Parametric Equations: Projectile Motion

Parametric equations express both $x$ and $y$ as functions of a third variable — a parameter, usually time $t$. Instead of $y = f(x)$, we have $x = f(t)$ and $y = g(t)$. As $t$ advances, the point $(x(t), y(t))$ traces a curve in the $xy$-plane.

Projectile motion is the classic example: $x(t) = v_{0}\cos\theta \cdot t$ (constant horizontal velocity) and $y(t) = v_{0}\sin\theta \cdot t - \frac{1}{2}gt^{2}$ (vertical: upward velocity minus gravity).

With $v_{0} = 20$ m/s and $\theta = 45°$: $x(t) = 14.14t$, $y(t) = 14.14t - 4.9t^{2}$. The projectile lands at $t = 2.89$ s, having traveled $x = 40.8$ m.

Why parametric?

The parametric form captures when the projectile is at each position, not just where the curve goes. Two projectiles can follow the same parabolic path at different speeds — same curve, different parametrizations.

Try it in the visualization

Watch the projectile trace its parabolic arc as $t$ advances. The two smaller graphs show $x(t)$ (linear) and $y(t)$ (quadratic) separately. Adjust initial speed and angle to change the trajectory. The time slider lets you freeze at any instant.