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That's right.
In Galileo's day (and maybe even today), most people thought what you thought—that the ball dropping straight down should reach the ground before the ball that was thrown out. Galileo proved that this is not the case. Since each ball falls the same vertical distance and the pull of gravity on both is the same, they hit the ground at the same time. It is true, though, that the overall speed of the ball moving away from the tower is faster, but the rate at which both accelerate toward the ground is still the same. A simple thought experiment reveals why...
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