Two-dimensional motion is analyzed by splitting vectors into $x$- and $y$-components. If a velocity vector has magnitude $v$ and angle $\theta$, then $v_x = v\cos\theta$ and $v_y = v\sin\theta$.

In the absence of air resistance, projectile motion separates into independent horizontal and vertical motions. Horizontal acceleration is $0$, while vertical acceleration is $-g$, where $g \approx 9.8\,\text{m/s}^2$ near Earth’s surface.

For a projectile launched from $y_0$ with initial speed $v_0$ at angle $\theta$, the position functions are $x(t) = x_0 + v_0\cos\theta\, t$ and $y(t) = y_0 + v_0\sin\theta\, t - \tfrac{1}{2}gt^2$. The trajectory is parabolic because $x$ changes linearly in time while $y$ changes quadratically.