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I've seen some science\history documentary in which they made a replica of Galileo's inclined plane experiment.
They rolled a ball down the plane; and it's progressed in length units each unit of time was - 1, 4, 9, 16 etc. I realized it was squares of natural numbers; but I didn't really understand why this happens. I expected it to be exponential; as in - 1, 2, 4, 8, 16 etc.
I've checked a few more videos since; and now I understand that this progression is "time squared". That makes sense; but I still don't exactly understand why it works this way.
Also in some other video, which is a segment from some science show, they said the ball suppose to roll - 1, 3, 5, 7 etc. Did they just got it wrong?
So why does it work this way - time squared; and what other basic things in physics progress this way; can you give some examples? What about energy needed to accelerate mass?