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Constant Acceleration

Kinematics deals primarily with situations where the acceleration of an object is constant, simplifying the analysis.
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The statement of the theorem

Let r(t)∈R3\mathbf{r}(t) \in \mathbb{R}^3 be the position vector of a particle at time t∈Rt \in \mathbb{R}. Define the acceleration vector a(t)\mathbf{a}(t) such that a(t)=a0\mathbf{a}(t) = \mathbf{a}_0, where a0∈R3\mathbf{a}_0 \in \mathbb{R}^3 is a constant vector. The velocity vector v(t)\mathbf{v}(t) and the position vector r(t)\mathbf{r}(t) are then determined by the following differential equations and initial conditions:\\begin{align*} \frac{d^2\mathbf{r}}{dt^2} &= \mathbf{a}_0 \\ \frac{d\mathbf{r}}{dt} &= \mathbf{v}(t) \\ \mathbf{r}(0) &= \mathbf{r}_0 \\ \mathbf{v}(0) &= \mathbf{v}_0 \end{align*}\newline\text{The unique solution to this system is given by the explicit kinematic equations:}\newline\mathbf{r}(t) &= \mathbf{r}_0 + \mathbf{v}_0 t + \frac{1}{2} \mathbf{a}_0 t^2 \\ \mathbf{v}(t) &= \mathbf{v}_0 + \mathbf{a}_0 t