Math 2.0

Mechanics of particles and systems Definition

Coefficient of restitution

Builds on Elastic collisionInelastic collisionMomentum

Definition

Let two particles of masses m1,m2>0m_1, m_2 > 0 move on a line and collide. They have velocities u1,u2∈ℝu_1, u_2 \in \R before the collision and v1,v2∈ℝv_1, v_2 \in \R after it, where u1>u2u_1 > u_2 so that they approach. The coefficient of restitution of the collision is e=v2−v1u1−u2,e = \frac{v_2 - v_1}{u_1 - u_2}, the ratio of the speed of separation to the speed of approach.

Remarks

If momentum is conserved, the kinetic energy lost equals 12μ(1−e2)(u1−u2)2\frac12 \mu (1 - e^2)(u_1 - u_2)^2, where μ=m1m2/(m1+m2)\mu = m_1 m_2/(m_1 + m_2). So e=1e = 1 for an elastic collision, e=0e = 0 for a perfectly inelastic collision, and 0≤e<10 \le e < 1 for other collisions that lose energy without the bodies passing through each other. Example: a ball dropped from height hh hits the floor at speed 2gh\sqrt{2gh}; if it rises to height h′h', then e=h′/he = \sqrt{h'/h}, so h=1h = 1 and h′=0.64h' = 0.64 give e=0.8e = 0.8. Non-example: if u1≤u2u_1 \le u_2, the particles do not approach, the denominator is not positive, and the quantity is not a coefficient of restitution. For oblique collisions, ee is defined with the components of the velocities along the line of contact.

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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 collisiondefinitionCoefficient of restitution, definition, no evidence levelCoefficient of restit…definition
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
  28. Coefficient of restitution definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Coefficient_of_restitution
About this page
TagK2QQ37JBS1
URIm2://K2QQ37JBS1
Kinddefinition
WritingW0 stub
MSC70F35, 70F10
DefinesCoefficient of restitution
UsesElastic collision, Inelastic collision, Momentum
Changed3 November 2026

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