Bibliography.
G.R. Fowles, “Analytical Mechanics”. (Chapter 1).
Position and velocity
Example. (Rectilinear motion) Consider the position of a particle moving on the horizontal axis
$$ \mathbf{r}(t) = \alpha t\, \hat{\imath}, $$
where $\alpha$ is a constant. The particle is moving on a straight line with velocity
$$ \bm{v}= \frac{d\mathbf{r}}{dt} =\alpha\,\hat{\imath}. \square $$
Example. (Circular motion) The position vector of a particle that moves in a circular motion is
$$ \mathbf{r}(t) = \alpha \left[ \sin(\omega t)\, \hat{\imath} + \cos(\omega t)\, \hat{\jmath} \right].
$$
The position vector is on a circle of radius
$$ |\mathbf{r}| = \sqrt{\alpha^2 \sin^2(\omega t) + \alpha^2 \cos^2(\omega t)} = \alpha. $$
The velocity of the particle is
$$ \bm{v}(t) \equiv \frac{d\mathbf{r}}{dt} = \omega \alpha \left[ \cos(\omega t) \hat{\imath} - \sin(\omega t) \hat{\jmath} \right]. $$
The speed is constant, $|\bm{v}| = \omega \alpha$. Also $\bm{v}\cdot\mathbf{r} = 0$, that is, $\bm{v}\perp\mathbf{r}$.
Example. (Cycloidal motion) Let
$$ \begin{aligned} \mathbf{r} & = \mathbf{r}_1 + \mathbf{r}_2 \\ \mathbf{r}_1(t) & = \alpha\omega t\, \hat{\imath} + \alpha\, \hat{\jmath} \\ \mathbf{r}_2(t) & = - \alpha \left[ \sin(\omega t)\, \hat{\imath} + \cos(\omega t)\, \hat{\jmath} \right]. \end{aligned} $$
The position vector as a combination of a vector representing rectilinear motion in the $x$ direction and a circular motion of radius $\alpha$. The curve that the particle is following (its trajectory) is called cycloidal and it has the parametric representation