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This question already has an answer here:

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?

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marked as duplicate by John Rennie, Kyle Kanos, ACuriousMind, Floris, Martin Jul 8 '15 at 8:51

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ There is no requirement for integer (natural) squares, the video was doing that for the general public who would understand $x\propto t^2$ better with the more commonly known squares: 1, 2, 4, 9, etc. $\endgroup$ – Kyle Kanos Jul 7 '15 at 14:43
  • $\begingroup$ See also Can you completely explain acceleration to me? $\endgroup$ – John Rennie Jul 7 '15 at 14:44
  • $\begingroup$ @John Rennie - My question is different from the other question. I ask - why is it squares of every natural number and not exponential. The other question is - "Why is it proportional to the square of the time and not just time?" $\endgroup$ – MrAccident Jul 7 '15 at 15:34
  • $\begingroup$ @Kyle Kanos - Maybe; but my question is - why does it work this way; and specifically - not exponential. $\endgroup$ – MrAccident Jul 7 '15 at 15:35
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    $\begingroup$ The natural numbers are the times i.e. 1 second, 2 seconds, 3 seconds etc, and since distance is propeortional to $t^2$ the distances are $1^2$, $2^2$, $3^2$, and so on. The question I've linked explains why $s \propto t^2$ rather than $s \propto e^t$. That's why it's a duplicate. $\endgroup$ – John Rennie Jul 7 '15 at 15:37
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The distance between successive lines is 1,3,5... making their absolute position 1,4 (1+3), 9 (1+3+5)...

This follows from double integration of the equation of motion with a constant force:

$$v = \int \frac{F}{m} dt = a\; t\\ x = \int v\; dt = \int a\; t\; dt = \frac12 a t^2$$

So the position increases quadratically with time for constant acceleration (constant force)

As for your question about "other basic things in physics": this applies to any situation where there is a constant force. The energy needed to go faster increases (since kinetic energy goes as velocity squared) but the power is velocity times force, so as the speed goes up the power goes up as well.

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  • $\begingroup$ Can't say I understand the equations; don't even know what the symbols stand for; but after persuing the issue the whole day and thinking about it - I started to understand. There's linear, exponential and quadratic rates all in the same relationship here; fascinating. Thank you for the answer. $\endgroup$ – MrAccident Jul 7 '15 at 20:38

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