Math 2.0

Mechanics of particles and systems Variable mass systems and rockets Proposition

Tsiolkovsky rocket equation

Builds on Variable-mass systemConservation of momentum

Statement

Let a rocket move along a line with no external force. Assume that it ejects mass backwards at a constant speed ve>0v_e > 0 relative to itself. So it is a variable-mass system, and its mass decreases from m0m_0 to mfm_f, with m0>mf>0m_0 > m_f > 0. Then its change of velocity is Δv=veln⁡m0mf.\Delta v = v_e \ln \frac{m_0}{m_f}.

Remarks

The equation of motion of the variable-mass system is m dv/dt=−ve dm/dtm\, dv/dt = -v_e\, dm/dt, a consequence of conservation of momentum. So dv=−ve dm/mdv = -v_e\, dm / m, and integrating from m0m_0 to mfm_f gives the result. The result does not depend on how fast the fuel is burnt. Example: to reach Δv=ve\Delta v = v_e the mass ratio must be m0/mf=e≈2.72m_0 / m_f = e \approx 2.72; to reach Δv=3ve\Delta v = 3 v_e it must be e3≈20e^3 \approx 20. With gravity or drag the hypothesis of no external force fails, and the actual gain in speed is smaller, by the time integral of these forces divided by the mass. Tsiolkovsky published the equation in 1903.

Going deeper

For a student

The equation is the budget rule of spaceflight: every manoeuvre has a Δv\Delta v, computed with the vis-viva equation, and the sum fixes the needed mass ratio. For a multistage rocket, the Δv\Delta v of the stages add, which lets the total exceed what one stage of the same propellant could reach. It is the simplest application of the general equation m dv/dt=F+u dm/dtm\, dv/dt = F + u\, dm/dt of a variable-mass system, with F=0F = 0 and u=−veu = -v_e.

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…conceptVariable-mass system, concept, no evidence levelVariable-mass systemconceptVelocity, 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…statementTsiolkovsky rocket equation, statement, no evidence levelTsiolkovsky rocket eq…statement
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. Variable-mass system concept
  9. Velocity concept
  10. Ordinary differential equation definition
  11. Classical mechanics concept
  12. Lipschitz condition definition
  13. Momentum concept
  14. Inertial frame of reference concept
  15. Solution of an ordinary differential equation definition
  16. Newton's laws of motion statement
  17. Initial value problem definition
  18. Picard–Lindelöf hypotheses definition
  19. Picard operator definition
  20. An iterate of the Picard operator is a contraction statement
  21. Integral equation form of an initial value problem statement
  22. Picard–Lindelöf theorem statement
  23. Classical mechanics definition
  24. Momentum definition
  25. Conservation of momentum statement
  26. Tsiolkovsky rocket equation statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Tsiolkovsky_rocket_equation
About this page
TagFN7ER4VE95
URIm2://FN7ER4VE95
Kindstatement, proposition
WritingW0 stub
MSC70F99, 70M20
StatusThis copy of the site has no build data. Run m2 status FN7ER4VE95.
AboutVariable-mass system
UsesVariable-mass system, Conservation of momentum
Same as
Changed3 November 2026

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