From what I know, the Moon is accelerating away from the Earth. Do we know when it will reach escape velocity? How do we calculate this?
It will never reach such a high velocity. The moon is drifting further from the earth due to tidal acceleration. This process is, at the same time, slowing the rotation of the earth. Once the earth's rotational period matches the moon's orbital period, the earth-moon system will be tidally locked to each other (note: the moon is already tidally locked to the earth), and the acceleration will cease.
To briefly explain the mechanism, the gravitational pull between the earth and the moon causes tidal "bulges" to extend out on both bodies (just like the ocean tides, except that the entire surface moves slightly, not just the water). Since the earth rotates faster than the moon completes one orbit, the bulge on the earth lies slightly ahead of the earth-moon line, because the earth is rotating so quickly. This bulge gravitationally pulls on the moon, speeding it up, while at the same time the moon pulls on the bulge, creating a torque on the earth and slowing down its rotation.
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