Walking in Galileo’s footstepswith digital shoes

Matteo Siccardi

based on work done in collaboration with S. Arnone and F. Moauro

Brussels, October 25th, 2014

Galileo’s 450th birthday

1564–2014

Work on MechanicsGalilean relativity,uniformly accelerated motion,

I falling bodies, . . .

What did Galileo find?

M. Siccardi (SCIENTIX, 25.10.2014) Walking in Galileo’s footsteps 1 / 6

(Re-)reading a classic

1638 Discourses and Mathematical Demonstrations Relating to Two NewSciences

Three characters discussing (in Italian) propositions putforward (in Latin) by the Author.

3 Reproduce the original experiments!

7 Lack of a (fully-fledged) lab7 Targeting digital natives

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From theory. . . to the experimentSAGREDO [. . . ] It is thus evident by simple computation that (for) a moving bodystarting from rest and acquiring velocity at a rate proportional to the time, [. . . ] thespaces traversed are in the duplicate ratio of the times, i.e., in the ratio of the squaresof the times.

s ∝ t2

SIMPLICIO In truth, I find more pleasure in this simple and clear argument of Sagredothan in the Author’s demonstration which to me appears rather obscure; so that I amconvinced that matters are as described, once having accepted the definition ofuniformly accelerated motion. But as to whether this acceleration is that which onemeets in nature in the case of falling bodies, I am still doubtful; and it seems to me, notonly for my own sake but also for all those who think as I do, that this would be theproper moment to introduce one of those experiments — and there are many of them,I understand — which illustrate in several ways the conclusions reached.SAGREDO The request which you, as a man of science, make, is a very reasonableone; for this is the custom — and properly so — in those sciences where mathematicaldemonstrations are applied to natural phenomena, [. . . ] So far as experiments go theyhave not been neglected by the Author; and often, in his company, I have attempted inthe following manner to assure myself that the acceleration actually experienced byfalling bodies is that above described.

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. . . to the experiment

(back then)

“A piece of wooden moulding or scantling, about 12 cubits long, half a cubitwide, and three finger-breadths thick, was taken; on its edge was cut achannel a little more than one finger in breadth; having made this groovevery straight, smooth, and polished, and having lined it with parchment, alsoas smooth and polished as possible, we rolled along it a hard, smooth, andvery round bronze ball. Having placed this board in a sloping position, bylifting one end some one or two cubits above the other, we rolled the ball, asI was just saying, along the channel, noting, in a manner presently to bedescribed, the time required to make the descent.”

1638 2014M. Siccardi (SCIENTIX, 25.10.2014) Walking in Galileo’s footsteps 3 / 6

The GEOGEBRA applet: a theorem from 1638

THEOREM II, PROPOSITION II

The spaces described by a body falling from rest with a uniformly acceleratedmotion are to each other as the squares of the time-intervals employed intraversing these distances.

exact reproduction“experimental” errorspoint-particlevs. finite size ballrepeatability(data analysis)

We repeated this experiment more than once in order to measure the time with anaccuracy such that the deviation between two observations never exceeded one-tenthof a pulse-beat.

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A possible outcome

f ∆t(s).1 1.0.2 1.3.35 1.8.5 2.0.65 2.3.8 2.61 3.0

Check Galileo’s result

3 with OpenOffice3 or gnuplot3 or on log-log paper

Extra bonus:

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The GEOGEBRA applet: some theorems from 1638

THEOREM II, PROPOSITION II

The spaces described by a body falling from rest with a uniformly acceleratedmotion are to each other as the squares of the time-intervals employed intraversing these distances.

THEOREM IV, PROPOSITION IV

The times of descent along planes of the same length but of differentinclinations are to each other in the inverse ratio of the square roots of theirheights.

THEOREM V, PROPOSITION V

The times of descent along planes of different length, slope and height bear toone another a ratio which is equal to the product of the ratio of the lengths bythe square root of the inverse ratio of their heights.

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Conclusions

we made use of GEOGEBRA as a physics lab

CC-BY-SA applet→ free to download @

http://tube.geogebra.org/material/show/id/111630. . . and modify it (flexibility)

inexpensiveappealing to digital nativesreverse engineering

Thanks for your attention!

@:

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