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A reasonable guess at the max depth would be 10000 m (no doubt Google will find a better number).
1 year = 365.25*24*3600 seconds
So if it took one year to fall, the average speed would be 0.3 mm/sec.
A "stone" the size of a dust particle might possibly fall that slowly (because viscous drag affects small objects more than large ones) but a "normal sized" stone would not.
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