First slide

Motion of planets and satellites

Question

A satellite of mass m revolves around the earth of radius R at a height x from its surface. If g is the acceleration due to gravity on the surface of the earth, the orbital speed of the satellite is

Moderate

A

$gx$
B

$\frac{gR}{R - x}$
C

$\frac{{gR}^{2}}{R + x}$
D

${(\frac{{gR}^{2}}{R + x})}^{\frac{1}{2}}$

Solution

For the satellite, the gravitational force provides the necessary centripetal force, i.e.,
$\frac{{GM}_{e} m}{(R + x)^{2}} = \frac{{mv}_{0}^{2}}{(R + x)} or \frac{{GM}_{e}}{R^{2}} = g$
$v_{0} = \sqrt{\frac{{gR}^{2}}{R + x}}$

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