## Philosophy 167: Class 12 - Part 8 - Newton's Mechanics: a General Theory of Impact, and some Experimental Difficulties.

Smith, George E. (George Edwin), 1938-

2014-11-25

There is another early notebook, it's called the Waste Book. It was something from Barnaby Smith, and you know a bound book, with all empty pages that Newton started taking notes in. And he does something really extraordinary with impact to spheres in this. It's where the line is drawn.
If two equal and equally swift bodies meet one another, they shall be reflected so as to move as swiftly from one another after the reflection as they did to one another before it. Now this was definitely before Huygens' had published. There's no issue about this. For first opposed the spherical bodies, e and f, to have a springing or elastic force, so that meeting one another, they will relent and be pressed into a spheroidical figure as drawn.
And in that moment in which there is a period put to their motion towards one another, their figure will be most spheroidical, and their pressure, pression upon one another is at its greatest. And if the endeavor to restore their spherical figure be as much vigorous and forcible as their pressure upon one another was to destroy it, they will gain as much motion from one another after they're parted as they had towards one another before their reflection.
Secondly, suppose they be spherical and absolutely solid, then at the period of their motion toward one another, that is, at the moment of their meeting, their oppression is at the greatest, or rather it's done with the whole force by which their motion is stopped. For their whole motion was stopped by the force of their pressure upon one another in this moment, and there cannot be class succeed diverse degrees of pressure twixt two bodies in one moment.
In other words, the two pressures are equal and opposite. We're seeing the birth of the third law of motion here. Now, so long as neither of these two bodies yield to one another, they will retain the same forcible pressure towards one another, that is so much force as deprived the bodies of their motion towards one another so much to now urge them from one.
And therefore, they shall move from one another as much as they did towards one another before their reflection. This is the same reason when equal, unequal and unequally moved body's reflect, that they would separate from one another with as much motion as they came together. Now this is very early on.
It's probably 1664, 1665, in that period. And the striking thing is, I don't know of any others who saw that when two bodies collide and are perfectly elastic, they deform enormously. In fact, the only thing he doesn't see, and this is very subtle, where's the motion go when that happens?
Why don't they impact perfectly like hard bodies? Because they vibrate. That vibration absorbs motion. That's why we don't have any perfectly elastic bodies. We have perfectly elastic bodies, but they don't rebound the way these people called hard bodies. Because just as he says, that compression happens, but when it relaxes is of course you get a vibration and that absorbs push.
Go ahead. By the way, that's 19th century understanding of what happens on impact. All right. I included that again, to drive home the sense of physical intuitions he had about motion. At the end, and there are two versions of this manuscript, one of which has the section at the end, and the other does not.
Otherwise they're same. The section at the end describes some attempts to measure radius of circulation, et cetera, and he gets very disappointed. And he lists the reasons. And these are all quotes. The bodies here among us being an aggregate of smaller bodies, he is a corpuscularian, have a relenting softness and springiness which makes their contact be for some time in more points than one.
Now he knows they change shape on impact, but he hadn't thought until now that the point of impact doesn't stay the same. When they change shape, there's movement along them. The touching surfaces during the time of contact do slide upon one another more or less, and not at all according to their roughness.
Third, few or none of the bodies have a springiness so strong as to force them from one another with the same vigor that they came together. That's actually because of the vibration I just told you about. And finally, their motions are continually impeded and slackened by the mediums in which they move, air resistance.
So he tries to do experiments like this, he gets nowhere, he drops the topic. That's all we have for the loss of motion paper. But I repeat, the principle thing I want you to see. He's taking on the general problem when everybody else is taking on the problem of perfect spheres.
It's sort of typical of him. Why take on the simple version of the problem if you think you can solve the general one? I'll throw this in. This you've already seen earlier in conjunction with the Huygens solution. This is from the Lectures of the Algebra, the late 1670s, where he gives a total solution for head on spheres, perfectly elastic, based on two principles.
What we now call the Third Law of Motion and this relative speed of separation equals the relative speed of approach. You've seen him using that principle all through here. You saw Huygens proving that principle from fundamental principles. But Newton didn't know Huygens had proved that. That didn't appear until posthumously.