Kepler and Conservation of Energy

Question

I need help wtih the following question:

Consider the motion of a satellite with mass  m around a planet with mass  M (assume M>m). The orbit is elliptical.

When the satellite is closest to the planet the distance is R.

When the satellite is farthest to the planet the distance is 3R.

a.) Using Kepler's area law, find the speed of the satellite when it is closest to the planet. (Express this in terms of v.)

b.) Express the conservation of the mechanical energy for the satellite for the two extreme points on its orbit (i.e., when it is closest and when it is farthest.)

Note: This must be an equation involving only v  and R for the variables and G, M  and m for the parameters. (Do not simplify; write the equation as the first expression everyone would write whenever they hear the phrase "conservation of energy".)

c.) Find the speed v in terms of other parameters.

Answer 1

a. We denote the speed in the closest point as v_c and in the farthest point as v_f. The Kepler's area law states

v_cR = v_f (3R) or v_c = 3v_f = 3v

b. The total energy E = V + K. Here the kinetic energy K = mv²/2 and the potential energy V = GmM/r.

At the closest and farthest points

E_c = GmM/R + m(3v)²/2 , E_f = GmM/(3R) + mv²/2.

c. Using the conservation of energy law E_c =E_f and we get v = √^¯3Gm/(2R).