# On the Physics of Kayaking
## 5 The 10 \(\times\) 50 Meter Trial: Kayaking at Constant Velocity
### Velocity-Stroke Rate Relationship Model  
In this section, we model the relationship observed during the progressive 10 \(\times\) 50 m: \(\overline{V}=Af^{1/3}\) (Figure 14a). We start from Equation (7), corresponding to the average of Equation (2) over one cycle. Once the permanent regime is reached, \(\frac{\mathrm{d}\overline{V}}{\mathrm{d}t}=0\) (as \(V(t)=V(t+T)\)):  
\[\overline{F_{m}}=K\frac{1}{2}\rho SC_{D}\overline{V}^{2}. \tag{9}\]  
Using \(t_{p}=t_{out}-t_{in}\), the propulsion time, as there are two propulsion phases (left and right assumed symmetric) during one cycle, Equation (9) yields:  
\[\frac{2t_{P}}{T}\cdot\frac{1}{t_{p}}\int_{t_{in}}^{t_{out}}F(t)n(\theta(t)) \mathrm{d}t=K\frac{1}{2}\rho SC_{D}\overline{V}^{2}. \tag{10}\]  
As observed in Figure 14, the mean projected force along the axis of the motion \(\widetilde{F}=\int_{t_{in}}^{t_{out}}F(t)n(\theta(t))\mathrm{d}t/t_{p}\) is approximately constant, except for the first two trials. Therefore, we define \(\widetilde{F}_{0}\) as this constant value. As the stroke rate \(f\) is the inverse of the period, we obtain:  
\[f\cdot t_{P}\widetilde{F}_{0}=K\frac{1}{4}\rho SC_{D}\overline{V}^{2}. \tag{11}\]  
At this stage, we need to find an expression for the evolution of \(t_{P}\) as a function of the velocity \(\overline{V}\). The ratio between the hydrodynamic coefficients \(SC_{D}\) of the boat and the paddle is small (\(SC_{D}=6\times 10^{-3}\) m\({}^{2}\) for the boat and \(SC_{D}=1.2\times 10^{-1}\) m\({}^{2}\) for a paddle). For this reason, we expect little drift of the paddles in the water compared to the boat during the propulsion. In the limit where the paddle is anchored in the water, the velocity of the paddle in the frame of the boat \(V_{P/B}\) equals the velocity of the boat \(\overline{V}\). Defining \(L_{p}\) as the length of the propulsive path of the paddle in the frame of the boat, it comes \(V_{P/B}=L_{p}/t_{P}\). Therefore, the relationship between \(\overline{V}\) and \(t_{P}\) should be close to \(1/t_{P}=\overline{V}/L_{p}\). We used the  
Figure 14: (**a**) Velocity–stroke rate relationship in the steady state regime for the two kayakers. (**b**) Evolution of the average propulsion force in the direction of paddling with trial N for Athlete A1. The force is averaged over the propulsion time. data of the 10 \(\times\) 50 m of Athlete A1 to analyze the evolution of the inverse of the propulsion time and the mean velocity (Figure 15). In Figure 15, the relationship between \(1/t_{P}\) and \(\overline{V}\) is linear for velocities larger than 2 m/s for both arms. We found a propulsive length of \(L_{p}=1.64\) m for Athlete A1. This length can be compared to the arc length described by the center of a paddle, where we expect the force resultant to be applied. For Athlete A1, the distance between the two paddle centers was \(\ell\approx 1.76\) m and we measured paddle angles going from \(50^{\circ}\) to \(160^{\circ}\). This corresponds to a traveled distance \(L_{p}\approx 0.5\ell\times\Delta\theta\approx 1.69\) m. This represents a 3% difference with the previous measurement. Using the relation between \(t_{P}\) and \(\overline{V}\), Equation (11) can be simplified:  
\[\overline{V}=\left(\frac{4L_{p}\widetilde{\widetilde{F}}_{0}}{K\rho SC_{D}} \right)^{\frac{1}{3}}f^{\frac{1}{3}}. \tag{12}\]  
This equation is compatible with the experimental data presented in Section 5.1. We found A = 4.2 for Athlete A1 (\(\widetilde{F}_{0}=86\) N, \(L_{p}=1.64\) m, \(K=1.25\), \(SC_{D}=5.85\times 10^{-3}\) m\({}^{2}\)). This value must be compared to A = 4.4 found experimentally. Thus, we recovered the experimental behavior with a precision of 5%. In this section, we assumed symmetry between the left and right propulsion. In the standing start experiments (Figure 9), we observed a difference in the patterns between the left and right propulsion.