Independent Motion Along x and y Axes

Write the Cartesian position vector $\vec{\mathbf{r}}(t) = x(t)\hat{\mathbf{i}} + y(t)\hat{\mathbf{j}}$ and record that each direction is then handled on its own.

State $v_x(t) = \frac{dx}{dt}$ and $v_y(t) = \frac{dy}{dt}$, and assemble $\vec{\mathbf{v}}(t) = v_x\hat{\mathbf{i}} + v_y\hat{\mathbf{j}}$ as the derivative of the position vector, in $\text{m}\cdot\text{s}^{-1}$.

Write $\vec{\mathbf{a}}(t) = a_x\hat{\mathbf{i}} + a_y\hat{\mathbf{j}}$ with $a_x = \frac{dv_x}{dt}$ and $a_y = \frac{dv_y}{dt}$, carrying units of $\text{m}\cdot\text{s}^{-2}$.