As a second example, suppose that A and B are found to be equal by doing the experiment with one strength ofexplosion, 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 showsthat it is true. So, here is another law, which might be stated: If twobodies have equal masses, as measured by equal velocities at one velocity, theywill have equal masses when measured at another velocity. From these examples wesee that what appeared to be only a definition really involved some laws ofphysics.
作为第二个例子，假定我们的实验，是用同样强度的爆炸，来产生一定的速度，A和B，就是通过这个实验，被发现是相等的；然后，如果我们使用一个更强的爆炸，那么，现在得到的速度还会相等，这是真的还是假的？再者，逻辑上，没有任何东西，可以决定这个问题，但实验证明，是真的。于是，这里就有另外一条规律，可陈述为：如果两个物体，被用某一个速度来测量，其速度相等，那么，它们就拥有相等的质量；当用另外一个速度来测时，它们依然拥有相等的质量。从这些例子，我们可以看到，有些东西，似乎只是定义，但实际上，它们可能也包含着一些物理规律。

In the development that follows we shallassume it is true that equal masses have equal and opposite velocities when anexplosion 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 movingafter the collision? This is again a symmetrical situation, with no preference betweenright and left, so we assume that they stand still. We shall also suppose thatany 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 inopposite directions will come to rest after the collision.
在接下来的发展中，我们将假定，对于两个同等的质量，当一个爆炸发生在它们之间后，它们将拥有相同且相反的速度。我们还要做另外一个相反的假定，如果两个完全同等的对象，以相反的方向、同样的速度运动，相撞并被某种胶之类的东西，粘在一起，那么碰撞后，它们将会向哪个方向运动呢？这又是一个对称的情况，对左和右，并无倾向，所以，我们假定它们将会静止。我们还将假设，任何两个质量相等的对象，即便它们是由不同材料所组成，当它们以同等的速度、相反的方向运动时，那么，当它们碰撞并粘在一起后，也将会停止。

10–3Momentum is conserved! 10-3 动量是守恒的！

Fig. 10–1.End view of linear air trough. 图10-1 一个线性空气槽的端视图

Fig. 10–2.Sectional view of gliders with explosiveinteraction cylinder attachment. 滑翔器的断面图，中间有爆炸交互作用气缸
We can verify the above assumptions experimentally:first, that if two stationary objects of equal mass are separated by anexplosion they will move apart with the same speed, and second, if two objectsof equal mass, coming together with the same speed, collide and stick togetherthey will stop. This we can do by means of a marvelous invention called an airtrough,1which gets rid of friction, the thing which continually bothered Galileo (Fig. 10–1). Hecould 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 objectswill slide without difficulty, on and on at a constant velocity, as advertisedby Galileo. This is done by supporting the objects on air. Because air has verylow friction, an object glides along with practically constant velocity whenthere is no applied force. First, we use two glide blocks which have been madecarefully to have the same weight, or mass (their weight was measured really,but we know that this weight is proportional to the mass), and we place a smallexplosive cap in a closed cylinder between the two blocks (Fig. 10–2). Weshall start the blocks from rest at the center point of the track and forcethem apart by exploding the cap with an electric spark. What should happen? Ifthe speeds are equal when they fly apart, they should arrive at the ends of thetrough at the same time. On reaching the ends they will both bounce back withpractically opposite velocity, and will come together and stop at the center wherethey started. It is a good test; when it is actually done the result is just aswe have described (Fig. 10–3).
我们可以通过实验，验证上面的假设：第一，如果两个静止的、同等质量的对象，被一个爆炸分开，它们将以同样的速度，分开移动，第二，如果两个同等质量的对象，以同样的速度，相向而行，碰撞并粘在一起，它们将会停止。有一个了不起的发明，被称为空气槽（脚注1），通过它，可以完成此实验，它剔除了摩擦，这是一直困扰着伽利略的事情（图10-1）。伽利略不能通过滑动的东西，来做这个实验，因为，它们不能自由滑动，但是，通过这个神奇的槽，我们今天就可以剔除摩擦。我们的对象，会以一个恒速，毫无困难地滑开，正如伽利略所宣传的那样。这是通过在空气中支撑对象，来完成的。因为空气的摩擦力很低，在空气中，当没有力作用于其上时，一个对象实际上会以恒速滑动。首先，我们使用两个滑翔块，他们的被仔细地制作，以保证有同样的重量，或者质量，（它们的重量，被仔细地测量过，但我们知道，重量正比于质量），两个块之间，有一个气缸，我们的把一个小的爆炸物，放在气缸中（图10-2）。开始时，两个块在轨道的中间，静止着，我们用一个电子打火装置，点燃爆炸物，迫使两个块分开。如果它们飞开时，速度相同，它们就应该同时到达槽的尽头。到尽头后，会被反弹回来，以实际上同样的速度，方向相反，会在中间相会并停止，中间是它们开始的地方。这是一个好的实验；实际做时，结果正如所期。（图10-3）

Fig. 10–3.Schematic view of action-reactionexperiment with equal masses. 图10-3 用相等质量所做的作用力与反作用力的示意图

Now the next thing we would like to figureout is what happens in a less simple situation. Suppose we have two equalmasses, one moving with velocity v and the other standing still, and they collide and stick; what isgoing to happen? There is a mass 2m altogether when we are finished, drifting with an unknown velocity.What velocity? That is the problem. To find the answer, we make the assumptionthat if we ride along in a car, physics will look the same as if we arestanding still. We start with the knowledge that two equal masses, moving inopposite directions with equal speeds v , will stop dead when they collide. Now suppose that while thishappens, we are riding by in an automobile, at a velocity −v . Then what does it look like? Since we are riding along with one ofthe two masses which are coming together, that one appears to us to have zerovelocity. 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 withvelocity v . We therefore conclude that an object with velocity 2v , hitting an equal one at rest, will end up with velocity v, or what is mathematically exactly the same, an object withvelocity v hitting and sticking to one at rest will produce an object moving withvelocity v/2 . Note that if we multiply the mass and the velocity beforehand and addthem together, mv+0 , we get the same answer as when we multiply the mass and the velocityof everything afterwards, 2m times v/2 . So that tells us what happens when a mass of velocity v hits one standing still.
下一个我们希望搞清的事情，就是在一种简单的情况下，会发生什么。假设我们有两个相等的质量，一个以速度v运动，另一个静止，它们碰撞并粘连；下面会发生什么呢？碰撞后，总的质量是2m，以一个未知的速度在漂移。速度是什么呢？这就是问题。要找到答案，我们做一个假设，如果我们乘一辆车，那么，我们看这个速度为v的质量，与我们静止时，看一个速度为0的质量，物理上是一样的。开始时，我们的知识就是：两个相等的质量，以同等速度v，相对而行，碰撞之后，就会停止。现在假设，当这事发生时，我们在旁边，乘着一辆汽车，以-v的速度跟着。那么，看上去像什么呢？由于我们是陪着一个质量一起走，该质量对我们来说，速度为零。另外一个质量，从另一个方向过来，速度对我们来说，就是2v。（图10-4）。最后，碰撞后的混合质量，速度显得就是v。因此，我们得出结论，一个速度为2v的对象，撞上另一个同等的、静止的对象，最后速度是v，或者，用数学表达也是同样，一个速度为v的对象，碰撞并粘上另外一个静止的对象，将会产生一个移动速度为v/2的对象。注意，如果我们把碰撞前的质量乘以速度，加起来，即mv+0，那么，它与碰撞后的质量乘以速度，即2m乘以v/2，是同样的。所以，这就告诉了我们：当一个速度为v的质量，撞上另一个静止的质量，会发生什么。

Fig. 10–4.Two views of an inelasticcollision between equal masses. 图10-4 两个视图：两个相等质量，非弹性碰撞。

In exactly the same manner we can deducewhat happens when equal objects having any two velocities hit eachother.
以几乎同样的方式，我们可以推出，当两个相等的对象，以任意速度相撞，会发生什么。

Fig. 10–5.Two views of another inelasticcollision between equal masses. 图10-5 两个视图:两个相等质量的另一种非弹性碰撞
Suppose we have two equal bodies withvelocities v1 and v2 , respectively, which collide and stick together. What is theirvelocity v after the collision? Again we ride by in an automobile, say atvelocity v2 , so that one body appears to be at rest. The other then appears tohave a velocity v1− v2 , and we have the same case that we had before. When it is allfinished they will be moving at 1/2(v1− v2)with respect to the car. What then is the actual speed on the ground?It is v=1/2(v1− v2)+ v2 or 1/2(v1+v2) (Fig. 10–5).Again we note that
m v1+m v2=2m(v1+ v2)/2. (10.6)
假设我们有两个相等的物体，速度分别为v1和v2，碰撞并粘在一起，它们碰撞后的速度v是什么呢？我们还是乘一辆车，按速度v2走，这样的话，一个物体对我们来说就是静止的，另外一个物体的速度，就显得是 v1− v2，这样，我们的情况，就与前面的一样。当碰撞结束后，相对于车而言，它们将以 1/2(v1− v2)的速度移动。那么，实际地面上的速度是什么呢？它是v=1/2(v1− v2)+v2 ，或者 1/2(v1+v2) (图 10–5). 我们还注意到：
m v1+mv2=2m(v1+ v2)/2. (10.6)

Thus, using this principle, we can analyze any kind of collision inwhich two bodies of equal mass hit each other and stick. In fact, although wehave worked only in one dimension, we can find out a great deal about much morecomplicated collisions by imagining that we are riding by in a car in someoblique direction. The principle is the same, but the details get somewhat complicated.
两个物体，质量相等，通过撞击及粘连，会产生各种类型的碰撞，对此，利用这个原理，我们都可进行分析。事实上，虽然我们研究的，只是一维的，对于更复杂的碰撞，我们可以通过想象：我们的乘着一辆车，在某种倾斜的方向运动，来找到更多的信息。原理一样，只是细节要更复杂一些。

The next problem that we want to work outis what happens if we have two different masses. Let us take a mass mand a mass 2m and apply our explosive interaction. What will happen then? If, as a resultof the explosion, m moves with velocity v , with what velocity does 2m move? The experiment we have just done may be repeated with zeroseparation between the second and third masses, and when we try it we get thesame result, namely, the reacting masses m and 2m attain velocities −v and v/2 . Thus the direct reaction between m and 2m gives the same result as the symmetrical reaction between m and m , followed by a collision between m and a third mass m in which they stick together. Furthermore, we find that the masses mand 2m returning from the ends of the trough, with their velocities (nearly)exactly reversed, stop dead if they stick together.
我们想搞清的下一个问题，就是如果两个对象的质量不一样，会发生什么？我们一个质量取m，另一个取2m，然后，应用我们的爆炸相互作用。会发生什么呢？作为爆炸的结果，如果m的速度是v，那么2m移动的速度是多少呢？我们可以重复刚做的那个实验，让第二和第三质量之间的间隔为零，然后，开始实验，结果相同，质量m和2m的速度分别为−v和 v/2。这样，m和2m之间的直接反应，与m和m之间的对称反应，结果一样，后者中，在m与第三个质量m相撞，并粘在一起。另外，我们发现质量，m和2m，从那个槽的端点返回，它们的速度，基本被反转，如果它们粘在一起的话，就会停下来。

Fig. 10–7.Two views of an inelasticcollision between m and 2m . 图10-7 两个视图：m与2m之间的一个非弹性碰撞。
Now the next question we may ask is this.What will happen if a mass m with velocity v , say, hits and sticks to another mass 2m at rest? This is very easy to answer using our principle of Galileanrelativity, for we simply watch the collision which we have just described froma car moving with velocity −v/2 (Fig. 10–7).From the car, the velocities are
v′1=v−v(car)=v+v/2=3v/2
and
v′2=−v/2−v(car)=−v/2+v/2=0.
现在，我们可以问的下一个问题就是，如果一个质量m，矢速为 v，撞击并粘上另一个静止的质量2m，会发生什么？用我们的伽利略相对性原理，这个问题，很容易回答，因为我们只是从一个汽车上，观察了我们刚刚描述过的碰撞，汽车速度为−v/2 (图 10–7)。从汽车上，速度是：
v′1=v−v(car)=v+v/2=3v/2
和：
v′2=−v/2−v(car)=−v/2+v/2=0.

After the collision, the mass 3m appears to us to be moving with velocity v/2 . Thus we have the answer, i.e., the ratio of velocities before andafter collision is 3 to 1 : if an object of mass m collides with a stationary object of mass 2m , then the whole thing moves off, stuck together, with a velocity 1/3 as much. The general rule again is that the sum of the productsof the masses and the velocities stays the same: mv+0 equals 3m times v/3 , so we are gradually building up the theorem of the conservation ofmomentum, piece by piece.
碰撞之后，质量3m，对我们显得就是以速度v/2在移动。这样，我们就有了答案，即碰撞前与碰撞后，速度的比例是3比1：如果一个质量为m的对象，碰撞一个静止的、质量为2m的对象，那么，整个质量都移动，且粘在一起，速度最多为1/3。普遍规则就又是，质量与速度的乘积的总和，保持不变：mv+0 等于 3m乘以 v/3，就这样，一步一步，我们逐渐地建立起了动量守恒的原理。

Now we have one against two. Using the samearguments, we can predict the result of one against three, two against three,etc. The case of two against three, starting from rest, is shown in Fig. 10–8.
现在，我们有一个撞二个。用同样的论证，我们可以预计一撞三、二撞三等等的结果。二撞三的情况，从静止开始的，如图10-8。

Fig. 10–8.Action and reaction between 2mand 3m . 图 10-8 2m 和 3m 之间的作用与反作用。

In every case we find that the mass of thefirst object times its velocity, plus the mass of the second object times itsvelocity, is equal to the total mass of the final object times its velocity. Theseare all examples, then, of the conservation of momentum. Starting from simple,symmetrical cases, we have demonstrated the law for more complex cases. Wecould, in fact, do it for any rational mass ratio, and since every ratio isexceedingly close to a rational ratio, we can handle every ratio as preciselyas we wish.
在每种情况，我们都发现，第一个对象的质量乘以其速度，加上第二个对象的质量乘以其速度，等于最终对象的质量乘以其速度。因此，这些全都是动量守恒的例子。从简单的、对称的情况开始，我们为更复杂的情况，演示了此规律。事实上，我们可以为任何合理的质量比例，做此事；由于每种比例，都极为接近合理的比例，所以，每种比例，我们都可掌控，想多精确，就多精确。

10–4Momentum and energy 10-4 动量与能量
All the foregoing examples are simple caseswhere the bodies collide and stick together, or were initially stuck togetherand later separated by an explosion. However, there are situations in which thebodies do not cohere, as, for example, two bodies of equal mass whichcollide with equal speeds and then rebound. For a brief moment they are incontact and both are compressed. At the instant of maximum compression they bothhave zero velocity and energy is stored in the elastic bodies, as in a compressedspring. This energy is derived from the kinetic energy the bodies had beforethe collision, which becomes zero at the instant their velocity is zero. The lossof kinetic energy is only momentary, however. The compressed condition isanalogous to the cap that releases energy in an explosion. The bodies are immediatelydecompressed in a kind of explosion, and fly apart again; but we already knowthat case—the bodies fly apart with equal speeds. However, this speed ofrebound is less, in general, than the initial speed, because not all the energyis available for the explosion, depending on the material. If the material isputty no kinetic energy is recovered, but if it is something more rigid, some kineticenergy is usually regained. In the collision the rest of the kinetic energy istransformed into heat and vibrational energy—the bodies are hot and vibrating.The vibrational energy also is soon transformed into heat. It is possible tomake the colliding bodies from highly elastic materials, such as steel, withcarefully designed spring bumpers, so that the collision generates very littleheat and vibration. In these circumstances the velocities of rebound are practicallyequal to the initial velocities; such a collision is called elastic.
前面所有的例子，都比较简单，即物体相撞，并粘在一起，或者，最初被贴在一起，然后被爆炸分开。然而，在有些情况下，物体并不粘在一起，例如，两个同等质量的物体，以同等的速度碰撞，然后又弹开。在一个很短的瞬间，它们接触，双方都被压缩。在最大压缩的刹那，两者速度都为零，能量被存储在弹性的物体中，就像被压缩的弹簧那样。这个能量，来自碰撞前物体所具有的动能，在速度为零的刹那，动能也为零。然而，动能的消失，只是瞬间的。压缩的条件，可以类比于那个在爆炸中释放能量的帽子装置。类似于爆炸，物体立即被解压缩，然后，又飞出去，但是，我们已经知道了这个情况—物体以同等速度飞出去。然而，这个反弹的速度，一般要比初始速度小，因为，对这个爆炸来说，并不是所有能量，都能被得到，这依赖于材料。如果材料是腻子，那么，就没有动能，会被恢复，但是，如果它是某种更刚性的材料，那么，有些动能，就会重被得到。在碰撞中，剩下的动能，被转化成热能和振动能，物体变热，且在振动。振动能，很快也会转化成热能。用高弹性的材料，例如刚，来制造相撞的物体，再加上仔细设计的弹性保险杠，这是可能的；这样，碰撞所产生的热和振动，就会很小。在这种情况下，反弹的速度，几乎就等于初始的速度；这种碰撞，被称为弹性碰撞。

That the speeds before and afteran elastic collision are equal is not a matter of conservation of momentum, buta matter of conservation of kinetic energy. That the velocities of thebodies rebounding after a symmetrical collision are equal to and opposite eachother, however, is a matter of conservation of momentum.
对于一个弹性碰撞，碰撞前的速度，与碰撞后的速度相等，并不是一个动量守恒的事情，而是一个动能守恒的事情。然而，对于一个对称性的碰撞，物体的在碰撞前后的矢量速度，相等且相反，这是一个动量守恒的事情。

We might similarly analyze collisionsbetween bodies of different masses, different initial velocities, and variousdegrees of elasticity, and determine the final velocities and the loss ofkinetic energy, but we shall not go into the details of these processes.
对于质量不同、初始矢量速度不同、和弹性反弹度数{系数}不同的物体，我们可以类似地分析：它们之间的碰撞，并得到最终矢量速度、和消失的动能，但是，我们不会进入到这些过程的细节中去。

Elastic collisions are especiallyinteresting for systems that have no internal “gears, wheels, or parts.” Thenwhen there is a collision there is nowhere for the energy to be impounded,because the objects that move apart are in the same condition as when theycollided. Therefore, between very elementary objects, the collisions are alwayselastic or very nearly elastic. For instance, the collisions between atoms ormolecules in a gas are said to be perfectly elastic. Although this is an excellentapproximation, even such collisions are not perfectly elastic; otherwiseone could not understand how energy in the form of light or heat radiation couldcome out of a gas. Once in a while, in a gas collision, a low-energy infraredray is emitted, but this occurrence is very rare and the energy emitted is verysmall. So, for most purposes, collisions of molecules in gases are consideredto be perfectly elastic.
有些系统，没有内部的“传动装置、轮子，或部分”，对于这些系统来说，弹性碰撞，尤其有趣。因此，当碰撞发生时，没有地方，可以收容这些能量，因为，被撞开的对象，与它们在碰撞时，处于同样的条件。因此，对于很基础的对象，它们之间的碰撞，总是弹性的，或非常接近弹性。例如，气体中的原子或分子间的碰撞，就被认为是完全弹性的。就算这种碰撞，并非完全弹性的，也是一种精彩的近似；否则的话，你不能理解：在光和热辐射的形式中的能量，是如何从气体中出来的。偶尔，在气体的碰撞中，低能量的红外线，被发射出来，但是，这种现象非常罕见，且发射出来的能量也很小。所以，对大多数的目的来说，气体中的分子碰撞，被认为是完全弹性的。