stances, so it is good to study these patterns in one particular circumstance. For example, we shall study collisions; different kinds of collisions have much in common. In the flow of fluids, it does not make much difference what the fluld is the laws of the fow are similar. Other problems that we shall study are vibrations and oscillations and, in particular, the peculiar phenomena of mechanical waves- sound vibrations of rods, and so on In our discussion of Newton's laws it was explained that these laws are a kind of program that says"Pay attention to the forces, " and that Newton told us only two things about the nature of forces. In the case of gravitation, he gave us the complete law of the force. In the case of the very complicated forces between he was not the right laws for the for he d one rule, one general property of forces, which is expressed in his Third Law, and that is the total knowledge that Newton had about the nature of forces-the law gravitation and this principle, but no other details This principle is that action equals reaction What is meant is something of this kind Suppose we have two small bodies, say particles, and suppose that the first one exerts a force on the second one, pushing it with a certain force. Then, simultaneously, according to Newton's Third Law, the second particle will push on the first with an equal force, in the opposite direction; furthermore, these forces effectively act in the same line This is the hypothesis, or law, that Newton proposed, and it seems to be quite te, though ne ot exact (we shall discuss the errors later ). For the moment ye shall take it to be true that action equals reaction. Of course, if there is a third for instance, exerts its own push on each of the other two. The result is that the total effect on the first two is in some other direction, and the forces on the first two particles are, in general, neither equal nor opposite. However, the forces on each particle can be resolved into parts, there being one contribution or part due to each other interacting particle. Then each pair of particles has corresponding components of mutual interaction that are equal in magnitude and opposite in irection 10-2 Conservation of momentum Now what are the interesting consequences of the above relationship? Sup pose, for simplicity, that we have just two interacting particles, possibly of different mass, and numbered I and 2. The forces between them are equal and opposite what are the consequences? According to Newton's Second Law, force is the time rate of change of the momentum, so we conclude that the rate of change of momen tum p, of particle l is equal to minus the rate of change of momentum p2 of particle dp1/dt=-dp 2/dt (10.1) Now if the rate of change is always equal and opposite, it follows that the total change in the momentum of particle I is equal and opposite to the total change in the momentum of particle 2; this means that if we add the momentum of particle to the momentum of particle 2, the rate of change of the sum of these, due to the mutual forces(called internal forces) between particles, is zero; that is d(pi p2)/dt (10.2) There is assumed to be no other force in the problem. If the rate of change of this m is always zero, that is just does not change. (This quantity is also written m U1 m2U2, and is called the total momentum of the two particles. We have now obtained the result that the total momentum of the two particles does not change because of any mutual interactions between them. This statement expresses the law of conservation of