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These equations model circular motion. Equation R is the position vector given in polar coordinates. What I've done is represent this vector onto the complex plane via equation (1). Equation (2) and (3) are the first and second time-derivatives, respectively.

Now, the question I have is this one: It seems "magical" to me that these derivatives actually give me the radial and tangential components for velocity and acceleration. How do I demystify this? Why is this obvious, apart from the fact that e^(itheta) and ie^(itheta) model the polar unit vectors?

In other words, why doesn't projecting vectors given in polar coordinates onto the complex plane produce contradictions? Why does it work? Why is it that the terms actually represent the tangential and radial components?

Also, do these equations imply that the typical introductory physics problem where we have to find the centripetal force at the bottom of a ditch as simply mv^2/r wrong? Because of the term given for radial acceleration by equation (3).

Thanks. I hope my question is clear.

If it's useful I got this from http://farside.ph.utexas.edu/teaching/301/lectures/node89.html