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Linear momentum as state

Momentum combines mass and velocity into one vector quantity. It is especially useful when several bodies interact because an isolated system's total momentum is conserved.

Think like a programmer

Momentum is state you can sum across objects. Its conservation is a strong integration test because individual velocities can change dramatically while the system total remains stable.

Model checklist

Inputs
Masses in kilograms and velocity vectors in m/s.
State
Each body's momentum.
Rule
Multiply mass by velocity and sum all bodies.
Output
System momentum in kg·m/s.
Check
Total momentum before and after an isolated interaction agrees.
\[\\mathbf{p}=m\\mathbf{v}\]

Elastic collision check

Change the incoming body. The collision model calculates outgoing velocities and checks total momentum before and after.

Momentum before 8.00 kg·m/s; after 8.00 kg·m/s. Outgoing velocities: 1.33 and 5.33 m/s.

Try this experiment

Prediction: Changing the incoming mass changes outgoing velocities, but not the total momentum.

Change mass and velocity. Compare the before/after status values and identify which quantity is the system-level invariant.

Where this model breaks

Momentum conservation requires an isolated system or accounting for external impulse. A track, wall, or frictional surface can transfer momentum to objects outside the chosen boundary.

Summary

Store momentum as vector state when interactions matter. Sum it across the declared system and use conservation as a test rather than a visual guess.

Glossary

Self-check

  1. What are the units of momentum?
  2. Which total stays constant in an isolated collision?
  3. What external effect can break the test?

Sources

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 Linear Momentum as State, write down the quantities you can control, the values your program must retain, and the result a reader could inspect. In Momentum and Collisions, 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 Linear Momentum as State 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

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

Model review: turn Linear Momentum as State into a test

Represent momentum as mass times velocity and use total momentum as a testable property of an isolated system.

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