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What is the greatest speed reached by a falling object?

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It depends by what object it is. The highest speed it reaches is called Terminal Velocity.

From wikipedia http://en.wikipedia.org/wiki/Terminal_velocity :

"A free-falling object achieves its terminal velocity when the downward force of gravity (Fg) equals the upward force of drag (Fd). This causes the net force on the object to be zero, resulting in an acceleration of zero.[1]"

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