Precalculus Lesson

Decomposing Motion: Parametric Equations & Planar Tracking

Shared by A STEM Educator

11th GradeBTC Style / Thinking Classrooms50 min

Learning Objective

Students will interpret parametric equations as decomposed horizontal and vertical motion, plot oriented parametric curves, analyze domain restrictions imposed by the parameter, and eliminate the parameter algebraically and trigonometrically to convert to Cartesian form.

Lesson Overview

Students discover how continuous planar motion (projectile paths, roller coaster cars) can be decomposed into independent horizontal and vertical component functions of time, then work toward eliminating the parameter to reveal the underlying Cartesian relationship.

Materials

Vertical whiteboards or large paper taped to wall
Dry-erase markers (multiple colors)
Pencils and blank paper
Basic calculator or phone calculator

Scaffolded Task Progression

1Task 1

A roller coaster car's position at time

t

seconds is given by

x (t) = 3 t

and

y (t) = 2 t

, for

0 \leq t \leq 4

.

1. Make a table of values for

t = 0, 1, 2, 3, 4

.
2. Plot the five points

(x, y)

on your whiteboard axes.
3. Draw arrows on your path showing the direction of travel as

t

increases.

What shape is the path? What does the orientation tell you about the car's motion?

2Task 2

Same roller coaster, same equations:

x (t) = 3 t

y (t) = 2 t

.

Now eliminate the parameter

t

algebraically to write a single equation relating

x

and

y

.

Does your Cartesian equation match the path you drew? What information is lost when you eliminate

t

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