Orbital speed of a satellite
1. Orbital speed of a satellite is the speed required to put the satellite into orbit. When a satellite (such as moon) revolves in a circular orbit around the earth, a centripetal force acts upon the satellite. This force is actually the gravitational force exerted by the earth on the satellite.
2. Suppose a satellite of mass m has to be put circular orbit around the earth at height h above its surface. Consider earth to be a sphere of mass Me and radius Re. Then radius of the orbit of the satellite will be r (=Re + h). If v0 is the required orbital velocity for the satellite, then centripetal force exerted by the earth on the satellite is mv02/r.
3. Now the gravitational force exerted by the earth on the satellite will be GMem/r2. Because the gravitational force provides the required centripetal force, hence we can write
G(Mem/r2)=mv02/r or GMe = v02/r
Or v0= √(GMe/r)= √(GMe/Re+ h)………..(1)
If the acceleration due to gravity on the earth’s surface is g, then
g = GMe/ Re2 or GMe=gRe2
Putting this value of GMe is eq. (1), we get
Thus eq(1) and (2) provide the speed of revolution of the satellite in its orbit.
Note:a) Orbital speed of the satellite depends only upon its height above the earth’s surface. Greater is the height h of the satellite above earth’s surface, smaller is the speed of the satellite.
b) Because the speed of a satellite is independent of the mass of the satellite, therefore, two satellites of different masses revolving in same orbit around the earth will have the same speed.