A family of equations that describe the motion of any object's [[Position]].
The case with constant [[Acceleration]]:
$\huge
\begin{align}
\mathbf{\vec x} (t) &=
\mathbf{\vec x}_{0} + \vec v(t) + \frac{1}{2} \vec a(t)t \\
\vec v(t) &= \vec v_{0} + \vec a_{0}t
\end{align}
$
>[!note] This is just a [[Maclaurin Series]] for [[Position]]
With non-constant acceleration, you simply pick a higher order [[Maclaurin Series]]:
$\huge
\begin{align}
\mathbf{\vec x} (t) &=
\mathbf{\vec x}_{0} + \vec v(t)t + \frac{1}{2} {\vec a(t)t^2}/
+ \frac{1}{3!}\vec j(t)t^2 + \cdots \\
&=
\mathbf{\vec x_{0}} +\sum^{\infty}_{n=1} \frac{ \mathrm{d}^{n}\mathbf{\vec x}}{\mathrm{d}t^n}(t) \frac{t^n}{n!}
\end{align}
$
To simplify, this is the case for where the $n^{th}$ [[../Math/Derivative|Derivative]] of [[Position]] is cosntant:
$\huge
\begin{align}
\mathbf{\vec x} (t) &=
\mathbf{\vec x_{0}} +\sum^{N}_{n=1} \frac{ \mathrm{d}^{n}\mathbf{\vec x}}{\mathrm{d}t^n}(t) \frac{t^n}{n!}
\end{align}
$