When taking the displacement between two points along a circular path to calculate its velocity, do you take the length of a chord connecting the two points or do you take the length of the arc connecting them?
When deriving the formula $a_c=\frac{v^2}{r}$, you presume that the displacement is a chord ($d$): $$\frac{\Delta v}{v}=\frac{d}{r}$$ $$a_c=\frac{\Delta v}{t}=\frac{v}{r}\frac{d}{t}=\frac{v}{r}v=\frac{v^2}{r}$$
However, other equations seem to suggest that displacement is an arc ($s$). For example, the equation $v=\omega r$, where $s=\theta r$ and $\omega=\frac{\theta}{t}$, suggests that $v=\frac{s}{t}$ rather than $\frac{d}{t}$.
Moreover, common equations like $a_c=4π^2rf^2$ are derived from the equation $v=\frac{2πr}{T}$, which suggests also that $v=\frac{s}{t}$ rather than $\frac{d}{t}$ (since $d=0$ in the reference frame of a full revolution).
If the formula $v=\omega r$ describes only speed, not velocity, and is based on distance instead of displacement (which would then be unequivocally a chord), then I’d understand. But the second equation, $a_c=4π^2rf^2$ involves acceleration, which definitely is the change in velocity, not speed. How can you derive an equation for acceleration based on one for speed if speed does not represent the magnitude of velocity?
