Question

Show that a satellite may speed down: that is, show that if frictional forces or reverse thrust cause a satellite to lose total energy, it will move into an orbit closer to the earth and may have increased kinetic energy.

Answer 1

using a newton's second law & the law of gravity to the satellite,

F_radial = G M m / r²                      { eq. 1 }

m v² / r = G M m / r²

where, M = mass of the earth

m = mass of the satellite

solving above eq,

v² = G M / r                                   { eq. 2 }

v = G M / r                          

the speed of the satellite is independent of its mass.

As the satellite loses speed, it begins to fall in towards the Earth. It begins to pick up speed due to its change in gravitational potential.

the kinetic energy of the satellite in a circular orbit is given as :

K.E = 1/2 m v²                            { eq. 3 }

inserting the value of 'v' in eq.3

K.E = 1/2 m (G M / r)

K.E = G M m / 2r