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In space, or the space between the earth and the moon, there is no "down" so the question asked does not give enough information for an adequate answer.

The sidereal period of the moon is 27.321682 days or 655.720368 hours

The daily period of the earth is 24 hours.

The equatorial circumference of the earth is 6378.1 km

So, if you could dangle a rope from the moon down to the earth the end would speed across the face of the earth at

speed of the earth's rotation - speed of the moon's orbit.

speed of the earth's rotation 6378.1 km/ 24 hours = 265.75 km/hr

speed of the moon's orbit 6378.1 km/655.72 hours = 9.7269 km/hr

265.75 - 9.73 = 256.02 km/hr

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