The Second Law of Motion

Proposition 28, RULE 4

If a body A is completely at rest and is a little larger than B, with whatever speed B moves toward A it will never move A, but will be repelled by A in the opposite direction, retaining its original motion.

Note 7 that the contrariety of these bodies is removed in three ways.

1. When one takes the other along with it, and they thereafter proceed to move at the same speed in the same direction.

2. When one is reflected in the opposite direction and the other retains its original rest.

3. When one is reflected in the opposite direction and transfers some of its motion to the other, which was at rest.

There can be no fourth possibility (from Prop. 13 Part 2). So we must now demonstrate (by Prop. 23 Part 2) that according to our hypothesis the least change occurs in these bodies.

Proof: If B were to move A until they both proceeded to move atthe same speed, it would have to transfer to A as much of its motion as A acquires (Prop. 20 Part 2) and would have to lose more than half of its motion (Prop. 21 Part 2), and consequently (Cor. Prop. 27 Part 2) more than half of its determination as well.

And so (Cor. Prop. 26 Part 2) it would undergo more change than if it were merely to lose its determination.

And if A were to lose some of its rest, but not so much that it finally proceeded to move with equal speed with B, then the opposition of these two bodies would not be removed.

For A by its slowness, insofar as that participates in rest, will be opposed to B’s speed (Cor. I Prop. 22 Part 2).

And so B will still have to be reflected in the opposite direction and will lose all its determination and part of its motion, which it has transferred to A. This, too, is a greater change than if it were merely to lose its determination.

Therefore, because the change is only in the determination, in accordance with our hypothesis, it will be the least that there can be in these bodies, and therefore (Prop. 23 Part 2) no other change will occur. Q.E.D.

In the proof of this proposition and also in the case of other proofs, we have not quoted Prop. 19 Part 2, in which it is demonstrated that the whole determination can be changed while yet the motion remains unaltered.

Yet attention should be paid to this proposition, so that the force of the proof may be rightly perceived. For in Prop. 23 Part 2 we did not say that the variation will always be the least absolutely, but the least that there can be. But that there can be such a change as we have supposed in this proof, one consisting solely in determination, is evident from Props. 18 and 19 with Cor. Part 2.

Proposition 29, RULE 5

If a body A at rest is smaller than B, them however slowly B moves toward A, it will move it along with it, transferring to it such a part of its motion that both bodies thereafter move at the same speed. (Read Art. 50 Part 2 of the Principia.)

In this rule as in the previous one, only three cases could be conceived in which this opposition would be removed. But we shall demonstrate that, according to our hypothesis, the least change occurs in these bodies.

And so (Prop. 23 Part 2) their variation, too, must occur in this way.

Proof: According to our hypothesis B transfers to A (Prop. 21 Part 2) less than half of its motion and (Cor. Prop. 27 Part 2)8 less than half of its determination. Now if B were not to take A along with it but were to be reflected in the opposite direction, it would lose all its determination, and a greater variation would occur (Cor. Prop. 26 Part 2).

Even greater would be the variation if it lost all its determination and at the same time a part of its motion, as is supposed in the third case. Therefore the variation, in accordance with our hypothesis, is the least. Q.E.D.

Proposition 30, RULE 6

If a body A at rest were exactly equal to a body B, which is moving toward it, to some degree A would be impelled by B, and to some degree B would be repelled by A in the opposite direction.

Here again, as in the preceding Prop., only three cases could be conceived.

And so it must be demonstrated that we are here positing the least variation that there can be.

Proof: If body B takes body A along with it until both are proceeding to move at the same speed, then there will be the same amount of motion in the one as in the other (Prop. 22 Part 2), and (Cor. Prop. 27 Part 2) B will have to lose half its determination and also (Prop. 20 Part 2) half its motion.

But if it is repelled by A in the opposite direction, then it II " will lose all its determination and will retain all its motion (Prop. 18 Part 2). This variation is equal to the former (Cor. I Prop. 26 Part 2). But neither of these possibilities can occur.

For if A were to retain its own state and could change the determination of B, it would necessarily be stronger than B (Ax. 20), which would be contrary to the hypothesis.

If B were to take A along with it until they were both moving at the same speed, B would be stronger than A, which is also contrary to the hypothesis.

Because both of these cases are ruled out, the third case will occur; B will give a slight impulse to A and will be repelled by A. Q.E.D.

Read Art. 51 Part 2 of the Principia.

Proposition 31, RULE 7 (See Diagram Prop. 30)

If B and A are moving in the same direction, A more slowly and B following it more quickly so that it finally overtakes A, and if A is bigger than B, but B’s excess of speed is greater than Ns excess of magnitude, then B will transfer to A so much of its motion that both will thereafter move at the same speed in the same direction.

But if on the other hand, A’s excess of magnitude should be greater than B’s excess of speed, B would be reflected by it in the opposite direction, retaining all its motion.

Read Art. 52 Part 2 of the Principia. Here again, as in the preceding propositions, only three cases can be conceived.

Proof Part 1: B being supposed to be stronger than A (Props. 21 and 22 Part 2) cannot be reflected in the opposite direction by A (Ax. 20).

Therefore, because B is stronger, itwill take A along with it, and in such a way that they proceed to move at the same speed. For then the least change will occur, as can easily be seen from the preceding propositions.

Part 2. B being supposed to be less strong than A (Props. 21 and 22 Part 2) cannot impel A (Ax. 20), nor give it any of its own motion.

Thus (Cor. Prop. 14 Part 2) it will retain all its motion, but not in the same direction , for it is supposed to be impeded by A.

Therefore (according to Chap. 2 Dioptrics) it will be reflected in the opposite direction, not in any other direction, retaining its original motion (Prop. 18 Part 2). Q.E.D.

Note that here and in the preceding propositions we have taken as proved that any body meeting from the opposite direction another body by which it is absolutely impeded from advancing further in the same direction, must be reflected in the opposite direction, not in any other direction. For the understanding of this, read Chap. 2 Dioptrics.

Scholium

Up to this point, to explain the changes of bodies resulting from their impact on each other, we have considered the two bodies as though isolated from all other bodies, that is, without taking into account bodies that surround them on all sides.

But now we shall consider their state and their changes wh ile taking into account bodies that surround them on all sides.