Philosophy 167: Class 10 - Part 7 - Impact of Hard Spheres: Newton's Algebraic Solution, and Huygens' Relativity Principle.
Smith, George E. (George Edwin), 1938-
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Newton has a series of lectures on algebra teaching his students. I can't believe he had any students but we know he had a few. Teaching them algebra. And mixed among the problems. The problem right after this has cows eating in a field at two different rates, and you're supposed to work out how long the field is gonna last.
Is the collision problem. But he does it algebraically. So if you look down here, this is the solution I just gave you. Okay, it's the very same algebraic solution as on the prior page, prior sheet. But he's doing it from the following. Let's read the problem. Having given the magnitude and motion of spherical bodies perfectly elastic, moving in the same right line, and striking against one another to determine their motions after reflection.
That's the problem. The before and after problem. And here's how he says to do it. The resolution of this question depends on their, on these conditions. That each body will suffer as much reaction as the action of each is upon the other. That's the first occurrence that we know of.
Not quite. It's virtually the first occurrence of Newton's Third Law of Motion. And second, and that they will recoil from each other after reflection. With the same velocity or swiftness as they met before. What we mean here is take the relative speed of approach. It equals the relative speed of separation after collision.
Those are the two principles. And he derives the same solution quite concisely from those two principles. Okay? So Newton is getting the same result. Already knowing Huygens' result of course, but getting it slightly differently. The actual paper that Huygens sent, I'm only showing you the first two pages, in supplementary material I show you the whole thing.
This is Christian Huygens' handwriting in Latin. You'll notice at the top in a different handwriting, read January 7th 1668. In other words this paper was read to a meeting of the Royal Society on that day. So everybody at the Royal Society heard it, but they didn't publish the paper.
Nevertheless, they only purchased the Redman Wallace paper. The number three down here is a very important principle I'm going to get to in just a moment. I'll give it to you in English. I wanted you to see also his drawings. Since you've seen the ones that appeared in the Opus Posthumous with the two boats moving.
He generally didn't draw boats. He just drew the hands. But you can see, it's relatively clear. Now, this paper sat at the Royal Society. It's still at the Royal Society. So if you go to the Royal Society and request it, what you'll probably get is currently a copy of it cuz they're rather protective of originals unless you've got very special credentials for wanting to see something like a watermark in it to date it, or something like that.
But Newton had at least two, probably three visits to the Royal Society during the 1670s. He may have asked to read this paper. And the reason it's interesting is this principle number three which I am now giving you as, this is my translation of that principle. The motion of their bodies and their equal and unequal speeds are to be understood respectively in relation to other bodies which are considered as at rest, even though perhaps both the former and the latter are involved in the common motion.
So we have to refer motion to a third body. And accordingly, when two bodies collide with one another, even if both together are further subject to another uniform motion. They will move each other with respect to a body that is carried by the same common motion no differently than if this motion coming from outside were absent to wall.
This is a generic statement of Galileo's claim that motion inside the hull of a ship is going to be the same whether the ship is standing or not. But this is a very general statement of a relativity principle for all collision. Okay, and the reason it's so interesting is when Newton first gets going on the Principia, he states almost exactly this principle.
Principle of relativity. And, the interesting question is whether he had seen it at the Royal Society. And then the very next paragraph, this is of course, now the figure, this is not his figure. This is a figure other people put in on this posthumous work. Thus, if someone conveyed on a boat that is moving with a uniform motion were to cause equal balls to strike one another at equal speeds with respect to himself and the parts of the boat.
We say that both should rebound also at equal speeds with respect to the same passenger. Just as would clearly happen if he were to cause the same balls to collide at equal speeds in a boat at rest and while standing on the ground. Okay, so, not only has he stated the relativity principle, he's giving you a way to work with it.