Velocity is the derivative of position, and acceleration is the derivative of velocity. Integration reverses this.

Displacement is the change in position; distance traveled is the total path length.

Arc length calculates the total length of the path, giving the distance traveled.

Integrating a velocity function gives you the displacement.

The magnitude of the velocity vector represents the speed of the object.

Differentiate each component of the position vector with respect to time.

Differentiate each component of the velocity vector with respect to time.

The total distance traveled over the given time interval.

The area under the velocity-time curve.

Parametric equations describe motion by defining the x and y coordinates as functions of time, $t$.

$v(t) = \frac{ds}{dt}$

$a(t) = \frac{dv}{dt}$

$\Delta s = s_{final} - s_{initial}$

$S = \int_a^b |\mathbf{r}'(t)| dt$

$S = \int_a^b \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} dt$

$\Delta \mathbf{r} = \int_{t_1}^{t_2} \mathbf{v}(t) dt$

$\sqrt{x^2 + y^2}$

$a(t) = \frac{d^2s}{dt^2}$

$\mathbf{v}(t) = \langle f'(t), g'(t) \rangle$

$\mathbf{a}(t) = \langle f'(t), g'(t) \rangle$

Location of an object at a given time, denoted as $s(t)$.

Rate of change of position with respect to time; $v(t) = \frac{ds}{dt}$.

Rate of change of velocity with respect to time; $a(t) = \frac{dv}{dt} = \frac{d^2s}{dt^2}$.

Change in position of an object; $\Delta s = s_{final} - s_{initial}$.

Total path length covered by an object.

A function represented as $\textbf{r}(t) = \langle f(t), g(t) \rangle$, defining an object's position at time $t$.

Functions where $x$ and $y$ are defined in terms of a parameter $t$: $x(t) = f(t), y(t) = g(t)$.

The length of a curve.

The derivative of the position vector $\mathbf{r}(t)$, which is the velocity vector.

The magnitude of the velocity vector, which is the speed.