Parametric motion

Some paths are easier to describe by giving each coordinate as a function of time. The parameter is the clock that moves one state around the curve.

Think like a programmer

A parametric path is a function that returns a vector for one input. Its derivative is another function that returns the tangent velocity. Keep both as pure functions so the scene only renders their result.

Model checklist

Inputs: Time and a circle radius in metres.
State: The current time parameter.
Rule: Evaluate sine and cosine for position; differentiate for tangent velocity.
Output: Point on the curve and velocity arrow.
Check: Position and velocity are perpendicular on a circle.

\[\\mathbf{r}(t)=\\begin{bmatrix}R\\cos t\\\\R\\sin t\\end{bmatrix},\\qquad\\mathbf{v}(t)=\\begin{bmatrix}-R\\sin t\\\\R\\cos t\\end{bmatrix}\]

Loading the interactive visual. The lesson text and model remain available while it starts.

Try this experiment

Prediction: The velocity arrow stays tangent to the circle and never points directly outward.

Move time through one turn. At several positions, compare the velocity arrow with a line from the circle center to the body.

Where this model breaks

This example uses a fixed radius and one radian per second. Real circular motion may have changing speed, changing radius, or forces that make the parameter no longer equal to clock time.

Summary

Use parametric functions when one scalar parameter naturally drives a multi-coordinate path. Derive velocity from the path and test its geometry.

Glossary

Parameter: input that selects a point on a curve.
Tangent: direction that touches a curve at one point.
Parametric equation: coordinates expressed as functions of a shared parameter.

Self-check

What input drives both coordinates?
Why is velocity tangent to this circle?
Which invariant can test the example?

Sources

OpenStax, University Physics Volume 1, motion in two dimensions chapter.

Model contract

Treat the lesson as a small function before treating it as a fact to memorize. Give every value a unit, keep only the state needed for the next step, and make the output easy to inspect.

\[\text{observable output} = f(\text{inputs},\,\text{state})\]

Inputs: Quantities you set or measure, with units and useful bounds.
State: Values the program must retain to reproduce the next result.
Rule: The relationship or update that turns inputs and state into a result.
Check: A known limit, unit check, invariant, or measured result that can expose a bad model.

Implement the idea as a model

For Parametric Motion, write down the quantities you can control, the values your program must retain, and the result a reader could inspect. In Motion in 2D and 3D, the useful program is not the drawing: it is the smallest explicit model that makes a prediction you can test.

Guided experiment

Prediction: changing one declared input while holding the others fixed should change only the outputs that the model connects to that input. Choose one input, predict the direction of change, then check a limiting case such as zero, a symmetric arrangement, or a familiar low-speed or small-change approximation.

Where this model breaks

This lesson is a teaching model, not a complete simulator. Before using it outside the stated question, check which interactions, scales, uncertainties, boundary conditions, and measurement limits it leaves out.

Summary

Treat Parametric Motion as a contract: named inputs and units enter a rule, the rule produces an observable result, and a known limit or invariant checks whether the implementation deserves trust.

Glossary

Input: a measured value or chosen parameter supplied to a model.
State: the smallest set of values needed to continue or reproduce a model.
Validation: comparing an output with a known result, limit, invariant, or measurement.

Self-check

Which values are inputs, and which values must remain state?
What observable result would tell you the model is behaving as expected?
Which assumption would you test first before applying the model to a real system?

Model review: turn Parametric Motion into a test

Scrub time along a circular parametric path and connect position, tangent velocity, and the parameter that drives both.

Name the inputs and units that the motion in 2d and 3d model needs.
Separate the state you must keep from values you can calculate when needed.
Write one rule that maps the current state and inputs to an observable result.
Choose a limiting case, unit check, invariant, or known result before trusting an output.
State one assumption you would change before using this simplified model for a real decision.

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