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I am in need of some clarification on tangential velocity in polar coordinates. As far as I know, the tangential velocity vector is ##\vec{v} = v\vec{e_t}##, where ##\vec{e_t} = \frac{\vec{v}}{v}##. This gives us the ##\vec{e_r}## and ##\vec{e_\varphi}## coordinates of the tangential velocity vector.

For example, the velocity for the Archimedes spiral is ##\vec{v} = s\omega \vec{e_p} + s\omega^2 t\vec{e_\varphi}##. But the tangential velocity is ##\frac{1}{\sqrt{1+\omega^2 t^2}}\vec{e_p} + \frac{\omega t}{\sqrt{1+\omega^2 t^2}}\vec{e_\varphi}##

What I do not understand is why are the two different? I was under the impression that the velocity (ie. the derivative of the location) isalready and by definitiontangential to the curve/trajectory. What is the difference between the two in this case?

Thank you.

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# I Tangential velocity in polar coordinates

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