Question

3. What is Orbital velocity? Obtain an Expression for it?​

Answer 1

Answer:

Orbital velocity is defined as the minimum velocity a body must maintain to stay in orbit.

Explanation:

For the derivation, let us consider a satellite of mass m revolving around the Earth in a circular orbit of radius r at a height h from the surface of the Earth. Suppose M and R are the mass and radius of the Earth respectively, then r = R + h.

To revolve the satellite, a centripetal force of mv2or is needed which is provided by the gravitational force GMmr2 between the satellite and the Earth.

Therefore, equating both the equations, we get

mv20r=GMmr2

v20=GMr=GMR+h

Simplifying the above equation further, we get

v0=GMR+h−−−−√……(eqn 1)

But GM=gR2, where g is the acceleration due to gravity.

Therefore,

V0=gR2R+h−−−−√

Simplifying the above equation, we get

v0=RgR+h−−−−√

Let g’ be the acceleration due to gravity in the ( at a height h from the surface)

g′=GM(R+h)2

Simplifying further, we get

GM(R+h)=g′(R+h)=g′r……(eqn 2)

Substituting (2) in (1), we get

v0=g′r−−−√=g′(R+h)−−−−−−−−√