momentum in that particular example. We conclude that if there is any kind of force, no matter how complicated, between two particles, and we measure or calculate m101 m202, that is, the sum of the two momenta, both before and after the forces act, the results should be equal, i.e., the total momentum is a constant. If we extend the argument to three or more interacting particles in more com- plicated circumstances, it is evident that so far as internal forces are concerned, the total momentum of all the particles stays constant, since an increase in momentum of one, due to another, is exactly compensated by the decrease of the second due to the first. That is, all the internal forces will balance out and therefore annot change the total momentum of the particles. Then if there are no forces from the outside(external forces), there are no forces that can change the total momentum: hence the total momentum is a constant It is worth describing what happens if there are forces that do not come from the mutual actions of the particles in question: suppose we isolate the interacting particles. If there are only mutual forces, then, as before, the total momentum of the particles does not change, no matter how complicated the forces. On the other hand, suppose there are also forces coming from the particles outside the isolated group. Any force exerted by outside bodies on inside bodies, we call an external force. We shall later demonstrate that the sum of all external forces equals the rate of change of the total momentum of all the particles inside, a very useful theorem The conservation of the total momentum of a number of interacting particl can be expressed as m11+m22+m303 a constant (10.3) if there are no net external forces. Here the masses and corresponding velocities of the particles are numbered 1, 2, 3, 4, .. the general statement of Newtons Second law for each particle (10.4) is true specifically for the components of force and momentum in any given direc- tion: thus the x-component of the force on a particle is equal to the x-component of the rate of change of momentum of that particle, or and similarly for the y- and z-directions. Therefore Eq. (10.3)is really three equations, one for each direction In addition to the law of conservation of momentum there is another inter esting consequence of Newton's Second Law, to be proved later, but merely stated now. This principle is that the laws of physics will look the same whether we are standing still or moving with a uniform speed in a straight line. For example, a child bouncing a ball in an airplane finds that the ball bounces the same as though he were bouncing it on the ground. Even though the airplane is moving very high velocity, unless it changes its velocity, the laws look the same to the child as they do when the airplane is standing still. This is the so-called relativity principle. As we use it here we shall call it"Galilean relativity"to distinguish it from the more careful analysis made by Einstein, which we shall study later We have just derived the law of conservation of momentum from Newton laws, and we could go on from here to find the special laws that describe Impacts and collisions. But for the sake of variety, and also as an illustration of a kind of reasoning that can be used in physics in other circumstances where, for example, one might not know Newtons laws and might take a different approach, we shall discuss the laws of impacts and collisions from a completely different point of iew. We shall base our discussion on the principle of Galilean relativity, stated above, and shall end up with the law of conservation of momentum We shall start by assuming that nature would look the same if we run along at a certain speed and watch it as it would if we were standing still. before dis-