10–1Newton’s Third Law 10-1 牛顿第三规律
On the basis of Newton’s second law ofmotion, which gives the relation between the acceleration of any body and theforce acting on it, any problem in mechanics can be solved in principle. For example,to determine the motion of a few particles, one can use the numerical methoddeveloped in the preceding chapter. But there are good reasons to make afurther study of Newton’s laws. First, there are quite simple cases of motionwhich can be analyzed not only by numerical methods, but also by directmathematical analysis. For example, although we know that the acceleration of afalling body is 32 ft/sec², and from this fact could calculate the motion bynumerical methods, it is much easier and more satisfactory to analyze themotion and find the general solution, s=s0+v0t+16t2. In the same way, although we can work out the positions of aharmonic oscillator by numerical methods, it is also possible to showanalytically that the general solution is a simple cosine function of t, and so it is unnecessary to go to all that arithmetical trouble whenthere 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, canbe 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 exactshape, which analysis reveals as a perfect ellipse.
牛顿第二规律，给出了一个物体的加速度，与作用于其上的力的关系，基于该规律，任何力学的问题，原则上都可以解决。例如，要得到几个粒子的运动，可以使用上一章研发出来的数字方法。但是，对牛顿规律，做深入的研究，还是有很好的理由。首先，有些相当简单的运动情形，不仅可以通过数字方法来分析，而且，也可以通过直接的数学方法，来分析。例如，虽然我们知道，一个下落物体的加速度是32 ft/sec²，从此事实出发，通过数字方法，可计算出其运动，但是，通过分析这个运动，找到普遍的方程s=s0+v0t+16t2，会更简单，且更令人满意。以同样的方式，对于一个和谐振荡，虽然我们可以通过数字分析，得到其位置，但是，还是有可能分析性地指出，其普遍方程，是一个简单的t的cosine函数，于是，如果有一个简单的和更准确的方式，能得到结果的话，那么，就没有必要，去受这种算数运算的麻烦了。以同样的方式，一个绕着太阳的物体，其运动，由万有引力决定，虽然其运动，可以通过第9章的数字方式，逐点地计算的出来，以指出其轨道的一般形状，但是，能得到一个准确的形状，当然更好，通过分析，可揭示出，它是一个完美的椭圆。

In the situations in which we cannot followdetails, we need to know some general properties, that is, general theorems orprinciples which are consequences of Newton’s laws. One of these is theprinciple 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 certainpatterns of motion that are repeated in many different circumstances, so it isgood to study these patterns in one particular circumstance. For example, we shallstudy collisions; different kinds of collisions have much in common. In theflow of fluids, it does not make much difference what the fluid is, the laws ofthe flow are similar. Other problems that we shall study are vibrations andoscillations and, in particular, the peculiar phenomena of mechanicalwaves—sound, vibrations of rods, and so on.
在有些情况下，我们无法跟踪细节，所以，我们就需要知道一些一般的属性，也就是说，一般的定理或原理，它们是牛顿规律的推论。其中之一，就是能量守恒原理，我们在第四章讨论过。另一个，就是动量守恒原理，即本章的题目。进一步研究力学的另外一个原因就是，有一些运动的模式，在很多不同的环境中，都被重复着，所以，在同一个具体的环境中，研究它们比较好。例如，我们将研究碰撞；不同类型的碰撞，有很多共同点。在流体的流动中，流体是什么，差别并不大，流动的规律是类似的。我们将研究的另外问题，就是震动和振荡，尤其是，特殊的机械波的现象，机械波包括声音的、棒的震动，等等。

In our discussion of Newton’s laws it wasexplained that these laws are a kind of program that says “Pay attention to theforces,” and that Newton told us only two things about the nature of forces. Inthe case of gravitation, he gave us the complete law of the force. In the caseof the very complicated forces between atoms, he was not aware of the rightlaws for the forces; however, he discovered one rule, one general property of forces,which is expressed in his Third Law, and that is the total knowledge thatNewton had about the nature of forces—the law of gravitation and this principle,but no other details.
This principle is that action equals reaction.
在我们关于牛顿规律的讨论中，已经解释了，这些规律就是某种程序，它们说：“对力要注意”，关于力的本质，牛顿只告诉了我们的两件事情。在万有引力的情况下，他给我们提供了完整的力的规律。对于非常复杂的原子之间的力，他并没有意识到力的正确规律；然而，他发现了一条规则，一条关于力的普遍属性，这在他的第三规律中，得到表达，牛顿关于力的本质的全部知识就是：万有引力的规律，和这个原理，但没有更多的细节了。
这条原理就是：作用力等于反作用力。

What is meant is something of this kind: Supposewe have two small bodies, say particles, and suppose that the first one exertsa force on the second one, pushing it with a certain force. Then,simultaneously, according to Newton’s Third Law, the second particle will pushon the first with an equal force, in the opposite direction; furthermore, theseforces effectively act in the same line. This is the hypothesis, or law, thatNewton proposed, and it seems to be quite accurate, though not exact (we shall discussthe errors later). For the moment we shall take it to be true that action equalsreaction. Of course, if there is a third particle, not on the same line as theother two, the law does not mean that the total force on the first oneis equal to the total force on the second, since the third particle, forinstance, exerts its own push on each of the other two. The result is that thetotal effect on the first two is in some other direction, and the forces on thefirst two particles are, in general, neither equal nor opposite. However, the forceson each particle can be resolved into parts, there being one contribution orpart due to each other interacting particle. Then each pair of particleshas corresponding components of mutual interaction that are equal in magnitudeand opposite in direction.
我们的意思，大致如此：假设我们有两个小的物体，比如说粒子，且假设，第一个粒子，有力作用于第二个之上，即以一定的力推它。那么，同时，依据牛顿第三规律，第二个粒子也会在相反的方向、用同等的力，推第一个；另外，这些力，是在同一条直线上起作用。这个假设或规律，就是牛顿提出的，它似乎是相当准确，虽然不精确（稍后，我们将讨论误差）。目前，我们将把作用力等于反作用力，当作是真的。当然，如果有第三个粒子，不在另外两个的连线上，那么，这个规律并不意味着，作用于在第一个粒子上的总的力，就等于作用于第二个上的总的力，因为第三个力，把它的推力，分别作用于另外两个之上。结果就是，在前面两个粒子上产生的总的效果，方向有所不同，而作用前面两个粒子的力，一般来说，既不相等，也不相反。然而，作用于每个粒子上的力，可被分解成部分，{对这些力的}有些贡献、或者部分贡献，可归于相互作用着的粒子。因此，每对粒子，都有相应的相互作用的分量，这些分量，大小相等，方向相反。

10–2Conservation of momentum 10-2 动量的守恒
Now what are the interesting consequences ofthe above relationship? Suppose, for simplicity, that we have just two interactingparticles, 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 themomentum, so we conclude that the rate of change of momentum p1of particle 1 is equal to minus the rate of change of momentum p2of particle 2 , or
dp1 /dt=−dp2/dt. (10.1)
现在，上面的关系，会带来什么样有趣的后果呢？为了简单，假设我们只有两个粒子，相互作用，质量大概也不同，标号为1和2。它们之间的力，相等且相反；那么，后果是什么呢？根据牛顿第二规律，力就是动量随时间的变化率，于是，我们得出结论，粒子1的动量的时间变化率，与负的、粒子2的动量的时间变化率，相等，或者：
dp1 /dt=−dp2/dt. (10.1)

Now if the rate of change is always equal and opposite, itfollows that the total change in the momentum of particle 1 is equal and opposite to the total change in the momentum ofparticle 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
d(p1+ p2)/dt=0. (10.2)
现在，如果变化率，总是相等、且相反，那么，粒子1的动量的总变化，就是相反且等于：粒子2的动量的总变化；这意味着，如果我们把粒子1的动量与粒子2的动量相加，那么，这个和的变化率，由于粒子间的相互作用力（被称为内部作用力），就是零，也就是说：
d(p1+ p2)/dt=0. (10.2)

There is assumed to be no other force in the problem. If the rate ofchange of this sum is always zero, that is just another way of saying that thequantity (p1 + p2) does not change. (This quantity is also written m1v1+m2v2, and is called the total momentum of the two particles.) Wehave now obtained the result that the total momentum of the two particles doesnot change because of any mutual interactions between them. This statement expressesthe law of conservation of momentum in that particular example. We concludethat if there is any kind of force, no matter how complicated, between twoparticles, and we measure or calculate m1v1+m2v2, that is, the sum of the two momenta, both before and after theforces act, the results should be equal, i.e., the total momentum is aconstant.
假设在这个问题中，没有任何其他的力。如果这个总和的变化率总是零，那正是：‘量(p1 + p2)并不变化’这一说法的另一种说法。（这个量也被写为 m1v1+m2v2，且被称为此两个粒子的总动量。）我们现在已经获得一个结果，即对于两个粒子，不论它们之间的相互作用为何，其总动量不变。这一表述，表达了这个特殊例子中的动量守恒规律。我们得出结论说，在两个粒子之间，如果有任何力，无论它多么复杂，我们测量或计算m1v1+m2v2，也就是说，在力作用于它们之前和之后，测两个动量之和，结果应该相等，亦即，总动量是一个常数。

If we extend the argument to three or moreinteracting particles in more complicated circumstances, it is evident that sofar as internal forces are concerned, the total momentum of all the particlesstays constant, since an increase in momentum of one, due to another, is exactlycompensated by the decrease of the second, due to the first. That is, all theinternal forces will balance out, and therefore cannot change the totalmomentum 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.
如果我们把这种论证，扩展到有三个以上粒子的更复杂的情况，则很明显，只要所考虑的只是内部的力，那么，所有粒子的总动量，就是常数，由于一个粒子动量的增加，正是被另一粒子动量的减少所补偿，前一粒子动量的增加，可归于后一粒子，而后一粒子动量的减少，则又可归于前一粒子。也就是说，所有的内部力量，都会互相平衡，因此，不会改变粒子的总动量。因此，如果没有来自外部的力（外力），则将没有任何力，可以改变总动量，所以，总动量就是个常数。

It is worth describing what happens ifthere are forces that do not come from the mutual actions of theparticles in question: suppose we isolate the interacting particles. If thereare only mutual forces, then, as before, the total momentum of the particlesdoes not change, no matter how complicated the forces. On the other hand,suppose there are also forces coming from the particles outside the isolatedgroup. Any force exerted by outside bodies on inside bodies, we call an externalforce. We shall later demonstrate that the sum of all external forces equals therate of change of the total momentum of all the particles inside, a very usefultheorem.
假设我们把相互作用着的粒子，隔离开来，那么，如果有力，不是来自这些粒子间的相互作用，那么，这时究竟会发生什么，就值得描述了。如果像以前一样，只有相互的力，那么，不论这些力，多么复杂，粒子间的总动量并不变化。另一方面，假设还有力，来自于其他粒子，这些粒子，在这个被隔离的组之外。任何外部物体作用于内部物体之上的力，我们都称之为：外部的力。稍后，我们将演示，所有外部力的总和，等于所有内部粒子的总动量的变化率，一个非常有用的定理。

The conservation of the total momentum of anumber of interacting particles can be expressed as
m1v1+m2v2+m3v3+⋯=aconstant, (10.3)
if there are no net external forces. Here the masses and correspondingvelocities of the particles are numbered 1 , 2 , 3 , 4 , … The general statement of Newton’s Second Law for each particle,
F=d (mv)/dt, (10.4)
is true specifically for the components of force and momentumin 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
Fx=d (mvx)/dt, (10.5)
and similarly for the y - and z -directions. Therefore Eq. (10.3)is really three equations, one for each direction.
若干相互作用着的粒子，如果没有净外部的力，则其总动量的守恒，可表示为：
m1v1+m2v2+m3v3+⋯=a 常数, (10.3)
这里，粒子的质量和矢量速度，被标识为1 , 2 , 3 , 4 , … 。每个粒子的牛顿第二规律的普遍声明为：
F=d (mv)/dt, (10.4)
它在任何被给予的方向上的力和动量的具体的分量，都是真的；这样，一个粒子的x分量，就等于该粒子的动量变化率的x分量，或：
Fx=d (mvx)/dt, (10.5)
y方向、z方向类似。所以，方程（10.3）真正来说是3个方程，每个方向一个。

In addition to the law of conservation ofmomentum, there is another interesting consequence of Newton’s Second Law, to beproved later, but merely stated now. This principle is that the laws of physicswill look the same whether we are standing still or moving with a uniform speedin a straight line. For example, a child bouncing a ball in an airplane finds thatthe ball bounces the same as though he were bouncing it on the ground. Eventhough the airplane is moving with a very high velocity, unless it changes itsvelocity, the laws look the same to the child as they do when the airplane isstanding still. This is the so-called relativity principle. As we use ithere we shall call it “Galilean relativity” to distinguish it from the morecareful analysis made by Einstein, which we shall study later.
除了能量守恒规律之外，牛顿第二规律，还有另一有趣后果，这里先提一下，后面我们会证明它。这个原理就是：无论我们是静止的，还是在做匀速直线运运动，物理学的规律，看上去都一样。例如，一个小孩在飞机上拍一个球，与它在地面上拍一个球，是一样的，虽然飞机是在高速运行，只要它不转弯，那么，对于小孩来说，这个规律看上去，与飞机是静止的时候，是一样的。这就是所谓的相对原理。由于我们这里要用它，我们将称它为“伽利略的相对性原理”，以区别于爱因斯坦的，后者经过更仔细的分析，稍后我们会学习。

We have just derived the law of conservationof momentum from Newton’s laws, and we could go on from here to find thespecial 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 inother circumstances where, for example, one might not know Newton’s laws andmight take a different approach, we shall discuss the laws of impacts andcollisions from a completely different point of view. We shall base ourdiscussion on the principle of Galilean relativity, stated above, and shall endup with the law of conservation of momentum.
我们刚从牛顿规律出发，导出了动量守恒规律，我们可以继续从这里出发，找出那些用来描述碰撞和撞击的特殊规律。但是，为了有所变化，以及为了展示某种推理，我们将从一个完全不同的角度，来讨论碰撞和撞击的规律；这里所说的某种推理，是指可用于物理学中的其他情形中的推理，例如，某人可能不知道牛顿规律，从而，在解决问题时，可能会采取不同的解决办法。我们的讨论，将基于上面提到的伽利略的相对性原理，结束于动量守恒规律。

We shall start by assuming that naturewould look the same if we run along at a certain speed and watch it as it wouldif we were standing still. Before discussing collisions in which two bodiescollide and stick together, or come together and bounce apart, we shall firstconsider two bodies that are held together by a spring or something else, andare then suddenly released and pushed by the spring or perhaps by a littleexplosion. Further, we shall consider motion in only one direction. First, letus suppose that the two objects are exactly the same, are nice symmetricalobjects, 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 avelocity v . Then it appears reasonable that the other body is moving toward theleft with a velocity v , because if the objects are alike there is no reason for right orleft to be preferred and so the bodies would do something that is symmetrical. Thisis an illustration of a kind of thinking that is very useful in many problemsbut would not be brought out if we just started with the formulas.
我们将这样开始，即假设我们是这样看自然的，要么是以某种速度跑着看，要么我们是静止地看，且无论以哪种方法看，自然都是一样的。碰撞，就是两个物体相撞后，粘在一起、或者又分开了；在我们讨论碰撞之前，我们将首先考虑一种情况：两个物体是由一个弹簧或其他某种东西，连在一起，然后，突然地被释放，被弹簧或一个小的爆炸推动。另外，我们将只考虑一个方向上的运动。首先，我们假设，两个对象，几乎一样，是对称性非常好的对象，然后，它们之间，有了一个小的爆炸。爆炸之后，一个对象开始移动，我们说它向右，速度为v。因此，似乎完全有理由说，另外一个物体，向左运动，速度为v，因为，如果对象是一样的，那么，就没有理由说：对象会倾向于向右或向左，所以，物体将会做某种对称性的运动。这里所展示的，是一种思维方式，它在很多问题中，都非常有用，但是，如果开始我们只讲公式的话，则此思维，无法开启。

The first result from our experiment is thatequal objects will have equal speed, but now suppose that we have two objectsmade of different materials, say copper and aluminum, and we make the two massesequal. We shall now suppose that if we do the experiment with two masses thatare 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 nothave to suppose that. You could define equal masses to mean twomasses that acquire equal velocities in this experiment.” We follow that suggestionand make a little explosion between the copper and a very large piece ofaluminum, 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 tinypiece, then when we make the explosion the aluminum goes flying away, and the copperhardly budges. That is not enough aluminum. Evidently there is some rightamount in between; so we keep adjusting the amount until the velocities comeout equal. Very well then—let us turn it around, and say that when thevelocities are equal, the masses are equal. This appears to be just adefinition, and it seems remarkable that we can transform physical laws into meredefinitions. Nevertheless, there are some physical laws involved, and ifwe accept this definition of equal masses, we immediately find one of the laws,as follows.
从我们的实验，得到的第一个结果，就是相等的对象，将有相等的速度，但是，现在假设我们有两个对象，由不同材料组成，比如铜和铝，我们要让这两个对象的质量相等。我们现在要假设，如果我们做实验，用两个对象，其质量相等，尽管其材料不同，但其矢量速度是相等的。有人会反对说：“但是，你不用那样假设，你也可以反过来做。你可以把相等的质量，定义为：在这个实验中，获得相等速度的两个质量。”我们采纳这个建议，然后，在铜和一大块铝块之间，做了一个爆炸，这块铝太重，以至于铜飞出去了，而铝几乎都没挪动一点。铝太多了，于是我们减少其量，直到变成非常小的一块，然后，我们再做爆炸，结果铝飞出去了，而铜却几乎没挪动一点。这是铝不够。很明显，它们之间，有一个正确的量；于是，我们持续调整，直到飞出去的速度相等。很好，现在我们反过来说，当速度相等的时候，质量就是相等的！这似乎只是一个定义，并且，我们能把物理规律转化成单纯的定义，似乎是值得注意的。尽管如此，还是牵扯到了一些物理规律，如果我们接受这个相等质量的定义，那么，我们立即就会发现这些规律中的一个，见下面。

Suppose we know from the foregoing experimentthat two pieces of matter, A and B (of copper and aluminum), have equal masses, and we compare a thirdbody, say a piece of gold, with the copper in the same manner as above, makingsure that its mass is equal to the mass of the copper. If we now make theexperiment between the aluminum and the gold, there is nothing in logic thatsays these masses must be equal; however, the experiment showsthat they actually are. So now, by experiment, we have found a new law. Astatement 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 toeach other. (This statement does not follow at all from a similarstatement used as a postulate regarding mathematical quantities.) Fromthis example we can see how quickly we start to infer things if we arecareless. It is not just a definition to say the masses are equal whenthe velocities are equal, because to say the masses are equal is to imply themathematical laws of equality, which in turn makes a prediction about anexperiment.
假设从上述实验出发，我们知道了，两块物体，A和B（铜和铝），质量相等；我们再拿第三个物体，比如一块金子，以同样的方式，与铜进行比较，保证其质量与铜的质量相等。如果然后，我们在铝和金之间，做此实验，那么，没有任何逻辑可以说，其质量应该相等；然而，实验指出，其质量确实相等。于是现在，通过实验，我们发现了一条新的规律。这条规律，可如此陈述：如果两个质量，分别等于第三个质量（如在这个实验中，通过相等的矢量速度），那么，它们就是相等的。（在数学中，就数学的量而言，有类似的公设，但是这个陈述，并不遵循这种公设）从这个例子，我们可以看出，如果我们粗心大意的话，我们{开始}推导事情，会有多快。说‘速度相等时质量就相等’，并不仅仅是一个定义，因为，说质量是相等的，就暗示着：数学相等的规律；而它{数学规律}反过来，就可对实验，做一个预告。