Derive an expression for critical velocity of a satellite revolving around the earth in a circular orbit.

Obtain an expression for critical velocity of a satellite orbiting around the earth

#### Solution 1

Consider a satellite of mass m revolving round the Earth at a height ‘h’ above the surface of the Earth.

Let M be the mass and R be the radius of the Earth.

The satellite is moving with velocity Vc and the radius of the circular orbit is r = R + h.

Centripetal force = Gravitational force

`:.(mV_c^2)/r=(GMm)/r^2`

`:.V_c^2=GM/r`

`:.V_c=sqrt((GM)/(R+h)) ...............(Equ. 1)`

This is the expression for critical velocity of a satellite moving in a circular orbit around the Earth.

We know that,

`g_h=(GM)/((R+h)^2`

`:.GM=g_h(R+h)^2`

Substituting in equation 1, we get

`:.V_c=sqrt(g_h(R+h)^2/(R+h)`

`:.V_c=sqrt(g_h(R+h)`

where g_{h} is the acceleration due to gravity at a height h above the surface of the Earth.

#### Solution 2

Let,

M = mass of the earth

R = radius of the earth

h = height of the satellite from the earth’s surface

m = mass of the satellite

vc

= critical velocity of the satellite in the given orbit

r = (R + h) = radius of the circular orbit

For the circular motion of the satellite, the necessary centripetal force is given as

`F_CP = (mv_c^2)/r` 1

Gravitational force provides the centripetal force necessary for the circular motion of the satellite

∴ F_{CP} = F_{G}

_{}

Equation (4) represents the expression for critical velocity.