Applications

A common application of parametric equations involves modeling the path or trajectory of projectile motion. Projectile motion is a term that describes the physical behavior of an object that is hit, thrown, shot, or launched and then allowed to travel without any additional forces impacting it (except gravity). The path that a projectile follows from its initial position until it impacts the ground will be shaped like a parabola opening downward.

For an object hit or thrown from a starting height \(h_{0}\) with an initial speed \(v_{0}\) at an initial angle of elevation \(\theta\), its horizontal distance \(x\) and height \(y\) as functions of time \(t\) can be described by the following parametric equations. Note that distance is measured in feet and time is measured in seconds.

\[x = \left(v_{0}\cos(\theta)\right)t, y = -16t^2 + \left(v_{0}\sin(\theta)\right)t + h_{0}\]

Let's look at an example.

Parametric Equations

Applications