Here is an earlier thread about
Blowing up the Moon;
it appears it would take 10[sup]13[/sup] megatons to blow up the moon,
or to do it on the cheap, 10[sup]7[/sup] megatons to bring it within the Roche limit, and see it disintegrate under the influence of tidal forces into a ring like Saturn’s.
That is ten million one-megaton bombs.
Quite a few.
SF worldbuilding at
http://www.orionsarm.com/main.html
And again, that’s only assuming ALL the force of your bombs is vectored in the direction you want, which obviously would not be the case if you just piled up all your bombs on the surface of the Moon and set them off. In fact, you would need many, many more bombs than that, at least five times as many.
Ok, so our Moon is too big. What about Mars’ moons?
They are much smaller, somewhere around ~20km iirc. Could we give them a nudge into a decaying orbit and crash them into the planet? What affect would a ~20km impact have on Mars?
-Pete.
Can I change this question please? Out of sheer vindictiveness, I’d like to know what would happen if we nudged Phobos out of orbit and collided it with France. 
(I’d just like to point out that I’m not really anti-French, but as a brit I’m culturally obliged to slate them at every opportunity. And even occasionally moon them).
Regrettably, you would almost certainly annihilate human civilization and all life on Earth. Phobos is quite a bit larger than that nasty comet in “Deep Impact.”
Just thought I’d point out that the moon is peppered with craters from the impact of big-ass Armageddon-sized rocks. If none of those were sufficient to deflect the moon substantially, then it’s doubtful we’ve got anything that would do the job.
I think we are on to something with the mass accelerator. We could build several mobile nuclear factories that crushed rocks into small chunks and ejected them in the right direction at several miles per second. That will kill two birds with one (heheheh) stone.
Not only will the moon’s orbit change, its mass will also diminish, making further pushing easier…voila 
I suggest leaving the moon where it is.
Without it the earth would no longer be tilting approximately 23.5 degrees. Instead of a nice stable climate we would get frequent dramatic changes. It is possible that we owe our very existance to the moon’s effects.
And now you want to get rid of it. That’s some gratitude. 
Let’s just write a massive “Chairface” on its surface and call it a day…
Then can we dig up France and fling it at Phobos?
The French heard your threat and are already preparing to surrender.
Where do you get that idea? The moon doesn’t keep the earth tilted - conservation of angular momentum ensures that the Earth will stay exactly the way it is whether the moon is there or not. My understanding is that the Earth is tilted because of a collision early on when it was being formed or shortly thereafter.
What we would lose are tides. Without tides, I wonder what would happen to features like the gulf stream?
NOVA did an episode on the moon not too long ago, but I was kinda dozing on the couch through a lot of it. I do remember these key points:
The moon was once an orbiting planet that collided with earth. It wasn’t a direct hit, it just sort of grazed us. The earth and the moon exchanged material.
This collision caused the earth to wobble.
Early on the moon took up most of the sky.
The moon moves about two inches a year away from the earth.
The earth still wobbles from this collision. The farther the moon moves away from the earth, the less the earth wobbles. Eventually the moon will be far enough away that the earth will no longer wobble. That will be the end of the seasons. That will be the end of life on earth.
Life probably existed when this collision occurred, but it is not clear if it originated on the runaway planet we call the moon, or it started here.
Life on earth is as dependent on the moon as it is on the sun. This should make every Wiccan happy. (OK, they didn’t really address neopaganism on NOVA)
If you’re really interested, you might want to check either NOVA’s or PBS’s web sites for better information than I just pulled out of my clouded memory.
The wobbles would be slight deviations in the earth’s tilt- losing the moon would not magically reset the earth’s axis to be 90 degrees from the orbital path. None of the planets’ axes are plumb to their orbits, whether they have moons or not.
According to some of the latest research, a planet’s angle of inclination can vary enourmously over very long periods of time. Scientists think that Mars was once tilted 90 degrees over, and that the strong influence of the moon’s gravity stops that happening to Earth. The Earth wobbles slightly on its axis, but its nothing compared to the other planets in the solar system.
Simply wrong I’m afraid. The seasons aren’t caused by wobbles in the Earth’s orbit. They are caused because the Earth’s axis of rotation is inclined about 23 degrees to the orbital plane. In the north’s summer, the norhtern hemisphere points towards the sun, so the days are longer. More heat is absorbed by the surface, so its hotter.
The moon stabilises the Earth’s rotation, so over millions of years the seasons haven’t varied much. There is a 41,000 year cycle called precession, but the effect is tiny. If the moon wasn’t there, its possible that the angle of inclination could vary enourmously. If it reached 90 degrees, one side of the Earth would point towards the sun for 6 months while the other half froze. Not good for life, which is why the moon is so important.
I thought the latest theory was that a 3rd Mars-sized object hit the Earth, ejecting from the Earth the matter that made up the moon.
Was that what you meant, or was Nova presenting a different theory?
I got this idea from our prominent scientists through such medias as books, television and various Internet sites. This is a subject that I was interested in and I did read a lot about it.
I attach a quote from http://www.bbc.co.uk/radio4/science/nature_20031027.shtml
Thanks for the link Panzar, thats what I was trying to explain.