Math 2.0

Mechanics of particles and systems Definition

Elastic collision

Builds on MomentumKinetic energy

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 elastic 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 is conserved, 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.

Remarks

For motion on a line, the solution with (v1,v2)≠(u1,u2)(v_1, v_2) \ne (u_1, u_2) is v1=(m1−m2)u1+2m2u2m1+m2,v2=(m2−m1)u2+2m1u1m1+m2.v_1 = \frac{(m_1 - m_2) u_1 + 2 m_2 u_2}{m_1 + m_2}, \qquad v_2 = \frac{(m_2 - m_1) u_2 + 2 m_1 u_1}{m_1 + m_2}. Example: equal masses with u1=1u_1 = 1, u2=0u_2 = 0 give v1=0v_1 = 0, v2=1v_2 = 1; momentum 11 and kinetic energy 12m\frac12 m are unchanged. Non-example: equal masses with u1=1u_1 = 1, u2=0u_2 = 0 that stick together give v1=v2=12v_1 = v_2 = \frac12; momentum is conserved but the kinetic energy drops from 12m\frac12 m to 14m\frac14 m, so clause 2 fails. On a line, if clause 1 holds and (v1,v2)≠(u1,u2)(v_1, v_2) \ne (u_1, u_2), then clause 2 is equivalent to v2−v1=−(u2−u1)v_2 - v_1 = -(u_2 - u_1): the relative velocity is reversed.

Reading path

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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…definitionLipschitz condition, definition, no evidence levelLipschitz conditiondefinitionSolution of an ordinary differential equation, definition, no evidence levelSolution of an ordina…definitionInitial value problem, definition, no evidence levelInitial value problemdefinitionPicard–Lindelöf hypotheses, definition, no evidence levelPicard–Lindelöf hypot…definitionPicard 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 collisiondefinition
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  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. Lipschitz condition definition
  11. Solution of an ordinary differential equation definition
  12. Initial value problem definition
  13. Picard–Lindelöf hypotheses definition
  14. Picard operator definition
  15. Integral equation form of an initial value problem statement
  16. An iterate of the Picard operator is a contraction statement
  17. Picard–Lindelöf theorem statement
  18. Classical mechanics definition
  19. Momentum definition
  20. Kinetic energy definition
  21. Elastic collision definition

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Elastic_collision
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DefinesElastic collision
UsesMomentum, Kinetic energy
Changed3 November 2026

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