The Second Law of Motion

Proposition 25, RULE 2 (See Diagram Prop. 20)

If A and B are unequal in mass, B being greater than A, other conditions being as previously stated, then A alone will be reflected, and each will continue to move at the same speed.

Proof: Because A is supposed to be smaller than B, it will also have less force than B (Prop. 21 Part 2).

But because in this hypothesis, as in the previous one, there is contrariety only in the determination, and so, as we have demonstrated in the previous proposition, variation must occur only in the determination, it will occur only in A and not in B (Ax. 20).

Therefore only A will be reflected in the opposite direction by the stronger B, while reta ining its speed undiminished. Q.E.D.

Proposition 26 (See Diagram Prop. 20)

If A and B are unequal in mass and speed, B being twice the size of A and the motion in A being twice the speed of that in B, other conditions being as before stated, they will both be reflected in the opposite direction, each retaining the speed that it possessed.

Proof: When A and B move toward each other, according to the hypothesis, there is the same amount of motion in the one as in the other (Cor. 2 Prop. 22 Part 2).

Therefore the motion of the one is not contrary to the motion of the other (Cor. Prop. 19 Part 2), and the forces are equal in both (Cor. 2 Prop. 22 Part 2).

Therefore this hypothesis is exactly similar to the hypothesis of Proposition 24 Part 2 and so, according to the same proof, A and B will be reflected in opposite directions, retaining their own motion undiminished. Q.E.D.

Corollary: From these three preceding propositions it is clear that to change the determination of one body requires equal force as to change its motion.

Hence it follows that a body that loses more than half its determination and more than half its motion undergoes more change than one that loses all its determination.

Proposition 27, RULE 3

If A and B are equal in mass but B moves a little faster than A, not only will A be reflected in the opposite direction, but also B will transfer to A half the difference of their speeds, and both will proceed to move in the same direction at the same speed.

Proof: By hypothesis, A is opposed to B not only by its determination but also by its slowness, insofar as it participates in rest (Cor. I Prop. 22 Part 2).

Therefore, even though it is reflected in the opposite direction and only its determination is changed, not all the contrariety of these two bodies is thereby removed.

1. Hence (Ax. 19) there must be a variation both in determination and in motion.

But because B, by hypothesis, moves faster than A, B will be stronger than A (Prop. 22 Part 2).

Therefore a change (Ax. 20) will be produced in A by B, by which it will be reflected in the opposite direction.

2. As long as it moves more slowly than B, A is opposed to B (Cor. 1 Prop. 22 Part 2).

Therefore a variation must occur (Ax. 19) until it does not move more slowly than B.

In this hypothesis there is no cause strong enough to compel it to move faster than B.

So because it can move neither more slowly nor faster than B when it is impelled by B, it will proceed to move at the same speed as B.

Again, if B transfers less than half its excess of speed to A, then A will proceed to move more slowly than B.

If it transfers more than half, then A will proceed to move more quickly than B. But both these possibilities are absurd, as we have just demonstrated.

Therefore, a variation will occur until a point is reached when B has transferred to A half its excess of speed, which B must lose (Prop. 20 Part 2).

And so both will proceed to move with equal speed in the same direction without any contrariety. Q.E.D.

Corollary: Hence it follows that, the greater the speed of a body, the more it is determined to move in the same straight line, and conversely, the more slowly it moves, the less its determination.

Scholium: Lest my readers should here confuse the force of determination with the force of motion, I think it advisable to add a few words wherein the force of determination is explained as distinct from the force of motion.

If bodies A and C are conceived as equal and moving in a straight line toward each other at equal speed, these two bodies (Prop. 24 Part 2) will be reflected in opposite directions, each preserving its own motion undiminished.

But if body C is at B, and moving at an oblique angle toward A, it is clear that it is now less determined to move along the line BD or CA.

So although it possesses motion equal to A’s, yet the force of C’s determination when it moves from directly opposite toward A-a force that is equal to body A’s force of determination-is greater than C’s force of determination when it moves from B toward A.

It is greater in proportion as the line BA is greater than the line CA. For in proportion as BA is greater than CA, so much more time does B require (with B and A moving at the same speed, as is here supposed) to be able to move along the line BD or CA, along which it opposes the determination of body A.

So when C moves from B to meetA at an obl ique angle, it will be determined as if it were to proceed to move along the line AB’ toward B’ (which I suppose, it being at a point where the line AB’ cuts BC produced, to be the same distance from C as C is from B).

But A, retaining its original motion and determination, will proceed to move toward C, and will push body B along with it, because B, as long as it is determined to motion along the diagonal AB’ and moves with the same speed as A, requires more time than A to describe by its motion any part of the line AC.

And to that extent it is opposed to the determination of body A, which is the stronger. But in order for C’s force of determination in moving from B to A, insofar as it participates in the direction CA, to be equal to C’s force of determination in moving directly toward A (or, by hypothesis, equal to A’s force of determination), B will have to have degrees of motion in excess of A in proportion as the line BA is greater than the line CA.

And then, when it meets body A at an oblique angle, A will be reflected in the opposite direction toward N and B toward B’, both retaining their original motion.

But if the excess of B over A is more than the excess of the line BA over the line CA, then B will repel A toward N, and will impart to it as much of its motion as will make the ratio of the motions of B to A the same as the ratio of the line BA to the line CA, and, losing as much motion as it has transferred to A, it will proceed to move in its original direction.

For example, if the line AC is to the line AB as I to 2, and the motion of body A is to that of body B as I to 5, then B will transfer to A one degree of its motion and will repel it in the opposite direction, and B with four remaining degrees of motion will continue to move in its original direction.