Then the point at which the two bodies are closest to one another, or the perigee, does that also correspond to max speed (v), acceleration (a), and force (F)?
Well, the maximum speed is easy. The kinetic energy at perigee is at a maximum, so clearly, the speed is also a maximum.
For the force and the acceleration, I'm not exactly sure. For a real gravitational system with
N number of particles, no closed-form solutions exist. Approximations for systems with three bodies (e.g., the sun, the earth, and a spacecraft) exist, but extending that to anything more complicated is something which can only be done numerically
. Which is to say, it's not something that you'd be able to do on the MCAT - such a thing is something that would likely be found in a graduate celestial mechanics course. The complexity all comes about because you have multiple objects all exerting a force on each other, which in turn changes the direction of the net force on the spacecraft.
In short, finding the maximum force and acceleration on a body where you have more than two bodies at work (even if one has a negligible mass) is far beyond the scope of anything that would be seen in undergraduate physics.
This really showcases the power of energy arguments. If you were to try to use Newton's law of gravitation and the kinematic equations to calculate acceleration, velocity, and position, and then figure out the maxima, you would be forced to do some pretty heavy numerical analysis. But, if you assume energy is conserved and are told the orbit is elliptical, you can easily find the maxima and minima of the velocity. Powerful, indeed.
I guess what I'm trying to say is that the nature of the net force and acceleration on a body in a system with more than two bodies is not straightforward in any way and far, far beyond the scope of the MCAT. I might even go so far as to say, if you see a question with elliptical orbits, think of energy and Kepler's empirical laws first.
Make sense?
