18gay Clubs , Union, New Jersey

From the Moon to Earth

This is the story of the home.


In previous post, we estimate that the breakeven point is 90% of the distance between the moon and Earth:

\\ [r = 0.9 d \\]

ever been asked: why it took a rocket like the Saturn V to reach to the moon, while to return only reached with small engine service module in lunar orbit?

calculate the gravitational potential. For a point mass, it is given by

\\ [E_G = - \\ frac {G M_1} {r} m \\]

where $ M_1 $ is the mass of the gravitating body, $ m $ is the mass of test body and $ r $ is the distance between them. As mentioned above, the problem we are discussing is a simplifiación the whole problem. Under this simplification, what is the gravitational potential porducido the Earth ($ M_ \\ Oplus $) and Moon ($ M_L $)? The superposition principle says:

\\ [E_G = - \\ left (\\ frac {G M_ \\ Oplus m} {r} + \\ frac {m} {G M_L dr} \\ right). \\]

grouping and rearranging we have:

\\ [E_G = - G m \\ left (\\ frac {M_ \\ Oplus} {r} + \\ frac {M_L} {dr} \\ right) . \\]

As expected, the maximum of this curve occurs for $ r = 0.9d $. Not to work with such large numbers, we can normalize this equation the value of Earth's gravitational potential on the surface of the Earth, ie

\\ [E_G (R_ \\ Oplus) \\ equiv E_0 = - G m \\ left (\\ frac {M_ \\ Oplus} {R_ \\ Oplus} + \\ frac {M_L} {d-R_ \\ Oplus} \\ right), \\]

and therefore

\\ [E_G = - \\ frac {G m} {

Finally, consider the potential wells

for some planets and moons in the Solar System, made by the genius creator of the comic xkcd

.