Quote from: IslandPlaya on 05/06/2014 03:54 pmAssuming the distances between the COM are different between the North and South poles and you have some way of converting linear motion into a storeable form then a round trip would take no energy at all. (Neglect losses in the system of course.)And, in fact, if the body is spherically symmetric and of uniform density, then travel along a straight line connecting any two points on the surface takes no energy, if friction can be avoided. And the trip time is always the same, regardless of the distance between the two points on the surface.

Assuming the distances between the COM are different between the North and South poles and you have some way of converting linear motion into a storeable form then a round trip would take no energy at all. (Neglect losses in the system of course.)

And, in fact, if the body is spherically symmetric and of uniform density, then travel along a straight line connecting any two points on the surface takes no energy, if friction can be avoided. And the trip time is always the same, regardless of the distance between the two points on the surface.

Wendy Kreiger also told me trip time is always the same. I'm willing to take your word as well as Wendy's. But I would be happier if I knew why.

Quote from: Proponent on 06/17/2014 06:46 pmAnd, in fact, if the body is spherically symmetric and of uniform density, then travel along a straight line connecting any two points on the surface takes no energy, if friction can be avoided. And the trip time is always the same, regardless of the distance between the two points on the surface.It's a free energy ride if we're willing to let gravity do all the work. But then we're stuck with trip time on order of half an hour to an hour for most rocks.But we can shorten trip time by investing energy to accelerate at the beginning of the trip. But then the payload would need to be slowed before it reaches its destination. The deceleration at the end of the trip could put energy back into the system. So it seems to me faster trips could also be done for very little energy.

See equation (9) of the analysis attached to this post.

Quote from: Hop_David on 06/17/2014 10:20 pmWendy Kreiger also told me trip time is always the same. I'm willing to take your word as well as Wendy's. But I would be happier if I knew why.Without using calculus, we can easily show how the problem reduces to simple harmonic motion.Consider a particle constrained to travel along chord parallel to the y-axis. The acceleration of a particle along the chord is -(GM/R^{3})y. (I'm sure you can figure this out yourself: just calculate the acceleration of gravity at a given x and y, and then determine its y-component.) The form of the acceleration is the same as for a mass on a spring or for a small-amplitude pendulum (restoring force is proportional to displacement). In both of those cases, the period of oscillation is independent of amplitude. The acceleration is also independent of x, so it doesn't matter which chord (i.e., between which two points on the surface) we are travelling.Does that help?

A quick geometrical demonstartion that second focus lies on center of chord connecting two points....

NASA Developing drones/hoppers for Mars, asteriods and Moon. http://www.parabolicarc.com/2015/08/09/nasa-mars-drones/For the Moon a small H2O2 hopper could operate off one the Xprize landers eg MoonExpress MX-1 which uses H2O2.One big plus of hopper using lander fuel is that all the lander's reserve fuel can be used for exploration. A 10kg hopper + 2kg of H202 (12Kg total) with 150 ISP thrusters would have 10Km radius.4kg fuel would give 30km radius, ideal for exploring around Shackleton crater (21km diameter). http://space.stackexchange.com/questions/4413/lunar-sub-orbital-trajectoriesNB 1degree is 30Km.

This might be of interest:"Developing an Aerial Transport Infrastructure for Lunar Exploration" by David L. AkinAbstract http://www.lpi.usra.edu/meetings/leagilewg2008/pdf/4096.pdfPresentation http://www.lpi.usra.edu/meetings/leagilewg2008/presentations/oct29pmSalonIII/Akin4096.pdf