4.2 Parametric Functions Modeling Planar Motion

Parametric motion basics

A parametric function gives x(t) and y(t) separately in terms of a parameter t, often representing time. The point (x(t), y(t)) traces a path in the plane as t increases. This is useful for planar motion, where an object's horizontal and vertical positions change independently but simultaneously.

Speed and direction from rates

The instantaneous rate of change of x and y with respect to t describe horizontal and vertical velocity. The object's overall speed is found using the distance formula applied to changes in x and y over a small time interval: √((Δx)² + (Δy)²)/Δt. Direction of motion relates to the ratio Δy/Δx.

Analyzing position over intervals

Average rate of change of x(t) or y(t) over [t₁, t₂] gives average horizontal or vertical velocity. Comparing positions at different t-values reveals whether motion is moving toward or away from a point, speeding up, or reversing direction — key for interpreting planar motion models in context.

Key terms

Parametric functions

Equations x(t) and y(t) that define position in the plane in terms of a parameter t, often time.

Path/curve of motion

The set of all points (x(t), y(t)) traced as t varies over its domain.

Average speed (parametric)

Distance between two positions divided by the elapsed time: √((Δx)² + (Δy)²)/Δt.

Independent x and y variation

Key advantage of parametric equations: x(t) and y(t) can change independently, allowing curves that fail the vertical line test.

Direction of motion

Determined by whether x(t) and y(t) are increasing or decreasing over an interval (e.g., increasing x, decreasing y = moving right and down).

Try an AP-style question

x(t) = t² − 4t. For t ≥ 0, when is the particle moving left?

1. t > 4
2. 0 ≤ t < 2
3. t > 2
4. 0 < t < 4