posted by sally .
how is this equation derived from the momentum principle?
In the near-Earth orbit the rate of change of momentum is (v/R)p = (v/R)mv
and how does it go from there to this
In the near-Earth orbit the rate of change of momentum is (v/R)p = (v/R)mv toward the center of the Earth, and this must equal the gravitational force GMm/R2, so
mv2/R = GMm/R2, and
mv2 = GMm/R.
Therefore in this orbit
K+U = (1/2)mv2 - GMm/R = -(1/2)GMm/R
This is a negative number reflecting the fact that this is a bound state. To minimize the work required to move the satellite far away, we want the final kinetic energy to be zero, and if it is far away the gravitational potential energy is also zero. Since the change in K+U is equal to the work we do, we have
0 - (-(1/2)GMm/R) = W, so
W = (1/2)GMm/R.