How can I demonstrate a simple derivation of the vis-viva formula (used for calculating orbital velocity in an elliptical orbit)?

Answer 1

a = semi-major axis of the ellipse

e = total orbital energy per unit mass (do not confuse this e with eccentricity)

M = mass of primary body at a focus of the ellipse

Using the relation

e = -GM/(2a),

e = v² / 2 - GM / r

-GM/(2a) = v² / 2 - GM / r

v² = 2GM [1/r - 1/(2a)]

v² = GM (2/r - 1/a)

This is the vis-viva equation.