Equation (11). From the Pickett data, the distance where the 99% of velocity is reached is 50 m. Therefore, by clearing in the expression, we find the following:  
\[m=\frac{50\cdot d}{\ln(cosh(\frac{1}{2a}\cdot\ln(\frac{1+a}{1-a})))}=\dots=96.0 1\text{ kg}.\]  
As we can see, both the resistance coefficient and the mass obtained offer reasonable values. Now, we can transfer these four parameters to the transient expression to compare on a graph experimental versus estimated values. The general expression for the transient is written as follows:  
\[v=\sqrt{\frac{f}{d}}tanh\Bigg{(}\sqrt{\frac{d}{f}}\left(\frac{f}{m}t\right) \Bigg{)}.\]With the integration constant \(K=0\) if the starting is from rest. However, to compare, we must replace time with abscissa distance. Its equivalence in our model can be obtained by integrating as in Equation (11) to obtain the following:  
\[dist=\frac{m}{d}\left[\ln\left(cosh\left(\sqrt{\frac{d}{f}}\cdot\frac{f}{m} \cdot t\right)\right)\right].\]  
With these data, the representation is presented in Figure 10. This adjustment is probably no more than an estimate based on reasonable assumptions, but although the first experimental point is not well reproduced, on the whole, it provides realistic results and the curve fits quite well the rest of the points. It must be taken into account that the oscillations due to strokes are specially relevant in the first part of acceleration and they are not recorded in the experimental data, so a significant noise amount must be supposed. Note that one could also proceed by a multiparameter fit (\(f,d,m\)) by minimization. In fact, by this method, one obtains curves very close to the experimental one but with unrealistic parameters. Moreover, several combinations of parameters are possible, which is not consistent and shows that our fitting strategy for this estimation is more appropriate. Note also that a more appropriate fit could be made if at least some of the parameters are previously known.  
Finally, if we resort to the optimal quadrature point (from Equation (12)), of value \(a\sim\)0.68, the estimated work and acceleration power for this curve are \(W=1525.4\) J and \(P=311.4\) W, respectively--higher values but not far from those obtained for the estimations of [PERSON] et al. [9] or [PERSON] et al. [15].