10 Consercation of Momentum 10-1 Newton's Third Law On the basis of Newton's second law of motion,which gives the relation 10-1 Newton's Third Law between the acceleration of any body and the force acting on it,any problem in 10-2 Conservation of momentum mechanics can be solved in principle.For example,to determine the motion of a few particles,one can use the numerical method developed in the preceding chapter. 10-3 Momentum is conserved! But there are good reasons to make a further study of Newton's laws.First,there 10-4 Momentum and energy are quite simple cases of motion which can be analyzed not only by numerical methods,but also by direct mathematical analysis.For example,although we 10-5 Relativistic momentum know that the acceleration of a falling body is 32 ft/sec2,and from this fact could calculate the motion by numerical methods,it is much easier and more satisfactory to analyze the motion and find the general solution,s=so+vot+1612.In the same way,although we can work out the positions of a harmonic oscillator by numerical methods,it is also possible to show analytically that the general solution is a simple cosine function of t,and so it is unnecessary to go to all that arithmetical trouble when there is a simple and more accurate way to get the result.In the same manner,although the motion of one body around the sun,determined by gravitation,can be calculated point by point by the numerical methods of Chapter 9,which show the general shape of the orbit,it is nice also to get the exact shape, which analysis reveals as a perfect ellipse. Unfortunately,there are really very few problems which can be solved exactly by analysis.In the case of the harmonic oscillator,for example,if the spring force is not proportional to the displacement,but is something more complicated,one must fall back on the numerical method.Or if there are two bodies going around the sun,so that the total number of bodies is three,then analysis cannot produce a simple formula for the motion,and in practice the problem must be done numeri- cally.That is the famous three-body problem,which so long challenged human powers of analysis;it is very interesting how long it took people to appreciate the fact that perhaps the powers of mathematical analysis were limited and it might be necessary to use the numerical methods.Today an enormous number of problems that cannot be done analytically are solved by numerical methods,and the old three-body problem,which was supposed to be so difficult,is solved as a matter of routine in exactly the same manner that was described in the preceding chapter,namely,by doing enough arithmetic.However,there are also situations where both methods fail:the simple problems we can do by analysis,and the moderately difficult problems by numerical,arithmetical methods,but the very complicated problems we cannot do by either method.A complicated problem is, for example,the collision of two automobiles,or even the motion of the molecules of a gas.There are countless particles in a cubic millimeter of gas,and it would be ridiculous to try to make calculations with so many variables (about 1017- a hundred million billion).Anything like the motion of the molecules or atoms of a gas or a block or iron,or the motion of the stars in a globular cluster,instead of just two or three planets going around the sun-such problems we cannot do directly,so we have to seek other means. In the situations in which we cannot follow details,we need to know some general properties,that is,general theorems or principles which are consequences of Newton's laws.One of these is the principle of conservation of energy,which was discussed in Chapter 4.Another is the principle of conservation of momentum, the subject of this chapter.Another reason for studying mechanics further is that there are certain patterns of motion that are repeated in many different circum- 10-1
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
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-
cussing collisions in which two bodies collide and stick together, or come together nd bounce apart, we shall first consider two bodies that are held together by a ring 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 erials speed, but now suppose that we have two objects made of different mater 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 alu minum. Evidently there is some right amount in between; so we keep adJusting he 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 foll 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 quanti From this example how quickly we start to infer things if we ar 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 hich in turn makes a prediction abo 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 red at another velocity. From these examples we see that what appeared to be only a definition really involved physic In the development that follows we shall assume it is true that equal mass 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 104
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 10-3 Momentum is conserved! We can verify the above assumptions experimentally: first, that if two station ary objects of equal mass are separated by an explosion they will move apart with the same speed, and second, if two objects of equal mass, coming together with the Fig. 10-1. End view of linear air same speed, collide and stick together they will stop. This we can do by means of trough marvelous invention called an air trough, *which gets rid of friction, the thing which continually bothered Galileo(Fig. 10-1). He could not do experiments by sliding things because they do not slide freely, but, by adding a magic touch, we can today get rid of friction. Our objects will slide without difficulty, on and on at a constant velocity, as advertised by Galileo. This is done by supporting the objects BUMPER SPRING OY PISot CA on air. Because air has very low friction, an object glides along with practically s constant velocity when there is no applied force. First, we use two glide blocks CNND was measured really, but we know that this weight is proportional to the mass), YLIND PISTON BUMPER SPRING and we place a small explosive cap in a closed cylinder between the two blocks (Fig. 10-2). We shall start the blocks from rest at the center point of the track and Fig. 10-2. Sectional view of gliders force them apart by exploding the cap with an electric spark. What should happen? with explosive interaction cylinder attach If the speeds are equal when they fly apart, they should arrive at the ends of the ment trough at the same time. On reaching the ends they will both bounce back with ractically opposite velocity, and will come together and stop at the center where they started. It is a good test; when it is actually done the result is just as we have described(Fig. 10-3) Now the next thing we would like to figure out is what happens in a less simple D situation. Suppose we have two equal masses, one moving with velocity v and the Mm3htme白9与m that if we ride along in a car, physics will look the same as if we are standing still. We start with the knowledge that two equal masses, moving in opposite directions with equal speeds v, will stop dead when they collide. Now suppose that while g.103. of action. this happens, we are riding by in an automobile, at a velocity -. Then what does reaction experiment with equal masses. it look like? Since we are riding along with one of the two masses which are coming together, that one appears to us to have zero velocity. the other mass, however, going the other way with velocity v, will appear to be coming toward us at a velocity 2v(Fig. 10-4). Finally, the combined masses after collision will seem to be passing by with velocity We therefore conclude that an object with velocity 2u, hitting ELocITY. -v) an equal one at rest, will end up with velocity v, or what is mathematically exactly he same, an object with velocity w hitting and sticking to one at rest will produce an object moving with velocity v/2. Note that if we multiply the mass and the locity beforehand and add them together, mo+o, we get the same answer as mm AFTER COLLISION [ mIm hen we multiply the mass and the velocity of everything afterwards, 2m times v/2. So that tells us what happens when a mass of velocity v hits one standing still Fig. 10-4. Two views of an inelastic In exactly the same manner we can deduce what happens when equal objects collision between equal masses. having any two velocities hit each other. Suppose we have two equal bodies with velocities vI and U2, respectively, hich collide and stick together. What is their velocity v after the collision Again we ride by in an automobile, say at velocity u2, so that one body appears to be at rest. The other then appears to have a velocity v1-v2, and we have the same case that we had before. When it is all finished they will be moving at vr-2)with respect to the car. What then is the actual speed on the ground? H. V. Neher and R. B. Leighton, Amer. Jour. of Phys. 31, 255(1963)