Vis-Viva Equation
The Vis-Viva Equation is a formula used to calculate the speed of an object in orbit based on its distance from the center of the body it is orbiting and the mass of that body.
It is the fundamental formula in orbital mechanics that describes the relationship between a satellite's speed, its distance from the central body, and the orbit's size. It applies to objects in circular, elliptical, parabolic, and hyperbolic orbits around a massive central body, such as Earth or the Sun. Orbital Mechanics is the branch of physics and engineering that deals with the study of the movement of objects in space, influenced primarily by gravity. It is a specialized area within celestial mechanics that focuses on understanding the paths, or orbits, that objects like satellites, spacecraft, and natural bodies follow as they are governed by gravitational forces, especially from celestial bodies such as planets, moons, and stars.
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v: Relative speed of the two bodies
r: Distance between the two bodies' centers of mass
a: Length of the semi-major axis (a > 0 for ellipses, a = ∞ or 1/a = 0 for parabolas, and a < 0 for hyperbolas)
G: Gravitational constant, and the value is 6.67384 ×10−20 Km3⋅Kg−1⋅s−2
M: Mass of the central body
This Vis-Viva equation helps with various deductions that can be drawn by analyzing the relationships between the semi-major axis and the object's positional distance.
For Example: If the value of r is 1000 Km, a is 4000 Km and M is 350 Kg, then the Vis-Viva Equation for any Keplerian orbit (elliptic, parabolic, hyperbolic, or radial) is
V2 = GM (2/r - 1/a)
V = √6.67384 ×10−20 *350 (2/1000- 1/4000)
V = 20218128004.343 Km/s
