Math 2.0

Mechanics of particles and systems Definition

Inelastic collision

Builds on MomentumKinetic energyElastic collisionReduced mass

Definition

Let two particles of masses m1,m2>0m_1, m_2 > 0 have velocities u1,u2∈ℝ3u_1, u_2 \in \R^3 before a collision and v1,v2∈ℝ3v_1, v_2 \in \R^3 after it. The collision is inelastic if both of the following hold:

  1. the total momentum is conserved, m1v1+m2v2=m1u1+m2u2m_1 v_1 + m_2 v_2 = m_1 u_1 + m_2 u_2;
  2. the total kinetic energy decreases strictly, 12m1∣v1∣2+12m2∣v2∣2<12m1∣u1∣2+12m2∣u2∣2\frac12 m_1 \abs{v_1}^2 + \frac12 m_2 \abs{v_2}^2 < \frac12 m_1 \abs{u_1}^2 + \frac12 m_2 \abs{u_2}^2.

The collision is perfectly inelastic if moreover v1=v2v_1 = v_2.

Remarks

In a perfectly inelastic collision, v1=v2=(m1u1+m2u2)/(m1+m2)v_1 = v_2 = (m_1 u_1 + m_2 u_2)/(m_1 + m_2). The kinetic energy lost is 12μ∣u1−u2∣2\frac12 \mu \abs{u_1 - u_2}^2, where μ=m1m2/(m1+m2)\mu = m_1 m_2/(m_1 + m_2) is the reduced mass; this is the largest loss compatible with clause 1. Example: m1=m2=1m_1 = m_2 = 1, u1=2u_1 = 2, u2=0u_2 = 0 and v1=v2=1v_1 = v_2 = 1 on a line; momentum stays 22 and kinetic energy drops from 22 to 11. Non-example: with the same data, v1=0v_1 = 0 and v2=2v_2 = 2 keep the kinetic energy at 22, so clause 2 fails; this collision is elastic.

Reading path

To understand this page from the ground up, read these 27 pages in order. The foundations come first.

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Derivative, concept, no evidence levelDerivativeconceptDifferential equation, concept, no evidence levelDifferential equationconceptEuclidean space, object, no evidence levelEuclidean spaceobjectEuclidean vector, concept, no evidence levelEuclidean vectorconceptFunction, concept, no evidence levelFunctionconceptInterval, concept, no evidence levelIntervalconceptOrdinary differential equation, concept, no evidence levelOrdinary differential…conceptVelocity, concept, no evidence levelVelocityconceptOrdinary differential equation, definition, no evidence levelOrdinary differential…definitionFunction of several variables, concept, no evidence levelFunction of several v…conceptLipschitz condition, definition, no evidence levelLipschitz conditiondefinitionDouble integral, concept, no evidence levelDouble integralconceptSolution of an ordinary differential equation, definition, no evidence levelSolution of an ordina…definitionInitial value problem, definition, no evidence levelInitial value problemdefinitionTriple integral, concept, no evidence levelTriple integralconceptCentre of mass, concept, no evidence levelCentre of massconceptPicard–Lindelöf hypotheses, definition, no evidence levelPicard–Lindelöf hypot…definitionReduced mass, concept, no evidence levelReduced massconceptPicard operator, definition, no evidence levelPicard operatordefinitionIntegral equation form of an initial value problem, statement, no evidence levelIntegral equation for…statementAn iterate of the Picard operator is a contraction, statement, no evidence levelAn iterate of the Pic…statementPicard–Lindelöf theorem, statement, no evidence levelPicard–Lindelöf theor…statementClassical mechanics, definition, no evidence levelClassical mechanicsdefinitionMomentum, definition, no evidence levelMomentumdefinitionKinetic energy, definition, no evidence levelKinetic energydefinitionElastic collision, definition, no evidence levelElastic collisiondefinitionInelastic collision, definition, no evidence levelInelastic collisiondefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Derivative concept
  2. Differential equation concept
  3. Euclidean space object
  4. Euclidean vector concept
  5. Function concept
  6. Interval concept
  7. Ordinary differential equation concept
  8. Velocity concept
  9. Ordinary differential equation definition
  10. Function of several variables concept
  11. Lipschitz condition definition
  12. Double integral concept
  13. Solution of an ordinary differential equation definition
  14. Initial value problem definition
  15. Triple integral concept
  16. Centre of mass concept
  17. Picard–Lindelöf hypotheses definition
  18. Reduced mass concept
  19. Picard operator definition
  20. Integral equation form of an initial value problem statement
  21. An iterate of the Picard operator is a contraction statement
  22. Picard–Lindelöf theorem statement
  23. Classical mechanics definition
  24. Momentum definition
  25. Kinetic energy definition
  26. Elastic collision definition
  27. Inelastic collision definition

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Inelastic_collision
About this page
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MSC70F35, 70F10
DefinesInelastic collision
UsesMomentum, Kinetic energy, Elastic collision, Reduced mass
Changed3 November 2026

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