We need a general approach if we want to obtain exact solutions.
One-dimensional models are useful in cases where we have symmetries to exploit, but we need to rewrite Newton's Equation using vector calculus in order to encode directional information for general cases.
$$ \begin{align} \vect{F} &= m \frac{dv}{dt} \\ &= m \frac{d \vect{v}}{d \vect{x}} \cancelto{v(t)}{\frac{dx}{dt}} \\ &= m \frac{d\vect{v}}{dx} \vect{v(t)} \\ &= m \frac{d\vect{v}}{dx} \frac{d ( \frac{\vect{v}^2}{2})}{d\vect{v}} \\ \vect{F} &= \frac{1}{2} m \frac{d}{dx} \left[\vect{v}^2\right] \end{align}$$