The vis-viva equation greatly simplifies some calculations in orbital mechanics. It is reminiscent of how conservation of energy can sometimes trivialize what appears to be a complicated problem. In fact, the vis-viva equation is derived from conservation of energy, but the derivation is not trivial. Which is good: the effort required in the derivation implies the equation is a shortcut to a place that might take longer to arrive at starting from first principles.
The term vis viva is Latin for “life force” and was applied to mechanics by Liebnitz around 350 years ago. The vis-viva equation is also known as the vis-viva law or the vis-viva integral.
The vis-viva equation says that for an object orbiting another object in a Keplerian orbit
Here v is the relative velocity of the two bodies, r is the distance between their centers of mass, and a is the semi-major axis of the orbit. The constant μ is the standard gravitational parameter. It equals the product of the gravitational constant G and the combined mass M of the two bodies.
In practice it is often accurate enough to let M be the mass of the larger object; this works for GPS satellites circling the earth but would not be adequate to describe Charon orbiting Pluto.
For a circular orbit, r = a and so v² = μ/r. Then r is constant and v is constant.
More generally a is constant but r is continually changing, and the vis-viva equation says how r and v relate at any given instant. This shows, for example, that velocity is smallest when distance is greatest.
For an object to escape the orbit of another, the ellipse of its orbit has to become infinitely large, i.e. a → ∞. This says that if v is escape velocity, v² = 2μ/r. For a rocket on the surface of a planet, r is the radius of the planet. But for an object already in a high orbit, escape velocity is less because r is larger.