45. Rules will be given for calculating how much the motion of a given body is altered by collision with other bodies.
For us to use these results to work out how individual bodies speed up, slow down, or change direction as a result of collision with other bodies, all we need is
• to calculate the power each body has to produce or resist motion, and
• to accept as a firm principle that the stronger power always produces its effects. This would be easy to calculate for the special case of
• a collision between two perfectly hard bodies in isolation from any other bodies that might affect the outcome.
In that class of special cases the following rules would apply.
46. The first rule.
When two perfectly hard bodies, x and y,
• of the same size
• moving at the same speed
• in opposite directions along a single line
collide head-on, they will come out of the collision still moving at the same speed with the direction of each precisely reversed.
47. The second rule.
When two perfectly hard bodies, x and y, of which
• x is slightly larger than y,
• moving at the same speed
• in opposite directions along a single line
collide head-on, they will come out of the collision still moving at the same speed as before, both moving in the direction in which x had been moving before the collision; that is, y would bounce back but x wouldn’t.
(Text obtained online, see works cited below)
[The following is quotation; my summary and commentary is in brackets.]
Baruch Spinoza
The Principles of Cartesian Philosophy
and Metaphysical Thoughts followed by
Lodewijk Meyer Inaugural Dissertation on Matter (1660)
Part II
Proposition 24
Rule 1
If two bodies, A and B, should be completely equal and should move in a straight line toward each other with equal velocity, on colliding with each other they will both be reflected in the opposite direction with no loss of speed.
In this hypothesis it is evident that, in order that the contrariety of these two bodies should be removed, either both must be reflected in the opposite direction or the one must take the other along with it. For they are contrary to each other only in respect of their determination, not in respect of motion.
[We have two equal bodies moving directly toward each other with equal speed. After colliding, they reflect in the opposite directions at the same speed as before. The two bodies only oppose each other in their direction, so their motions will not cancel each other.]
Proof:
When A and B collide, they must undergo some variation (Ax. 19). But because motion is not contrary to motion (Cor. Prop. 19 Part 2), they will not be compelled to lose any of their motion (Ax. 19). Therefore there will be change only in determination. But we cannot conceive that only the determination of the one, say B, is changed, unless we suppose that A, by which it would have to be changed, is the stronger (Ax. 20). But this would be contrary to the hypothesis. Therefore because there cannot be a change of determination in only the one, there will be a change in both, with A and B changing course in the opposite direction -- but not in any other direction (see what is said in Chap. 2 Dioptrics) -- and preserving their own motion undiminished. Q.E.D.
(Spinoza 73)
[Axiom 19: When bodies having opposite motion collide with each other, they are both -- or at least one of them -- compelled to undergo some change.
(49)
Proposition 19: Motion, regarded in itself, is different from its determination toward a certain direction; and there is no need for a moving body to be for any time at rest in order that it may travel or be repelled in an opposite direction.
Corollary: Hence it follows that motion is not contrary to motion.
(69)
Axiom 20: A change in anything proceeds from a stronger force.
(49)]
[We know that when bodies collide, either or both of their motions must alter in some way. We also know that a body's motion is not the same thing as its direction of motion, because when bodies collide, they can change their direction while maintaining their speeds.
When two bodies collide, for only one to change its direction, the other must have a stronger motion.
But here we hypothesize that both bodies have the same strength of motion. Thus both bodies must change direction, and they travel the opposite direction to their approach, at the same velocity of their approach.]
Spinoza, Principles of Cartesian Philosophy, Part II, Proposition 25, Rule 2:
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.
(73)
[Bodies A and B are moving directly toward each other at the same speed, but B's mass is greater than A's. After colliding, B will continue moving in the same direction at the same speed, but A will be reflected in the opposite direction, at its previous 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 retaining its speed undiminished.
(74)
[Axiom 20: A change in anything proceeds from a stronger force.]
[We know that an object that is moving as fast as another object, but is of greater mass, is moving with more motion, that is, with greater force. We also know that motion is not contrary to motion, which is to say, motion will not cancel other motion. However, the direction of motion can be inverted. So because B's mass is greater than A's mass, only A's direction will change, while B continues the same direction, both at the same speed as before.]
Descartes, René Principles of Philosophy.
Available online:
http://www.earlymoderntexts.com/pdf/descprin.pdf
Spinoza, Baruch. The Principles of Cartesian Philosophy and Metaphysical Thoughts followed by Lodewijk Meyer Inaugural Dissertation on Matter (1660). Transl. Samuel Shirley and Steven Barbone. Indianapolis: Hackett, 1998.
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