Two satellites are launched at a distance R from a planet of negligible radius. Both satellites are launched in the tangential direction. The first satellite launches correctly at a speed v0 and enters a circular orbit. The second satellite, however, is launched at a speed (1/2)v0. What is the minimum distance between the second satellite and the planet over the course of its orbit?
M = mass of planet, m = mass of satellite, R1 = minimum distance, v1 = speed at the minimum distance
Fg = G(m1)(m2)/(r^2)
Ug = -G(m1)(m2)/r
mvr = const. (conservation of momentum)
PE + KE = const.
The Attempt at a Solution
(Somehow, I got the right answer, but I am confused about the calculations.)
First, I wrote the sum of the potential and kinetic energies for the second satellite.
(1/2)m(v0)^2 - GMm/R ; using the fact that v0 is the speed for a circular orbit, I substituted for M and got U = -(7/8)m(v0)^2
Then, I set that value of U equal to the sum of the potential and kinetic energy at an arbitrary point on the orbit, with distance R1 and speed v1 at that point. My intention was to find the smallest possible value of R1 (the minimum distance.)
-(7/8)m(v0)^2 = (1/2)m(v1)^2 - GmM/(R1)
I substituted for M like I did previously, and for v1 using conservation of momentum
R(v0)/2 = (R1)(v1)
In the end, it was possible to cancel out v0 and m. The simplified equation turned out to be
7(R1)^2 -8R(R1) + R^2 = 0; the two roots of the equation are R1 = R and R1 = R/7; the latter is actually the answer, but I'm confused. It seems like R1 could correspond to any point on the orbit. I expected to get a range of possible values and then find the minimum. Why is it that only the perigee and apogee satisfy the equation? I can't figure out where I made an assumption that only applies to those points. Or, was my expression for mechanical energy incorrect? I noticed that if R1 is in between R/7 and R, my expression for total energy would be less than -(7/8)m(v0)^2. Perhaps there is a form of energy I forgot to consider?
I would really appreciate it if someone can clarify this.