Satellite moves in a circular orbit around the Earth at a speed of 5.1 km/s.
Determine the satellite's altitude above the surface of the Earth. Assume the Earth is a homogeneous sphere of radius 6370 km and mass 5.98 x 10^24 kg. The value of the universal gravitational constant is 6.67259 x 10^-11 N times m^2/kg^2. Answer in units of km.
We can assume that the altitude of the satellite is x from earth and so it is earths radius R +x meter distant from the centre of the earth.
The satellites speed is v . We also assume that the gravitational force F is given by:
F = GMm/(r+x)^2 and this is equal to the centrepetal force of the satellite, mv^2/(R+x), where G is the gravitational constant, M =mass of the earth and m is the mass of the satellite. Using the known values, we would detrmine the value of x, the altitude of the satellite.
GMm/(R+x)^2=mv^2/(R+x). Reducinf by m/(R+x) , we get:
GM/(R+x) = v^2 or
R+x = GM/v^2 or
x = GM/v^2 - R
=6.67259*10^-11*5.98*10^24/5100^2 - (6370000)
= 8971056.594 m
Thanks for neela's answer.
It is 8971.056594 km.