Figure Eight Trajectories for Spacecraft: Potential Energy Calculation

[dvvz2006]

Hi, I was wondering if someone could help with the following problem. I am having trouble setting up the equation and a) and by looking at b and c, am unsure how to do those as well. The problem is as follows:

Figure eight trajectories are used to send spacecraft form the Earth to the Moon. Moving along these trajectories, the spacecraft remain close to the line connecting Earth and Moon. Along this line, the potential energy is lower than elsewhere in space. Assume that Earth and Moon are stationary and that a spacecraft has a mass of 1000kg. '

a) Give the formula of that combines the potential energies of Earth and Moon in a coordinate system in which the centers of Earth and Moon are at r = 0 and r=R respectively. Here R is the distance between the Earth and Moon. As usual, the potential energy shall be zero at infinity.

b)Using calculus, find the position between Earth and Moon where the Potential Energy is at its maximum.
C) Calculate the potential energy that the spacecraft has at that point and express the result as a fraction of the energy that the spacecraft has when stationary at the Earth's surface.

Thanks very much,

 

[fargoth]

a) just add the potential of the Earth $$-\frac{GM}{r^2}$$
and the potential of the moon $$-\frac{Gm}{|r^2+R^2|}$$

b) now, you are required to find for what r, there's a extremum - meaning when does the derivative equals zero.

c) just put that point you found in the formula to find what's the energy of this point!