Motion with constraints

Some values are not physically allowed. A bead on a short rail cannot move past either stop, so the model needs a boundary rule as well as its ordinary motion rule.

Think like a programmer

A constraint is an invariant enforced after a proposed update. It is not just form validation: the correction represents an interaction with the modeled boundary.

Model checklist

Inputs: Proposed position and allowed lower and upper bounds in metres.
State: Current position on the segment.
Rule: Clamp the proposal to the allowed interval.
Output: Allowed position and boundary correction.
Check: The result always lies between both bounds.

const constrained = Math.max(lowerBound, Math.min(upperBound, proposedPosition));

A constrained position

Ask for a position outside a line segment and compare it with the allowed state.

Requested position: 12.0 m

0 m10 m

Requested 12.0 m; constrained position 10.0 m; boundary correction -2.0 m.

Try this experiment

Prediction: Requesting a position above 10 m leaves the object at 10 m.

Move the requested position across both stops. Identify the requested state, the corrected state, and the invariant the code guarantees.

Where this model breaks

Clamping position is a teaching simplification. A collision model also needs velocity, time of impact, restitution, and sometimes a reaction force to conserve the right quantities.

Summary

Make constraints explicit in the model, test the allowed range, and explain the physical interaction hidden by a simple clamp.

Glossary

Constraint: a rule limiting allowed states.
Invariant: a property that must remain true after updates.
Reaction: the effect of a boundary enforcing a constraint.

Self-check

What invariant does the clamp enforce?
Why is a constraint more than UI validation?
Which motion value is missing from this simplified collision?

Sources

D. E. Stewart, Rigid-Body Dynamics with Friction and Impact.

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 Motion with Constraints, write down the quantities you can control, the values your program must retain, and the result a reader could inspect. In One-Dimensional Motion, 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 Motion with Constraints 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 Motion with Constraints into a test

Constrain a moving state to an allowed line segment and distinguish a boundary rule from ordinary motion.

Name the inputs and units that the one-dimensional motion 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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