Mathematical Description of Motion

The position of a particle is expressed as:

$$ \vec{r}(t) = x(t)\hat{i} + y(t)\hat{j} + z(t)\hat{k} $$

Velocity

Velocity is defined as:

$$ \vec{v} = \frac{d\vec{r}}{dt} $$

It represents how fast and in what direction the position changes.

Acceleration

Acceleration is:

$$ \vec{a} = \frac{d^2 \vec{r}}{dt^2} $$

It represents how velocity changes with time.