3.1.2—Projectile motion

Eliminating time from the component equations gives y=x tanθ−gx²/(2u²cos²θ), a quadratic trajectory when gravity is uniform and air resistance is neglected.

Use the equation to find height, range or intersection with a target, but check that the chosen root corresponds to a future time and that the launch/landing geometry matches the question.

At a fixed horizontal distance, the quadratic may give two launch angles: a low path and a high path. Both can reach the point, but they have different flight times and maximum heights.

A parabolic path is an idealisation; drag, varying gravity or wind changes it, and an algebraic x-root is not automatically a physically valid time.