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I took a video of a swimmer pushing off of the wall and gliding through the water until they come to a stop. Using some video analysis software, I was able to plot data points for their velocity over time. enter image description here

I would like to fit this data to a curve, but am not sure what type of relationship it should be. I know that the only acceleration that the swimmer experiences is from drag, which is modeled by the equation $$F=\frac{1}{2}\rho v^{2}C_dA$$ Where $\rho$=density of the water, v=velocity of the swimmer relative to the water, $C_d$=drag coefficient, and A=reference area. This suggests that acceleration (since F=ma) is proportional to $v^2$. However, according to the SUVAT equations, $$v=u+at$$ Where u=initial velocity. I'm not too sure how to continue deriving the relationship between v and t. I'm thinking it could be exponential, inverse, or inverse squared, but I don't really have any reasoning behind those guesses.

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  • $\begingroup$ The drag formula can get a lot worse if you consider planing. I doubt a human body planes to any measurable amount, but certainly something like, say, a windsurf board planes at high speed and then enters into the standard drag formula as it slows and sinks. $\endgroup$
    – Carl Witthoft
    Commented Nov 23, 2021 at 17:55

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The equation $v=u+at$ is only valid for constant acceleration. To go ahead $$\begin{aligned} F&=ma \\\implies\frac{1}2\rho v^2C_dA &=m\dot v \end{aligned}$$ The above equation can be solved for the velocity as a function of time.

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