# One-Dimensional Mathematical Model for Kayak Propulsion
## 4 Estimations of the Model
### Drag Coefficient and Limiting Speed Estimations  
Studies of applied force versus speed in real paddlers allow us to analyze the limiting speed. Initially, we use the above-mentioned coefficient \(d=2.72\) kg/m, extracted from a field study [4]. It is analogous and of the same order as other hydrodynamic studies at constant speed and propulsion [1; 2; 3; 6], but it is a passive hydrodynamic drag coefficient. From the data of [PERSON] et al. [9], for the competition padding frequency, we have peak forces of 274 N and 153 N, for male and female categories, respectively, which transform to average forces of 274 - 0.666 - 0.726 = 132.48 N and 153 - 0.666 - 0.782 = 79.68 N, respectively, and where we have taken into account both the aquatic and aerial phases (factor 0.726 in men and 0.782 in women), as detailed in the previous Section 3.4. The expression for the limiting velocity allows us to either obtain velocities from a known \(d\) coefficient or to estimate the drag coefficient from the velocity data obtained in the study. Using the expression (32) and the passive resistance coefficient \(d=2.72\) kg/m, we obtain velocities of \(v_{M}=6.98\) m/s and \(v_{W}=5.41\) m/s for males and females, respectively, appreciably above the average velocities obtained in the study \(v_{M}=5.17\) m/s and \(v_{W}=4.45\) m/s. Similar results can be obtained for the rest of the data at lower stroke frequencies. Regarding the kinematic-dynamic study over 200 m by [PERSON] et al. [15], the conclusions are similar. From their data, we take the fastest interval (between 50 and 100 m) corresponding to a force of 276.31 N and average velocity of 5.2 m/s for the male category. Such force, averaged as before and with a coefficient of \(d=2.72\) kg/m, can lead us to a much higher velocity, of value \(v=7.01\) m/s, analogous to those obtained for [PERSON] et al. [9]. Similar results can also be obtained from more recent data [10]. If, on the other hand, we extract drag coefficients from field data, we obtain \(d=4.96\) kg/m and \(d=4.02\) kg/m for males and females, respectively, from the data of [PERSON] et al. [9] and \(d=4.94\) kg/m from that of [PERSON] et al. [15], being of the same order for the rest of the data or stroke frequencies and relatively constant by gender. We can evaluate the dynamic magnitudes from these force and drag coefficient estimates. We transfer the data to the expressions (16). Usingthe remaining ones in Figure 1, we obtain work to complete the acceleration interval of \(W_{M}=1201.94\) J and \(W_{W}=891.94\) J and their corresponding estimated average power at \(P_{M}=280.77\) W and \(P_{W}=145.47\) W for males and females, respectively. We have no data and cannot make any statements about the efficiency of the stroke. Additionally, the forces were obtained or correlated with accelerometry data measured in situ in the field studies. Therefore, the observed differences must have their origin in the simplifications assumed, which ultimately refer to the differences between active and passive hydrodynamic drag. From [PERSON]'s studies of human locomotion in water ([PERSON], 1977), it follows that active hydrodynamic drag is far superior to passive drag in all forms of human locomotion of this type. In particular for sprint canoeing, an estimate is made for active drag of the type \(f_{A}=3.2\cdot v^{2.46}\) with the particularity of being a _free_ exponent expression sometimes used for hydrodynamic drag adjustments ([PERSON] _et al._, 2018). A _phenomenological_ estimate tells us that in these settings, _higher exponents correspond to lower coefficients_, and the equivalent expression with squared exponent corresponds to a considerably higher coefficient. For example, for velocities of 5.17 and 4.45 m/s, the active hydrodynamic drag for the above expression is 181.10 and 125.93 N and corresponds to coefficients of \(d=6.81\) and \(d=6.36\) kg/m if we use quadratic-type hydrodynamic drag expressions.