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 × 106 m; G = 6.67 × 10–11 N m2 kg–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+(−G(Re+h)Mem) Orbital velocity of the satellite, v=Re+hGMe Total energy at height h =21Re+hGMem−Re+hGMem Total Energy=−21Re+hGMem The negative sign indicates that the satellite is bound to the Earth. Energy required to send the satellite out of its orbit = – (Bound energy) =2(Re+h)GMem =2(6.4×106+4×105)6.67×10−11×6×1024×200 =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.
