Parametric Functions Modeling Planar Motion

A parametric function is a natural way to describe something moving around a flat surface: a puck on ice, a robot on a floor, a ball in flight seen from the side. The parameter is time, and the two coordinates tell you where the object is at each moment. This page shows how to read the motion, one direction at a time.

Position at time $t$

In planar motion an object's position at time $t$ is the point $(x(t), y(t))$. The function $x(t)$ tracks the horizontal position and $y(t)$ tracks the vertical position. The two can be described, graphed and analyzed separately, then put back together.

Combine one row from each pair. If $x(t)$ is decreasing and $y(t)$ is increasing, the particle moves left and up.

Turning points

Where $x(t)$ changes from increasing to decreasing, the particle stops going right and starts going left: that point is a relative maximum of $x(t)$, the particle's farthest right position nearby. In the same way a maximum of $y(t)$ is a highest point and a minimum is a lowest point.

A turning point of one coordinate is not a stop. At a maximum of $x(t)$, if $y(t)$ is still changing, the particle is moving straight up or down for that moment. It is at rest only when neither coordinate is changing.

Motion with a calculator: a thrown ball

Seen from the side, a ball thrown from a height of 1.5 meters can be modeled with $x(t) = 15t$ meters across and $y(t)$$= -4.9t^2 + 12t + 1.5$ meters up, $t$ seconds after the throw. The horizontal position grows steadily; the height is a parabola in $t$.

Two graphs and one path

Three pictures go with planar motion: the graph of $x(t)$ against $t$, the graph of $y(t)$ against $t$, and the path in the $xy$-plane. The first two tell you when each coordinate rises or falls, so they answer "which way is it moving?". The path answers "where does it go?". Read the direction questions from the graphs against $t$, then check your answer on the path.