Orbital velocity and its expression

• The orbital velocity of a satellite is the velocity required to be given to the satellite in order to put it in an orbit around the earth.
• Once the satellite is put in an orbit with required velocity, the satellite keeps on revolving around the earth.
• For the continuous revolution of the satellite around the earth, the necessary centripetal force is provided by the gravitational force of the earth.

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Expression for orbital velocity

Let a satellite is put in an orbit of height h with an orbital velocity v_orb. Then, the necessary centripetal force for the satellite to move in the orbit is given by

Fc = mv_orb²/(R+h)

where m is the mass of the satellite and R is the radius of the earth.

If M is the mass of the earth and G is the gravitational constant, the gravitational force between the earth and the satellite is given by

Fg = GMm/(R+h)²

We have Fc = Fg

∴   mv²orb/(R+h) = GMm/(R+h)²

∴   v_orb² =  GM/(R+h)

If g is the value of acceleration due to gravity on the earth’s surface, we have

g = GM/R²

∴  GM =gR²

∴  v_orb² = gR²/ (R+h)

∴  v_orb = R × √ (g/(R+h))

This is the expression for orbital velocity.

References: 

i) https://www.britannica.com/science/orbital-velocity/additional-info#More-Articles

ii) https://byjus.com/physics/derivation-of-orbital-velocity/

Orbital velocity and its expression