Velocity and Acceleration with Parametric Position Functions

Velocity and acceleration follow the same conceptual framework as in one dimension. For instance, assume that said two dimensions are the x and y axes. Then

$v_y(t) = \frac{d}{dt}(y(t))$

$v_x(t) = \frac{d}{dt}(x(t))$

Similarly, acceleration works in much the same way:

$a_y(t) = \frac{d}{dt}(v_y(t)) = \frac{d}{dt}\left(\frac{d}{dt}(y(t))\right)$

$a_x(t) = \frac{d}{dt}(v_x(t)) = \frac{d}{dt}\left(\frac{d}{dt}(x(t))\right)$

Furthermore, as velocity and acceleration are vectors, the magnitudes of the velocity and acceleration can be found by means of the Pythagorean Theorem:

$|v(t)| = \sqrt{v_x(t)^2 + v_y(t)^2}$

$|a(t)| = \sqrt{a_x(t)^2 + a_y(t)^2}$

In summary, if you know the position vectors in both dimensions, you can find the velocity and acceleration in both dimensions, as well as the overall magnitude of each.