Vis viva

Vis Viva is the principle that the difference between the aggregate work of the accelerating forces of a system and that of the retarding forces is equal to one half the vis viva accumulated or lost in the system while the work is being done.

The Vis Viva Equations

Let a mass m be in orbit about a mass M. Let the velocity of m be v when it is at a distance r from M and let a be the semi-major axis of the orbit. Then set the kinetic energy equal to the gravitational potential energy to obtain

<math>E = (1/2)mv^2-GMm/r = -GMm/2a<math>

so the velocity is given by

<math>v = sqrt(GM((2/r)-(1/a))<math>