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Physics LibreTexts

57.10: The Vis Viva Equation

( \newcommand{\kernel}{\mathrm{null}\,}\)

When an object of small mass m orbits a body of much larger mass M, we can use conservation of energy considerations to find the smaller body's velocity v at radial distance r. We have for the small body m :

K=12mv2 (kinetic energy) U=GMmr (potential energy) E=GMm2a (total energy) 

where the quantity a is the radius for a circular orbit, the semi-major axis for an elliptical orbit, the negative of the semi-major axis for a hyperbolic orbit, or infinity for a parabolic orbit.

By conservation of energy,

E=K+UGMm2a=12mv2GMmr.

Solving for the orbit speed v, we find

v=GM(2r1a)

This result is known as the vis viva equation (Latin for "live force").


57.10: The Vis Viva Equation is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.

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