Conservation of momentum
Now what are the interesting consequences of the above relationship? Suppose, for simplicity, that we have just two interacting particles, possibly of different mass, and numbered $1$ 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 momentum $p_1$ of particle $1$ is equal to minus the rate of change of momentum $p_2$ of particle $2$, or \begin{equation} \label{Eq:I:10:1} dp_1/dt=-dp_2/dt. \end{equation} Now if the rate of change is always equal and opposite, it follows that the total change in the momentum of particle $1$ is equal and opposite to the total change in the momentum of particle $2$; this means that if we add the momentum of particle $1$ 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 \begin{equation} \label{Eq:I:10:2} d(p_1+p_2)/dt=0. \end{equation} There is assumed to be no other force in the problem. If the rate of change of this sum is always zero, that is just another way of saying that the quantity $(p_1+p_2)$ does not change. (This quantity is also written $m_1v_1+m_2v_2$, 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 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 $m_1v_1+m_2v_2$, 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 complicated 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 cannot 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 particles can be expressed as \begin{equation} \label{Eq:I:10:3} m_1v_1+m_2v_2+m_3v_3+\dotsb=\text{a constant}, \end{equation} 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 Newton’s Second Law for each particle, \begin{equation} \label{Eq:I:10:4} F=\ddt{}{t}(mv), \end{equation} is true specifically for the components of force and momentum in any given direction; 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 \begin{equation} \label{Eq:I:10:5} F_x=\ddt{}{t}(mv_x), \end{equation} 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 interesting 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 with a 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’s 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 Newton’s laws and might take a different approach, we shall discuss the laws of impacts and collisions from a completely different point of view. 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 discussing collisions in which two bodies collide and stick together, or come together and bounce apart, we shall first consider two bodies that are held together by a spring or something else, and are then suddenly released and pushed by the spring or perhaps by a little explosion. Further, we shall consider motion in only one direction. First, let us suppose that the two objects are exactly the same, are nice symmetrical objects, and then we have a little explosion between them. After the explosion, one of the bodies will be moving, let us say toward the right, with a velocity $v$. Then it appears reasonable that the other body is moving toward the left with a velocity $v$, because if the objects are alike there is no reason for right or left to be preferred and so the bodies would do something that is symmetrical. This is an illustration of a kind of thinking that is very useful in many problems but would not be brought out if we just started with the formulas. The first result from our experiment is that equal objects will have equal speed, but now suppose that we have two objects made of different materials, say copper and aluminum, and we make the two masses equal. We shall now suppose that if we do the experiment with two masses that are equal, even though the objects are not identical, the velocities will be equal. Someone might object: “But you know, you could do it backwards, you did not have to suppose that. You could define equal masses to mean two masses that acquire equal velocities in this experiment.” We follow that suggestion and make a little explosion between the copper and a very large piece of aluminum, so heavy that the copper flies out and the aluminum hardly budges. That is too much aluminum, so we reduce the amount until there is just a very tiny piece, then when we make the explosion the aluminum goes flying away, and the copper hardly budges. That is not enough aluminum. Evidently there is some right amount in between; so we keep adjusting the amount until the velocities come out equal. Very well then—let us turn it around, and say that when the velocities are equal, the masses are equal. This appears to be just a definition, and it seems remarkable that we can transform physical laws into mere definitions. Nevertheless, there are some physical laws involved, and if we accept this definition of equal masses, we immediately find one of the laws, as follows. Suppose we know from the foregoing experiment that two pieces of matter, $A$ and $B$ (of copper and aluminum), have equal masses, and we compare a third body, say a piece of gold, with the copper in the same manner as above, making sure that its mass is equal to the mass of the copper. If we now make the experiment between the aluminum and the gold, there is nothing in logic that says these masses must be equal; however, the experiment shows that they actually are. So now, by experiment, we have found a new law. A statement of this law might be: If two masses are each equal to a third mass (as determined by equal velocities in this experiment), then they are equal to each other. (This statement does not follow at all from a similar statement used as a postulate regarding mathematical quantities.) From this example we can see how quickly we start to infer things if we are careless. It is not just a definition to say the masses are equal when the velocities are equal, because to say the masses are equal is to imply the mathematical laws of equality, which in turn makes a prediction about an experiment. As a second example, suppose that $A$ and $B$ are found to be equal by doing the experiment with one strength of explosion, which gives a certain velocity; if we then use a stronger explosion, will it be true or not true that the velocities now obtained are equal? Again, in logic there is nothing that can decide this question, but experiment shows that it is true. So, here is another law, which might be stated: If two bodies have equal masses, as measured by equal velocities at one velocity, they will have equal masses when measured at another velocity. From these examples we see that what appeared to be only a definition really involved some laws of physics. In the development that follows we shall assume it is true that equal masses have equal and opposite velocities when an explosion occurs between them. We shall make another assumption in the inverse case: If two identical objects, moving in opposite directions with equal velocities, collide and stick together by some kind of glue, then which way will they be moving after the collision? This is again a symmetrical situation, with no preference between right and left, so we assume that they stand still. We shall also suppose that any two objects of equal mass, even if the objects are made of different materials, which collide and stick together, when moving with the same velocity in opposite directions will come to rest after the collision.