Is there a name for such a curve or can this even happen? I know when the velocity vector, $\mathbf{x'}$, and position vector, $\mathbf{x}$ are always orthogonal $\mathbf{x}(t)$ parametrizes a circle so I thought maybe in general it forms a spiral. But this is only an inkling.
The only equation I have relating two vectors and their angles is the standard $$\cos(\theta)=\frac{|\mathbf{x}\cdot \mathbf{x'}|}{||\mathbf{x}||||\mathbf{x'}||}$$
I tried converting this to polar, but I don't understand how to relate the notion of angle between $\mathbf{x}$ and $\mathbf{x'}$ in that context.
Best Answer
When expressed in polar form, the components of velocity are $\frac{dr}{dt}$ along the radius and $r\frac{d\theta}{dt}$ tangentially. Therefore if the velocity vector makes constant angle $\alpha$ with the radius vector, we have $$\tan\alpha=\frac{r\frac{d\theta}{dt}}{\frac{dr}{dt}}=r\frac{d\theta}{dr}$$ solving, we have $$\frac{1}{r}dr=\cot\alpha d\theta$$ this leads to the general solution $$r=ae^{\theta\cot\alpha},$$ which is indeed a spiral.
Note that when the velocity is perpendicular to the radius vector, $\cot\alpha=0$ which leaves us with the circle $r=a$.