Math 2.0

Mechanics of particles and systems Definition

Impulse

Builds on Riemann integralMomentumNewton's laws of motion

Definition

Let F ⁣:[t0,t1]→ℝ3F \colon [t_0, t_1] \to \R^3 be an integrable force acting on a particle during the time interval [t0,t1][t_0, t_1]. The impulse of FF over [t0,t1][t_0, t_1] is the vector J=∫t0t1F(t) dt,J = \int_{t_0}^{t_1} F(t)\, dt, where the integral is taken component by component.

Remarks

If FF is the net force on a particle of constant mass, Newton's second law gives J=p(t1)−p(t0)J = p(t_1) - p(t_0), the change of momentum. Example: the constant force F=(10,0,0)F = (10, 0, 0) over [0,0.5][0, 0.5] has impulse J=(5,0,0)J = (5, 0, 0). For a particle of mass 11 at rest at t=0t = 0, its velocity at t=0.5t = 0.5 is then (5,0,0)(5, 0, 0). Non-example: the line integral ∫F⋅dx\int F \cdot dx along the path is a scalar, the work, and not the impulse; it integrates over the path rather than over time. The unit of impulse is the newton-second, the same as the unit of momentum.

Reading path

To understand this page from the ground up, read these 29 pages in order. The foundations come first.

Show the path as a graph
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…conceptPartition of an interval, definition, no evidence levelPartition of an inter…definitionVelocity, concept, no evidence levelVelocityconceptOrdinary differential equation, definition, no evidence levelOrdinary differential…definitionClassical mechanics, concept, no evidence levelClassical mechanicsconceptLipschitz condition, definition, no evidence levelLipschitz conditiondefinitionMesh of a partition, definition, no evidence levelMesh of a partitiondefinitionRiemann sum, definition, no evidence levelRiemann sumdefinitionMomentum, 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…definitionRiemann integrable function, definition, no evidence levelRiemann integrable fu…definitionNewton's laws of motion, statement, no evidence levelNewton's laws of moti…statementInitial value problem, definition, no evidence levelInitial value problemdefinitionRiemann integral, definition, no evidence levelRiemann integraldefinitionPicard–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 levelMomentumdefinitionImpulse, definition, no evidence levelImpulsedefinition
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. Partition of an interval definition
  9. Velocity concept
  10. Ordinary differential equation definition
  11. Classical mechanics concept
  12. Lipschitz condition definition
  13. Mesh of a partition definition
  14. Riemann sum definition
  15. Momentum concept
  16. Inertial frame of reference concept
  17. Solution of an ordinary differential equation definition
  18. Riemann integrable function definition
  19. Newton's laws of motion statement
  20. Initial value problem definition
  21. Riemann integral definition
  22. Picard–Lindelöf hypotheses definition
  23. Picard operator definition
  24. An iterate of the Picard operator is a contraction statement
  25. Integral equation form of an initial value problem statement
  26. Picard–Lindelöf theorem statement
  27. Classical mechanics definition
  28. Momentum definition
  29. Impulse definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Impulse_(physics)
About this page
TagGT081TAH8C
URIm2://GT081TAH8C
Kinddefinition
WritingW0 stub
MSC70A05, 70F10
DefinesImpulse
UsesRiemann integral, Momentum, Newton's laws of motion
Changed3 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.