Understanding Velocity in Polar Coordinates

[srmeier]

Homework Statement

I don't understand why when we derive the velocity equation of motion in polar coordinates we start with position equal to R times R hat and not (theta times theta hat + R times R hat).

Homework Equations

none really..

The Attempt at a Solution

Is there an assumption I'm missing? or is it simply differentiating linear and angular velocity that is messing me up?

 

[vela]

It's because your position is given by

$$\vec{R} = \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} R\cos\theta \\ R\sin\theta \end{pmatrix} = R\begin{pmatrix} \cos\theta \\ \sin\theta \end{pmatrix} = R \hat{R}$$

Also, if you think about it, $$\theta \hat{\theta}$$ doesn't have units of length, so it's not a displacement and you can't add it to $$R\hat{R}$$.