What is the terminal velocity of the moon crashing into Earth?

The Newtonian equation is F = G x (m1.m2)/r^2. It’s hard to write equations on here, so see http://en.wikipedia.org/wiki/Newton's_law_of_universal_gravitation if that is unclear.

Basically since there is a factor for the mass of both objects, the gravitational force between the Earth and the Moon is dependent on both masses.

A quick calculation indicates that, if the moon were to stop dead in its orbit (not taking into account the negligible air resistance it would meet shortly before impact), classical physics says it would accelerate towards Earth at nearly 33 m/s/s, reaching a speed at impact of roughly 5025 m/s. That is 18,090 km/h, or 11,241 mph.

However General Relativity tells us that the Moon is in fact falling towards Earth with the same potential energy, but the Earth’s gravitational field has warped space-time such that the Moon appears to orbit the Earth. It doesn’t trace an elliptical path as such, but falls towards Earth in a straight line through space-time that is warped in an elliptical fashion.

The moon is already falling towards the Earth!

As it stands, the only way a collision would happen is if an outside force acted upon either the Earth or (more likely) the Moon. Once you add an outside force into the mix, the gravitational attraction between the two would no longer be the only factor. And without knowing the outside force, the equation is unsolvable.

That said, both bodies exert a considerable gravitational force on each other. That is why we have tides, and why the Moon goes around the Earth instead of hurtling off through space alone. The forces would combine as they always have.

@jerv Actually, the pull of the moon is slowing the Earth’s rotation, an effect known as tidal braking, which increases the length of our day by 2.3 milliseconds per century. The energy that Earth loses is picked up by the moon, increasing its distance from the Earth, which means the moon gets farther away by 3.8 centimeters annually.