Math 2.0

Mechanics of particles and systems Definition

Variable-mass system

Builds on Newton's laws of motionMomentumConservation of momentum

Definition

A variable-mass system is a body, treated as a point particle, whose mass m(t)>0m(t) > 0 changes with time because matter leaves it or joins it. Let v(t)v(t) be its velocity and let FF be the external force on it. Let u(t)u(t) be the velocity, relative to the body, of the matter that leaves or joins it at time tt. Its equation of motion is mdvdt=F+udmdt.m \frac{dv}{dt} = F + u \frac{dm}{dt}.

Remarks

The equation follows from conservation of momentum for the body together with the mass ∣dm∣\abs{dm} that it exchanges in a short time dtdt. The total momentum changes by F dtF\, dt, as in Newton's second law. The term u dm/dtu\, dm/dt is the thrust.

Example: a rocket ejects gas backwards at speed vev_e relative to itself, so u=−ve v^u = -v_e\, \hat{v} for the unit vector v^\hat v in the direction of motion, and dm/dt<0dm/dt < 0. The thrust −ve v^ dm/dt-v_e\, \hat{v}\, dm/dt then points forwards.

Non-example: a closed system made of a rocket together with all its exhaust has constant mass, so it is not a variable-mass system; its total momentum changes only by FF.

The naive formula F=d(mv)/dtF = d(mv)/dt is wrong in general, since it depends on the choice of inertial frame. It is right only when u=−vu = -v, that is, when the exchanged matter is at rest.

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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…definitionClassical mechanics, concept, no evidence levelClassical mechanicsconceptLipschitz condition, definition, no evidence levelLipschitz conditiondefinitionMomentum, concept, no evidence levelMomentumconceptInertial frame of reference, concept, no evidence levelInertial frame of ref…conceptSolution of an ordinary differential equation, definition, no evidence levelSolution of an ordina…definitionNewton's laws of motion, statement, no evidence levelNewton's laws of moti…statementInitial value problem, definition, no evidence levelInitial value problemdefinitionPicard–Lindelöf hypotheses, definition, no evidence levelPicard–Lindelöf hypot…definitionPicard operator, definition, no evidence levelPicard operatordefinitionAn iterate of the Picard operator is a contraction, statement, no evidence levelAn iterate of the Pic…statementIntegral equation form of an initial value problem, statement, no evidence levelIntegral equation for…statementPicard–Lindelöf theorem, statement, no evidence levelPicard–Lindelöf theor…statementClassical mechanics, definition, no evidence levelClassical mechanicsdefinitionMomentum, definition, no evidence levelMomentumdefinitionConservation of momentum, statement, no evidence levelConservation of momen…statementVariable-mass system, definition, no evidence levelVariable-mass systemdefinition
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. Classical mechanics concept
  11. Lipschitz condition definition
  12. Momentum concept
  13. Inertial frame of reference concept
  14. Solution of an ordinary differential equation definition
  15. Newton's laws of motion statement
  16. Initial value problem definition
  17. Picard–Lindelöf hypotheses definition
  18. Picard operator definition
  19. An iterate of the Picard operator is a contraction statement
  20. Integral equation form of an initial value problem statement
  21. Picard–Lindelöf theorem statement
  22. Classical mechanics definition
  23. Momentum definition
  24. Conservation of momentum statement
  25. Variable-mass system definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Variable-mass_system
About this page
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MSC70F99, 70M20
DefinesVariable-mass system
UsesNewton's laws of motion, Momentum, Conservation of momentum
Changed3 November 2026

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