Curves Defined by Parametric Equations

A baseball hit into the air follows a parabolic trajectory when air resistance is neglected. The motion can be described within a two-dimensional coordinate system, where both the horizontal displacement and vertical height are functions of time. Instead of expressing the trajectory as a single function of position, the motion is modeled using parametric equations: one function for the horizontal position and another for the vertical position as time progresses. Let the horizontal position be defined as f(t) and the vertical position as g(t); these functions represent the x- and y-coordinates of the baseball at any time t.

The use of parametric equations allows a complete description of the baseball's trajectory. The function f(t) gives the horizontal position based on the constant horizontal velocity, while g(t) incorporates both the initial vertical velocity and the downward acceleration due to gravity. This representation is essential because it accurately accounts for the baseball's position at each moment, resolving the ambiguity present in a single height corresponding to two different times—once during ascent and again during descent.

In the context of space alone, the baseball’s path appears parabolic; however, when represented parametrically, each point on the trajectory corresponds to a unique time value. So, the pair (f(t), g(t)) provides a one-to-one mapping between time and position in two-dimensional space. Although this method does not yield a single function y = f(x), it ensures a more complete and time-specific analysis of the motion, which is crucial for applications requiring precise tracking of position over time.