This question is an object of fascination of mine. There are a few different angles to take to it. My favorite is blasting the moon out of existence while its moving away from Earth.
What would happen if Earth's moon was removed?
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This has sat untouched for a while, so I did a quick search and found this article which sums it up nicely.
They key factors with regard to earth science would be:
1) Less extreme tides. The tides would only be linked to the sun and would not be as strong. They would also peak at noon. This has obvious implications for any coastal processes that rely on the tides. Many industries would be impacted. For one, the Bay of Fundy wouldn't be quite the tourist attraction it is.
2) Day length would potentially shorten at a greater rate. Our tidal lock to the moon helps to stabilize the slowing rotation of the Earth. In terms of Earth Science, this would have implications on any diurnal processes (weather), with compounding long-term effects (the climate).
3) The axial tilt of the Earth would change quite a bit. The moon helps to maintain our reasonably stable tilt. Greater tilt would result in more extreme seasonal changes, which in turn would affect all sorts of processes.
Those are the bigger ones from an Earth Science perspective.