A satellite orbits the earth at a height of 400 km above the surface. How much energy must be expended to rocket the satellite out of the earth's gravitational influence? Mass of the satellite =200 kg; mass of the earth =6.0×1024 kg; radius of the earth =6.4×106m; G = 6.67×10−11 Nm2kg−2.
Mass of the Earth, M=6.0×1024 kg
m=200 kg
Re=6.4×106 m
G=6.67×10−11 Nm2kg−2
Height of the satellite,h=400 km=4×105 m
Total energy of the satellite at height h=(1/2)mv2+(−GMem(Re+h))
Orbital velocity of the satellite, v=√GMeRe+h
Total energy at height h =12GMemRe+h−GMemRe+h
Total Energy=−12GMemRe+h
The negative sign indicates that the satellite is bound to the Earth.
Energy required to send the satellite out of its orbit = – (Bound energy)
=GMem2(Re+h)
=6.67×10−11×6×1024×2002(6.4×106+4×105)
=5.9×109 J
If the satellite just escapes from the gravitational field, then total energy of the satellite is zero. Therefore, we have to supply 5.9×109J of energy to just escape it.