Elastic and inelastic collisions

In an isolated collision, momentum is conserved. An elastic collision also conserves kinetic energy; an inelastic collision converts some kinetic energy into deformation, heat, sound, internal modes, or an explicitly modeled loss reservoir. Collision type is a model choice with testable invariants, not a visual impression from a smooth bounce.

Think like a programmer

Represent each body with mass, velocity, position/contact geometry when relevant, and a system-boundary convention. Calculate total momentum and kinetic energy before and after the collision from the same state data. Keep the collision rule, contact normal, restitution/material policy, external impulse, and timestep/contact-resolution policy explicit. Return an impulse or transfer ledger alongside outgoing velocities so a later pressure, energy, or momentum reducer does not guess what happened.

Model checklist

Inputs: Body masses, incoming velocities, contact normal/geometry, collision law or restitution, external impulse policy, rotational/deformation state policy, time/contact resolution, and units.
State: Pre/post momentum vectors, kinetic energies, contact impulse, relative normal velocity, loss/internal-energy ledger, and collision validity flags.
Rule: Resolve contact in the normal direction under the stated rule; apply equal and opposite impulse to bodies; update velocity and named transfer stores consistently.
Output: Outgoing velocities, impulse, momentum residual, kinetic-energy change, elastic/inelastic classification, and contact diagnostics.
Check: Isolated pair total momentum is conserved; elastic fixture preserves kinetic energy; perfectly inelastic fixture has shared outgoing velocity and non-increasing kinetic energy; reversing the contact normal changes signed impulse convention consistently; zero relative normal approach produces no impact impulse under the chosen policy; external impulse appears in the system ledger.

\[\mathbf p=\sum_i m_i\mathbf v_i,\qquad K=\sum_i\frac12m_i\lVert\mathbf v_i\rVert^2,\qquad \mathbf J_1=-\mathbf J_2\]

Elastic collision check

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

Incoming velocity: 4 m/sIncoming mass: 2 kg

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

Momentum and kinetic energy answer different questions. A perfectly inelastic collision can conserve total momentum while reducing translational kinetic energy. In an isolated model that reduction must be named as another store or a deliberate unresolved loss; it is not a momentum failure. Conversely, matching energy alone does not prove a collision rule is correct if momentum or contact geometry is wrong.

The one-dimensional direct-collision formula is a special fixture. In two or three dimensions, decompose relative velocity into contact-normal and tangential components. Friction, spin, contact duration, penetration handling, and shape affect the result. A coefficient of restitution is a simplified normal-velocity relation, not a full material model or a guarantee of energy conservation across every mode.

Try this experiment

Prediction: The displayed elastic calculation preserves total momentum for every valid mass and velocity, while an inelastic rule can preserve momentum but lower kinetic energy.

Change masses and incoming velocities. Record momentum and kinetic energy before/after for an elastic fixture. Then define a perfectly inelastic shared-velocity case and identify the missing translational energy as a named internal/deformation transfer. Add a hypothetical external wall impulse and explain why a two-body-only momentum test must then change its system boundary or expected ledger.

Where this model breaks

The one-dimensional elastic formula assumes a direct collision with no external impulse. Oblique 2D/3D collisions need normal directions, contact geometry, friction, rotation, and deformation energy. Finite contact time, soft materials, plasticity, adhesion, sound, numerical penetration correction, and multiple simultaneous contacts require richer models; a restitution coefficient alone is not enough.

Summary

Check momentum for an explicitly isolated system, then use a separate kinetic-energy and transfer ledger before claiming elasticity. Preserve contact geometry, impulse signs, and collision-law scope instead of inferring physics from a bounce animation.

Glossary

Elastic collision: collision conserving kinetic energy and momentum.
Inelastic collision: collision conserving momentum but not all kinetic energy.
Contact normal: direction perpendicular to a contact surface.
Impulse: time-integrated contact force that changes momentum.
Restitution: simplified rule for relative normal rebound speed.

Self-check

Which quantity is conserved in both isolated collision types?
What extra check identifies an elastic fixture?
Why can momentum conservation coexist with lost translational kinetic energy?
Why is this model one-dimensional?

Sources

OpenStax, University Physics Volume 1, collisions chapter.
J. B. Marion and S. T. Thornton, Classical Dynamics of Particles and Systems.

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 Elastic and Inelastic Collisions, 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 Elastic and Inelastic Collisions 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 Elastic and Inelastic Collisions into a test

Compare collision outcomes while checking momentum conservation and distinguishing kinetic-energy conservation from momentum conservation.

Name the inputs and units that the momentum and collisions 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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