Philosophy 167: Class 7 - Part 5 - Projectile Motion: Galileo's Response to the Difficulties.
Smith, George E. (George Edwin), 1938-
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And what's Galileo's reply, Salviati's reply? All the difficulties and objections you advance are so well-founded that I deem it impossible to remove them. For my part, I grant them all as I believe our author would also concede them. I admit, it's wonderful when you're writing a dialog that you put the words in the mouth into the author when you're the author of both, right?
I admit that the conclusions demonstrated so in the abstract are altered in the concrete. And are so falsified that horizontal motion is not equitable nor does natural acceleration occur and the word exactly is an insert in the ratio assumed, nor is the line of the projectile parabolic and so on.
But on the other hand I ask you not to reject in our author what other very great men have assumed despite it's falsity. The authority of Archimedes alone should satisfy everyone. Now I jump. Here I add that we may say that Archimedes and others imagined themselves in their theorizing, to be situated at infinite distance from the center, that is of the Earth.
In that case, their set assumptions would not be false and hence, their conclusions were drawn with absolute proof. Then if we wish later to put to use for a finite distance from the center, these conclusions prove by supposing immense remoteness therefrom we must remove from the demonstrated truth, whatever is significant in the fact that our distance from the center is not really infinite.
Though it is such that it can be called immense in comparison with the devices employed by us. And these shots coming to end on the surface of the terrestrial globe, shots, artillery, may alter in shape only insensibly whereas that shape is conceded to be enormously transformed in going to end at its center.
Also, the motion in the horizontal plane, all obstacles being removed ought to be equitable and perpetual, but it will be altered by the air and finally stop. Okay, so what are we gonna make of this theory? Before we even get approved for proposition one, we're conceded that it's very nice mathematically in the abstract, but of course the earth is not flat.
We're not infinitely far from the center, gravity doesn't function on parallel lines, but on lines directed toward the center. And therefore, what's to be said for a solution that assumes gravity does act along parallel lines, and the horizontal is flat? And I leave it to you to figure out exactly what Galileo thinks is going on here.
It's not straightforward. All right, that's the main reply, the reply continues. Next a more considerable disturbance arises from the impediment of the medium. By reason of it's multiple varieties, this is incapable of being subjected to firm rules understood and made into science. Motion in a resisting medium cannot be made into a science.
Considering nearly the impediment that the air makes to the motion in question here. It will be found to disturb them all in infinitive ways according to the infinitely many ways the shapes of movables vary and their heaviness and their speeds. As the speed the greater this is, the greater will be the opposition made to it by the air.
Which will also impede bodies the more, the less heavy they are. Those are correct. The heavier the body, the less deceleration from air. And the faster you go, the greater the deceleration from air. No firm science can be given of such accidenti, of heaviness, speeds, and shape which are variable in infinitely many ways.
Here to deal with such matters, scientifically, it is necessary to abstract away from them. We must find and demonstrate conclusions abstracted from the impediments, in order to make use of them in practice under those limitations that experience will teach us. Indeed, in projectiles that we find practicable, which are those of heavy material and spherical shape.
The deviations from exact parabolic paths will be quite insensible. The last claim, as I've already shown you, is just not true. I don't know if Galileo, they played tennis at the time but tennis wouldn't be the right sport because it was played more like squash than tennis. You had to use all the walls and things like that.
You had a net in the middle but the walls were in play. And if you go to Hampstead Heath, the as it were suburban palace of the British monarchs. Henry VIII tennis court is preserved there, and Newton even writes about how to curve a tennis ball. And so, but I don't know if they threw balls or anything like that, but they did have artillery and they knew perfectly well that artillery was not parabolic as I've already shown you.
As far as the claim though, the claim's much more interesting. Because Newton's gonna try to handle resistance. You'll see he gets in some trouble. But the best way I can say it, and by the way Descartes is going to make the same claim. Though not in public it's in a letter that Newton never saw.
That it's impossible to do a science of air resistance, too many variables are involved. I think I've already told this class one of my favorite stories. We use wind tunnels in the design and development of airplanes because we can't calculate the drag on an airplane. But it's slightly worse than that because the wind tunnel scale but there are three different scaling rules.
There's a fluid mechanical scaling rule. There's a boundary type scaling rule called Freda and there's acoustic scaling which is geometric. And the three are totally different. One's a Reynold's number scaring, another's numbered and a third is geometric. And so when you build the airplane in full scale after testing it in a wind tunnel, I told you the joke the other day from the 1985 paper.
Half of the ones that were studied in this particular study had way too much drag to be functional in as profitable airplanes when built full scale. And all we want to know is which ones not to build. I mean, that's the state we're in now. So one can make a case that Galileo wasn't crazy in saying this, okay?
It's a very, very complicated subject. And we'll keep coming back to it. That's one reason I'm stressing it. I give you here, this is a book you should, if any of you want to learn physics in conjunction with this course, my old friend Tony French, this happens to be Bernard Cohen's copy.
Not mine, but I bequeath it. I have the paperback of this. It's been in print every since the 60s. It's historically accurate. It's the only physics textbook I know of that anybody ever wrote where he took the trouble to be historically accurate. And it's a very, very good book from some points of view.
This is kind of a joke, but it's really an insult to all of you. I asked Tony French why Norton had kept it in print. And he said, because they're very good books. And I said, are they used anywhere as texts anymore? No. Why not? Because you have to be able to read to use them.
And what he meant is, they're not blocked off in four different colors, telling you what's important and what's not, etc. You really have to just read the text. But it's a great book, okay? It's a really wonderful book. Question?
Put the paper back on reserve?
I'll bring my copy of the paper back in.
I've had my own copy since I was a youngster, because it's such a good book. There are four of these books. One on quantum mechanics that's widely used in introductory courses. One on special relativity and one on vibration and waves. But this one is much, much larger and much more comprehensive.
He retired from MIT a few years ago because the person he was living with was Canadian and she wanted to go back to Canada. So he gave it up and he was someone I spent a lot of time with at the Dibner Institute and he was a good friend.
But you see the effects. The important thing here is, for spheres, the resistance, the amount of deceleration is proportional to velocity squared over the radius times the density. So the faster it is, the faster it decelerates. The larger the radius, other things remaining the same, the slower it decelerates.
The greater the density, the slower it decelerates. And you can see that at the top curve, that's just vertical fall for a one centimeter, ten gram. That's not much. That's a pretty small object, folks. One centimeter, ten grams. Ten grams is really very light. And you can see it reaches terminal velocity or very near terminal velocity within five seconds is very different.
The lower curve gives you velocity versus time. But it's a different world. Okay? It's a very, very dramatic effect. Now it's correct that if you have a high enough density object that's large so that you end up that it's very massive, then if the speeds are not to great, the resistance effects can be minimized.
But cannonballs are not the typical example where the speeds are not too great. In fact, those who may have noticed Galileo using the words supernatural speeds. Supernatural speeds are speeds you can't get in vertical fall. Okay. And those are cannonballs speeds, okay? So he refers to those as supernatural cause they can't be produced by nature.
So that's what he means by that? He's not-
Yeah, that's all he means by that.
That's right. Now of course in principle if you dropped it from high enough you might be able to do them, but that's besides the point.
So I hope everybody sees this, it's in effect. We'll come back to this. In next term, we'll spend three weeks on Newton's attempt to handle resistance forces because it's very important to the principia. And we will see it the last two weeks of this course, we'll see his first attempt at doing it.
Wagens tried like hell to do it and couldn't handle anything except resistance proportional to velocity, the math he couldn't handle. Actually, Newton couldn't quite handle the math either, and it took Johann Bernoulli to finally handle the math. But it's fairly extraordinary. Any questions on that? I mean, it's striking now, given your next paper will be about the law of inertia, the law of parabolic projection and then the law of free fall and I just let you see what the problem is with parabolic projection, as Galileo presents it, Isaac asked the right question, what's the standing of this theory?
I don't know how to answer that question. Other than, there's no claim to it being exact. Even in the absence of air resistance. Just due to the curvature of the Earth.