# Time derivatives of polar motion

1. Sep 9, 2015

### stumpoman

1. The problem statement, all variables and given/known data
A particle moves with v=constant along the curve
$$r = k(1+\cos \theta)$$Find $\mathbf{a}$

2. Relevant equations
$$\mathbf{r} = r\mathbf{e_r}$$ $$\mathbf{v} = \frac{\partial}{\partial t}(r\mathbf{e_r})$$ $$\mathbf{a} = \frac{\partial \mathbf{v}}{\partial t}$$ $$\mathbf{\dot e_r}=\dot \theta\mathbf{e_\theta}$$ $$\mathbf{\dot e_\theta}= -\dot \theta\mathbf{e_r}$$

3. The attempt at a solution
$$\mathbf{v} = \dot r\mathbf{e_r}+r\mathbf{\dot e_r} = \dot r\mathbf{e_r} + r\dot \theta\mathbf{e_\theta}$$ $$\mathbf{a} = \ddot r\mathbf{e_r}+\dot r\mathbf{\dot e_r}+\dot r\dot \theta\mathbf{e_\theta}+r\ddot \theta\mathbf{e_\theta}+r\dot \theta\mathbf{\dot e_\theta}$$ $$\mathbf{a} = \ddot r\mathbf{e_r}+\dot r\dot \theta\mathbf{e_\theta}+\dot r \dot \theta\mathbf{e_\theta}+r\ddot \theta\mathbf{e_\theta}-r\dot \theta ^2\mathbf{e_r}$$ $$\mathbf{a} = (\ddot r - r\dot \theta ^2)\mathbf{e_r} + (2\dot r\dot \theta + r\ddot \theta)\mathbf{e_\theta}$$
From here I am having some trouble. I cannot figure out how to get equations for $r$ and $\theta$ in terms of $t$. I would guess that it has something to do with v being constant. Because r is a cardioid I don't think I can just use $\theta=t$.

2. Sep 9, 2015

### andrewkirk

You have not yet used the fact that speed is constant. The first equation in your attempted solution gives the velocity. Use that to develop a formula for the speed and then set that equal to a constant. That should give you something to substitute into your last equation to get rid of some stuff.