A satellite of mass `10 kg` is placed initially in a temporary orbit `800 km` above the surface of earth. The satellite is to be placed now in a permanent orbit at `2000 km` above the surface of earth. Find the amount of workdone to move the satellite from temporary to permanent orbit. The radius of the earth is `6400 km` and `g = 10 ms^(-2)`.

Explanation

Consider a satellite of mass `m` orbiting around the earth with orbital velocity `upsilon`. Let `h` be the height of satellite from the surface of earth and `r` be the orbital radius of the satellite. Then `r = R + h` and
`upsilon = sqrt((GM)/(R + h))`
Total energy of obiting satellite is
`K = KE + PE = (1)/(2) m upsilon^(2) + ((-GM m)/(r))`
`= (1)/(2) (GM m)/(r) - (GM m)/(r) = (-GM m)/(2r) = (-mg R^(2))/(2(R + h))`
Workdone in shifting the satellite from the height `h` to `h_(1)` is
`W =` difference in energy of satellite in two orbits
`= (-mg R^(2))/(2(R + h_(1))) - [(-mg R^(2))/(2(R + h_(1)))]`
`= (100 xx 10 xx (6400 xx 10^(3))^(2))/(2)`
`xx[(1)/((6400 + 800) xx 10^(3)) - (1)/((6400 + 2000) xx 10^(3))]`
`= (100 xx 10xx(6400 xx 10^(3))^(2))/(2) xx (1200)/(7200 xx 8400 xx 10^(3))`
`4.06 xx 10^(8)J`