In this post, we will focus on the Orbital Velocity derivation. This derivation is equally applicable to the planets revolving around the Sun in their orbits and the satellites revolving around their specific planet.



The basic principle that is leveraged while going for this derivation is that the centripetal force required for the circular motion is supplied by the gravitational force acting between the revolving body and the central body (the entity at the center of the circular orbit)

Orbital Velocity Derivation | How to derive the orbital velocity equation?

First, we will derive the Orbital velocity expressions or equations (2 sets) and later will derive the Orbital Velocity for a nearby orbit. (so you get total 3 sets of equations)

Orbital Velocity expression #1 (step by step derivation)

The Gravitational Force between the earth and the satellite = F g = (G.M.m)/r2 ……………… (1)

G is the Gravitational Constant.



The centripetal force acting on the satellite = F c = mV2/r ……………………….. (2)

Here, M is the mass of earth and m is the mass of the satellite which is having a uniform circular motion in a circular track of radius r around the earth.

V is the linear velocity of the satellite at a point on its circular track.

Now this r is the sum of the radius of the earth(R) and the height(h) of the satellite from the surface of the earth.

r = R + h

Now equating, equation 1 and 2 we get,

F g = F c

=> (G.M.m)/r2 = mV2/r



V = [(GM)/r]1/2 …………(3)



This is the first equation or formula of Orbital Velocity of a satellite.



Here r = R +h

Orbital Velocity expression #2 (step by step derivation)

For a mass of m on earth’s surface, the following is true:

mg = (GMm)/R2 ………………………. (4)

Note, on earth surface h=0 and r = R. And gravitational force on a mass is equal to its weight on the surface.

From equation 4 we get this equation, GM = g. R2 …………………….. (5)

Substituting this expression of GM in equation 3, we get,

V = [(gR2)/r]1/2

V = R . (g/r)1/2 ……. (6)



This is the second formula.



Here, as said earlier,

r = R +h

Next we will derive the 3rd equation and that is for a NEARBY ORBIT, i.e. for an orbit which has negligible height above the earth’s surface.

Orbital Velocity expression for Near orbit (step by step derivation)

Let’s consider an orbit which is pretty close to the earth.

Now if the height of the satellite (h) from the surface of the earth is negligible with respect to the Radius of the earth, then we can write r=R+h = R (as h is negligible).

**From equation 3 (the fundamental form of orbital velocity equation), we get an equation of nearby orbit’s Orb. velocity:



V = [(GM)/R]1/2………….(3.1)

**And from equation 6 we get another form of equation for orbital speed (when h is negligible) at near earth orbit:

V orbital = R . (g/r)1/2



= R . (g/R)1/2



V orbital =(gR)1/2……….(6.1)



here, R = r

Orbital Velocity derived – list of what we derived

Here you get a set of Orbital Velocity expressions that are derived in this post. You can visit our post on quick listing and descriptions of these satellite velocity expressions. Also, there is also a post on the definition or explanation of this velocity.

V orbital = [(GM)/r]1/2

V orbital = R . (g/r)1/2

And for Nearby Orbit

V orbital = [(GM)/R]1/2

V orbital = (gR)1/2

Related study- highly suggested for you:

orbital velocity of satellite

We can use the equations of Orbital velocity to derive Kepler’s third law. Read the Post here Kepler’s 3rd Law. Also, here is another related & important article for you: Escape Velocity









Orbital Velocity derivation | How to derive the orbital velocity equation?