Question

derive an expression for critical velocity of satellite
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Answer 1

Answer:

The satellite is moving with velocity Vc and the radius of the circular orbit is r = R + h. This is the expression for critical velocity of a satellite moving in a circular orbit around the Earth. where gh is the acceleration due to gravity at a height h above the surface of the Earth.

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Answer 2

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 V and the radius of the circular orbit is r =R + hr=R+h. Centripetal force = Gravitational force

\therefore \dfrac{Mv^2_c}{r}=\dfrac{GMm}{r^2}∴

r

Mv

c

2

=

r

2

GMm

\therefore v^2_c=\dfrac{GM}{r}∴ v

c

2

=

r

GM

\therefore v_c=\sqrt{\dfrac{GM}{R+h}}∴ v

c

=

R+h

GM

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

We know that,

g_h=\dfrac{GM}{(R+h)^2}g

h

=

(R+h)

2

GM

GM=g_h(R+h)^2GM=g

h

(R+h)

2

Substituting in equation (1), we get

\therefore v_c=\sqrt{\dfrac{g_h(R+h)^2}{R+h}}∴ v

c

=

R+h

g

h

(R+h)

2

\therefore v_c=\sqrt{g_h(R+h)}∴ v

c

=

g

h

(R+h)

where g_hg

h

is the acceleration due to gravity at a height above the surface of the Earth.