Derive the equations of motion for a body moving in two dimensions: $\vec{v} = \vec{v_0} + \vec{a}t$ and $\vec{r} = \vec{r_0} + \vec{v_0}t + \frac{1}{2}\vec{a}t^2$.

Consider a particle moving in a plane with constant acceleration $\vec{a}$. At time $t=0$,the velocity is initial velocity $\vec{v_0}$ and the position is $\vec{r_0}$. At time $t=t$,the velocity is $\vec{v}$ and the position is $\vec{r}$.
$1$. Derivation of $\vec{v} = \vec{v_0} + \vec{a}t$:
Since the acceleration is constant,the instantaneous acceleration is given by:
$\vec{a} = \frac{\vec{v} - \vec{v_0}}{t - 0}$
$\vec{a} = \frac{\vec{v} - \vec{v_0}}{t}$
$\vec{v} = \vec{v_0} + \vec{a}t$
In component form:
$v_x = v_{0x} + a_x t$
$v_y = v_{0y} + a_y t$
$2$. Derivation of $\vec{r} = \vec{r_0} + \vec{v_0}t + \frac{1}{2}\vec{a}t^2$:
For constant acceleration,the average velocity is $\vec{v}_{avg} = \frac{\vec{v} + \vec{v_0}}{2}$.
The displacement is $\vec{r} - \vec{r_0} = \vec{v}_{avg} \cdot t = \left( \frac{\vec{v} + \vec{v_0}}{2} \right) t$.
Substituting $\vec{v} = \vec{v_0} + \vec{a}t$:
$\vec{r} - \vec{r_0} = \left( \frac{\vec{v_0} + \vec{a}t + \vec{v_0}}{2} \right) t$
$\vec{r} - \vec{r_0} = \left( \frac{2\vec{v_0} + \vec{a}t}{2} \right) t$
$\vec{r} = \vec{r_0} + \vec{v_0}t + \frac{1}{2}\vec{a}t^2$.