Math 2.0

Mechanics of particles and systems Definition

Kinetic energy

Builds on VelocityMomentum

Definition

Let a particle of mass m>0m > 0 have velocity x˙(t)∈ℝ3\dot x(t) \in \R^3 at time tt. Its kinetic energy at time tt is T(t)=12 m∣x˙(t)∣2,T(t) = \tfrac12\, m \abs{\dot x(t)}^2, where ∣⋅∣\abs{\cdot} is the Euclidean norm. For NN particles with masses mim_i and velocities x˙i\dot x_i, the kinetic energy is T=∑i=1N12mi∣x˙i∣2T = \sum_{i=1}^{N} \tfrac12 m_i \abs{\dot x_i}^2.

Remarks

Example: a particle of mass 22 with velocity (1,2,2)(1, 2, 2) has ∣x˙∣2=9\abs{\dot x}^2 = 9 and kinetic energy T=9T = 9. Non-example: the quantity m x˙m\, \dot x is a vector, the momentum pp, and not the kinetic energy. In terms of momentum, T=∣p∣2/(2m)T = \abs{p}^2/(2m). For a system with total mass MM and centre-of-mass velocity VV, the kinetic energy splits as T=12M∣V∣2+∑i12mi∣x˙i−V∣2T = \frac12 M \abs{V}^2 + \sum_i \frac12 m_i \abs{\dot x_i - V}^2.

Reading path

To understand this page from the ground up, read these 20 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…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 energydefinition
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.
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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

Where this leads

34 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Kinetic_energy
About this page
Tag4NMMPYCK2C
URIm2://4NMMPYCK2C
Kinddefinition
WritingW0 stub
MSC70A05, 70F10
DefinesKinetic energy
UsesVelocity, Momentum
Changed3 November 2026

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