Math 2.0

Mechanics of particles and systems Definition

Escape velocity

Builds on Newton's law of universal gravitationKinetic energyPotential energyConservation of energy

Definition

Let a body of mass M>0M > 0 be spherically symmetric, fixed, and the only source of force, and let G>0G > 0 be the gravitational constant. A test particle moves outside it by Newton's law of gravitation. Consider a particle at distance rr from the centre, moving in a direction that does not lead into the body. The escape velocity at distance rr is the least speed vev_e such that every such particle with speed v≥vev \ge v_e escapes. Here escaping means that the distance to the centre tends to ∞\infty. It equals ve=2GMr.v_e = \sqrt{\frac{2 G M}{r}}.

Remarks

The value follows from conservation of energy: the particle escapes exactly when its kinetic energy plus its potential energy is at least 00, that is 12mv2−GMm/r≥0\frac{1}{2} m v^2 - G M m / r \ge 0. The value does not depend on the mass mm of the particle or on the direction of launch.

Example: for the Earth, GM≈3.986×1014 m3/s2G M \approx 3.986 \times 10^{14}\ \mathrm{m^3/s^2} and r≈6.371×106r \approx 6.371 \times 10^6 m, so ve≈11.2v_e \approx 11.2 km/s.

Non-example: the speed GM/r\sqrt{G M / r} of a circular orbit of radius rr is smaller than vev_e by the factor 2\sqrt{2}, so the energy is negative and the particle stays bound.

One often writes μ=GM\mu = G M, so that ve=2μ/rv_e = \sqrt{2 \mu / r}.

Reading path

To understand this page from the ground up, read these 94 pages in order. The foundations come first.

  1. Derivative concept
  2. Differential equation concept
  3. Euclidean space object
  4. Euclidean vector concept
  5. Function concept
  6. Interval concept
  7. Limit point of a set definition
  8. Local maximum definition
  9. Ordinary differential equation concept
  10. Partition of an interval definition
  11. Sequence definition
  12. Velocity concept
  13. Fréchet derivative definition
  14. Total derivative concept
  15. Ordinary differential equation definition
  16. Cross product definition
  17. Function of several variables concept
  18. Vector-valued function concept
  19. Lipschitz condition definition
  20. Classical mechanics concept
  21. Limit of a function definition
  22. Local minimum definition
  23. Mesh of a partition definition
  24. Riemann sum definition
  25. Real sequence definition
  26. Subsequence definition
  27. Differentiable function of several variables concept
  28. Limit of a function of several variables concept
  29. Kinetic energy concept
  30. Momentum concept
  31. Parametrised curve concept
  32. Solution of an ordinary differential equation definition
  33. Vector field concept
  34. Partial derivative concept
  35. Inertial frame of reference concept
  36. Continuity at a point definition
  37. Derivative definition
  38. Riemann integrable function definition
  39. Bounded sequence definition
  40. Limit of a sequence definition
  41. Monotone sequence definition
  42. Continuity in R^n definition
  43. Differentiable function of several variables definition
  44. Partial derivative definition
  45. Gradient concept
  46. Initial value problem definition
  47. Line integral of a vector field concept
  48. Newton's laws of motion statement
  49. Continuous function definition
  50. Fermat's theorem on stationary points statement
  51. Antiderivative definition
  52. Riemann integral definition
  53. Convergent sequence definition
  54. Monotone convergence theorem for sequences statement
  55. Every sequence has a monotone subsequence statement
  56. Differentiable on an open set definition
  57. Higher partial derivatives definition
  58. Scalar potential concept
  59. Gradient definition
  60. Picard–Lindelöf hypotheses definition
  61. Work done by a force field concept
  62. Intermediate value theorem statement
  63. Definite integral definition
  64. Bolzano–Weierstrass theorem statement
  65. Iterated derivative definition
  66. Conservative vector field concept
  67. Picard operator definition
  68. Additivity of the integral over intervals statement
  69. Boundedness theorem statement
  70. Monotonicity of the integral statement
  71. C^k function characterised by continuous derivatives statement
  72. Potential energy concept
  73. Integral equation form of an initial value problem statement
  74. An iterate of the Picard operator is a contraction statement
  75. Extreme value theorem statement
  76. Vector field definition
  77. Picard–Lindelöf theorem statement
  78. Mean value theorem for integrals statement
  79. Rolle's theorem statement
  80. Conservative vector field definition
  81. Mean value theorem statement
  82. Classical mechanics definition
  83. First fundamental theorem of calculus statement
  84. Momentum definition
  85. Potential energy definition
  86. Second fundamental theorem of calculus statement
  87. Kinetic energy definition
  88. Angular momentum definition
  89. Fundamental theorem of calculus statement
  90. Central force definition
  91. Work–energy theorem statement
  92. Conservation of energy statement
  93. Newton's law of universal gravitation statement
  94. Escape velocity definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Escape_velocity
About this page
TagHZJBTDPJAS
URIm2://HZJBTDPJAS
Kinddefinition
WritingW0 stub
MSC70F05, 70M20
DefinesEscape velocity
UsesNewton's law of universal gravitation, Kinetic energy, Potential energy, Conservation of energy
Changed3 November 2026

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