# One-Dimensional Mathematical Model for Kayak Propulsion
## 3 Analytical Solutions for Periodic Propulsion
### General Stroke Model
\begin{table}
\begin{tabular}{l l l l l l} \hline \hline  & & \multicolumn{1}{c}{**Stroke Freq. \({}^{a}\)**} & \multicolumn{1}{c}{**Stroke Freq.**} & \multicolumn{1}{c}{**Stroke Freq.**} & \multicolumn{1}{c}{**Stroke Freq.**} \\  & **Gender** & **60** & **80** & **100** & **Race Pace** \\ \hline Frequency stroke (spm) & Men & \(63\pm 5\) & \(81\pm 3\) & \(99\pm 6\) & \(124\pm 7\) \\  & Women & \(60\pm 3\) & \(79\pm 6\) & \(100\pm 6\) & \(112\pm 3\) \\ Time in 200 m (s) & Men & \(54.35\pm 2.29\) & \(47.85\pm 2.00\) & \(43.67\pm 1.88\) & \(38.68\pm 0.83\) \\  & Women & \(61.35\pm 0.95\) & \(53.76\pm 1.73\) & \(48.08\pm 2.36\) & \(44.94\pm 1.21\) \\ Average velocity 200 m (m s\({}^{-1}\)) & Men & \(3.68\pm 0.15\) & \(4.18\pm 0.18\) & \(4.58\pm 0.20\) & \(5.17\pm 0.11\) \\  & Women & \(3.26\pm 0.05\) & \(3.72\pm 0.12\) & \(48.08\pm 2.36\) & \(44.94\pm 1.21\) \\ Aquatic phase duration (s) & Men & \(0.56\pm 0.03\) & \(0.50\pm 0.04\) & \(0.43\pm 0.03\) & \(0.37\pm 0.03\) \\  & Women & \(0.64\pm 0.03\) & \(0.55\pm 0.02\) & \(0.48\pm 0.01\) & \(0.43\pm 0.02\) \\ Aerial phase duration (s) & Men & \(0.40\pm 0.05\) & \(0.24\pm 0.03\) & \(0.18\pm 0.04\) & \(0.14\pm 0.03\) \\  & Women & \(0.35\pm 0.01\) & \(0.23\pm 0.05\) & \(0.17\pm 0.05\) & \(0.12\pm 0.02\) \\ Time until \(F_{max}\) (s) & Men & \(0.22\pm 0.03\) & \(0.21\pm 0.02\) & \(0.19\pm 0.02\) & \(0.16\pm 0.02\) \\  & Women & \(0.26\pm 0.03\) & \(0.24\pm 0.03\) & \(0.21\pm 0.02\) & \(0.20\pm 0.01\) \\ \(F_{max}\) (N) & Men & \(225\pm 31\) & \(234\pm 32\) & \(266\pm 33\) & \(274\pm 35\) \\  & Women & \(126\pm 11\) & \(130\pm 8\) & \(146\pm 7\) & \(153\pm 11\) \\ \(F_{ave}\) (N) & Men & \(118\pm 16\) & \(128\pm 18\) & \(157\pm 18\) \({}^{a}\) & \(171\pm 18\) \\  & Women & \(72\pm 6\) & \(80\pm 9\) & \(92\pm 13\) \({}^{a}\) & \(99\pm 15\) \\ Impulse (N\(\cdot\)s) & Men & \(66.3\pm 7.3\) & \(63.9\pm 7.3\) & \(67.7\pm 9.5\) & \(63.2\pm 8.4\) \\  & Women & \(46.5\pm 5.9\) & \(44.1\pm 5.5\) & \(44.2\pm 6.3\) & \(42.3\pm 6.6\) \\ \(F_{ave}/F_{max}\) (ratio \%) & Entire sample & \(53.3\pm 3.3\)\({}^{c,d,e}\) & \(57.2\pm 3.9\)\({}^{b,d,e}\) & \(61.0\pm 3.8\)\({}^{b,c,d}\) & \(64.8\pm 3.7\)\({}^{b,c,d}\) \\ \hline \hline \end{tabular} Note. Men \(n=5\), Women \(n=5\). All of them analyzed for each stroke frequency, mean \(\pm\) SD. spm = _stokes per minute. \({}^{a}\)_ Stroke frequency (spm). \({}^{b}\)\(p<0.05\) significantly different to 60 spm. \({}^{c}\)\(p<0.05\) significantly different to 80 spm. \({}^{d}\)\(p<0.05\) significantly different to 100 spm. \({}^{e}\)\(p<0.05\) significantly different to 100 spm.  
\end{table} 
Table 1: Table of mean \(\pm\) SD (standard deviation) for the variables of the force–time curve for the canoeing stroke (Figure 5), for each gender and \(F_{mean}/F_{max}\) for the whole sample. Extracted from [PERSON] et al. [9] and adapted. Figure 5: Average force–time curve for each gender and stroke frequency. Each curve represents the normalized mean curve (270, 300, 330 and 360 strokes from both sides at 60, 80, and 100 strokes per minute (spm) and at maximum frequency (max), respectively. M = men, F = women. Graph extracted from [PERSON] et al. [9].