每天来几段，现在更了多少了？

These examples may sound trivial, but manysubtleties enter into the description of change. Some changes are moredifficult to describe than the motion of a point on a solid object, for examplethe speed of drift of a cloud that is drifting very slowly, but rapidly formingor evaporating, or the change of a woman’s mind. We do not know a simple way toanalyze a change of mind, but since the cloud can be represented or describedby many molecules, perhaps we can describe the motion of the cloud in principleby describing the motion of all its individual molecules. Likewise, perhaps eventhe changes in the mind may have a parallel in changes of the atoms inside thebrain, but we have no such knowledge yet.
这些例子，听上去可能很琐碎，但是很多机敏，已经进入到了对变化的描述中。有些变化，要比固体上的一个点的运动，难描述的多，例如一个正在飘动的云，虽然它飘得很慢，但是，却在快速的形成或蒸发，或者，一个女人的心的变化。分析心的变化，我们并不不知道有什么简单的方法，但是，由于云可以通过很多分子来代表或描述，那么，或许我们可以通过从原理上描述其所有个别分子的运动，来描述云的运动。同样，或许，甚至心灵的变化，都在大脑的原子中，有着一个平行的变化，但是，我们还没有这种知识。

At any rate, that is why we begin with the motion of points; perhaps we should think of them as atoms, but it is probably better to be more rough in the beginning and simply to think of some kind of small objects—small, that is, compared with the distance moved. For instance, in describing the motion of a car that is going a hundred miles, we do not have to distinguish between the front and the back of the car. To be sure, there are slight differences, but for rough purposes we say “the car,” and likewise it does not matter that our points are not absolute points; for our present purposes it is not necessary to be extremely precise. Also, while we take a first look at this subject we are going to forget about the three dimensions of the world. We shall just concentrate on moving in one direction, as in a car on one road. We shall return to three dimensions after we see how to describe motion in one dimension. Now, you may say, “This is all some kind of trivia,” and indeed it is. How can we describe such a one-dimensional motion—let us say, of a car? Nothing could be simpler. Among many possible ways, one would be the following. To determine the position of the car at different times, we measure its distance from the starting point and record all the observations. In Table 8–1, s represents the distance of the car, in feet, from the starting point, and t represents the time in minutes. The first line in the table represents zero distance and zero time—the car has not started yet. After one minute it has started and has gone 1200 feet. Then in two minutes, it goes farther—notice that it picked up more distance in the second minute—it has accelerated; but something happened between 3 and 4 and even more so at 5 —it stopped at a light perhaps? Then it speeds up again and goes 13,000 feet by the end of 6 minutes, 18,000 feet at the end of 7 minutes, and 23,500 feet in 8 minutes; at 9 minutes it has advanced to only 24,000 feet, because in the last minute it was stopped by a cop.
以任何速率，那就是为什么我们要从点的运动开始；或许我们应该把它们想成原子，但是，开始的时候，粗糙一些，或许更好，只思考一些小的对象，所谓小，是指与移动的距离相比。例如，一辆汽车，走了上百英里，在描述其运动时，我们无需区分车头和车尾。当然了，可能会有细微的差别，但是，对于粗糙一点的目的，我们就说“这辆汽车”，当然，我们所说的点，并不是绝对的点，这一点同样并不重要；对于我们现在的目的，不需要那样极度精确。另外，当我们第一眼看这个主题时，我们将会忘掉，世界是三维的。我们将只集中于一个方向上的运动，就像车在一条路上一样。在我们知道了如何描述一维运动之后，我们将返回三维。现在，你可以说：“这些都是些零零碎碎的事情”，确实如此。我们如何才能描述这样一个一维的运动--汽车的运动呢？没有比这更简单的了。在很多可能的方法，下面就是其一。要决定汽车在不同时间的位置，我们测量它距起点的距离，并记录所有的观察结果。在表8-1中，s代表汽车距起点的距离，单位英尺，因此，t代表时间，单位分钟。表中的第一行，代表零距离和零时间，汽车尚未启动。一分钟之后，它启动并走了1200英尺。然后，在两分钟，它走得更远，注意它在第二分钟，走的更多，它加速了；但是，在3到4分钟、甚至在5分钟时，有什么事情发生了，停下来等信号灯？然后，它又加速，在6分钟末，达到13，000英尺，7分钟末，18，000英尺，8分钟，23，500英尺；在9分钟，它只走到24，000英尺，因为在最后一分钟，他被一个警察给拦住了。
Table 8–1 表8-1
t (min)
s (ft)
0 00000
1 01200
2 04000
3 09000
4 09500
5 09600
6 13000
7 18000
8 23500
9 24000

Fig. 8–1.Graph of distance versus time forthe car. 图8-1 汽车的距离对时间曲线
That is one way to describe the motion. Anotherway is by means of a graph. If we plot the time horizontally and the distancevertically, we obtain a curve something like that shown in Fig. 8–1. As thetime increases, the distance increases, at first very slowly and then morerapidly, and very slowly again for a little while at 4 minutes; then it increases again for a few minutes and finally,at 9 minutes, appears to have stopped increasing. These observationscan be made from the graph, without a table. Obviously, for a complete descriptionone would have to know where the car is at the half-minute marks, too, but wesuppose that the graph means something, that the car has some position at allthe intermediate times.
这是描述运动的一种方法。另一种方法，是凭借曲线图。如果我们水平画时间，垂直画距离，我们就得到一个曲线，如图8-1。随着时间的增长，距离也在增长，最初很慢，然后，就比较快，然后，在4分钟的时候，又非常慢了一阵，然后；又开始增长了几分钟，最后，在9分钟时，显然是停止增长了。这些观察，无需用表，都可从图中得到。很明显，对于一个完整的描述，也应该可以知道，在半分钟的标记处，汽车在什么地方，但是，我们认为曲线图应有某种意义，即汽车在所有的中间时间，都有相应的位置。

Table 8–2
t (sec) s (ft)
0 000
1 016
2 064
3 144
4 256
5 400
6 576

Fig. 8–2.Graph of distance versus time for a falling body. 图8-2 一个下落物体的距离对时间曲线图
The motion of a car is complicated. For another example we take something that moves in a simpler manner, following more simple laws: a falling ball. Table 8–2 gives the time in seconds and the distance in feet for a falling body. At zero seconds the ball starts out at zero feet, and at the end of 1 second it has fallen 16 feet. At the end of 2 seconds, it has fallen 64 feet, at the end of 3 seconds, 144 feet, and so on; if the tabulated numbers are plotted, we get the nice parabolic curve shown in Fig. 8–2. The formula for this curve can be written as
s=16t2.(8.1)
This formula enables us to calculate the distances at any time. You might say there ought to be a formula for the first graph too. Actually, one may write such a formula abstractly, as
s=f(t),(8.2)
meaning that s is some quantity depending on t or, in mathematical phraseology, s is a function of t . Since we do not know what the function is, there is no way we can write it in definite algebraic form.
这个汽车的运动，比较复杂。另一个例子，是一个正在下落的球，其运动方式，更为简单，也遵循更简单的规律。表8-2，是一个正在下降的球，时间为秒，距离为英尺。在零秒，球从零英尺处，开始下降，1秒末，下降了16英尺，2秒末，64英尺，3秒末，144英尺等，如果表中的数字，被画出来，就是一个精致的抛物线，如图8-2。这条曲线的公式，可写为：
s=16t*t.(8.1)
这个公式，可以让我们计算任何时间的距离。你可以说，对于第一张曲线图，也应该有一个公式。实际上，我们可以把这种公式，抽象为：
s=f(t),(8.2)
其意义就是，s是一个量，依赖于t，或者，用数学的语言表达，s是时间的函数。由于我们不知道这个函数是什么，所以，我们没有办法用确定的代数形式，把它写出来。

Another subtlety involved, and already mentioned,is that it should be possible to imagine that the moving point we are observingis always located somewhere. (Of course when we are looking at it, there it is,but maybe when we look away it isn’t there.) It turns out that in the motion ofatoms, that idea also is false—we cannot find a marker on an atom and watch itmove. That subtlety we shall have to get around in quantum mechanics. But we arefirst going to learn what the problems are before introducing the complications,and then we shall be in a better position to make corrections, in thelight of the more recent knowledge of the subject. We shall, therefore, take asimple point of view about time and space. We know what these concepts are in arough way, and those who have driven a car know what speed means.
所牵扯到的另外一个微妙，已经提到过，那就是，想象我们正在观察的点，总处在某个地方，是可能的。（当然，当我们看着它时，它在那里，但是，当我们转过脸去的时候，它或许不在那里。）结果就是，在原子的运动中，这个想法也还是错的，--在原子上，找不到一个标记，让我们能够观察它的运动。这个微妙，我们将会在量子力学中得到。但是，在引入此复杂问题之前，我们首先将去学习：问题是什么，然后，关于此主题的更新的知识，将有利于我们做出修改。因此，对于时间和空间，我们只取一个简单的观点。这些概念是什么，我们粗略地知道，并且，那些开车的人也知道，速度意味着什么。

8–2Speed 8-2 速度
Even though we know roughly what “speed” means,there are still some rather deep subtleties; consider that the learned Greekswere never able to adequately describe problems involving velocity. The subtletycomes when we try to comprehend exactly what is meant by “speed.” The Greeksgot very confused about this, and a new branch of mathematics had to bediscovered beyond the geometry and algebra of the Greeks, Arabs, andBabylonians. As an illustration of the difficulty, try to solve this problem bysheer algebra: A balloon is being inflated so that the volume of the balloon isincreasing at the rate of 100 cm³ per second; at what speed is the radius increasing when thevolume is 1000 cm³? The Greeks were somewhat confused by such problems, being helped,of course, by some very confusing Greeks. To show that there were difficultiesin reasoning about speed at the time, Zeno produced a large number ofparadoxes, of which we shall mention one to illustrate his point that there areobvious difficulties in thinking about motion. “Listen,” he says, “to thefollowing argument: Achilles runs 10 times as fast as a tortoise, nevertheless he can never catch thetortoise. For, suppose that they start in a race where the tortoise is 100 meters ahead of Achilles; then when Achilles has run the 100 meters to the place where the tortoise was, the tortoise hasproceeded 10 meters, having run one-tenth as fast. Now, Achilles has to runanother 10 meters to catch up with the tortoise, but on arriving at the endof that run, he finds that the tortoise is still 1 meter ahead of him; running another meter, he finds the tortoise10 centimeters ahead, and so on, ad infinitum. Therefore, atany moment the tortoise is always ahead of Achilles and Achilles can nevercatch up with the tortoise.” What is wrong with that? It is that a finite amountof time can be divided into an infinite number of pieces, just as a length ofline can be divided into an infinite number of pieces by dividing repeatedly bytwo. And so, although there are an infinite number of steps (in the argument)to the point at which Achilles reaches the tortoise, it doesn’t mean that thereis an infinite amount of time. We can see from this example that thereare indeed some subtleties in reasoning about speed.
尽管我们只是粗略地知道，“速度”是什么，但还是有一些深的微妙；比如，有学问的希腊人，永远也不能适当地描述，牵扯到速度的问题。当我们尝试去理解，“速度”究竟是什么时，这个微妙就来了。希腊人对这个问题，非常困惑，于是，在希腊人、阿拉伯人和巴比伦人的几何和代数学之外，一个新的数学分支，就应该被发现。吹一个气球，于是，它的体积，就是以每秒100立方厘米的速度在增加，那么，当体积达到1000立方厘米时，其半径增加的速度又是多少呢？希腊人对这类问题，非常困惑，当然，他们是被一些能困惑人的希腊人，给帮助了{忽悠}。为了指出，在那个时候，推理关于速度的问题，是有困难的，芝诺提出了大量的悖论，我们将讲其中之一，以展示他的观点，即在思考速度的时候，存在明显的困难。他说：“听下面的论证：阿喀琉斯跑步，比乌龟快10倍，但他永远也赶不上乌龟。假设比赛开始时，乌龟是在阿喀琉斯之前100米，然后，当阿喀琉斯跑过这100米时，由于乌龟的速度是阿喀琉斯的十分之一，所以它又前进了10米。现在，阿喀琉斯必须跑过这10米，才能赶上乌龟，但是，当他跑过这10米时，发现乌龟，又领先了1米；跑过这1米，他发现乌龟又领先了10厘米，如此以致无穷。因此，任何时候，乌龟总是领先于阿喀琉斯，阿喀琉斯永远也赶不上乌龟。这里究竟出了什么问题呢？那就是，时间的量，本来是有限的，但却被分成了无限的段，正如长度一定的线，通过不断的二分法，可以被分成无限的段一样。所以，阿喀琉斯在某个点，可以追上乌龟，虽然在此之前，步骤可以被分成无限（在论据中），但这并不意味着，时间也是无限的。从这个例子，我们可以看出，在论证速度时，确实有一些微妙。

In order to get to the subtleties in a clearerfashion, we remind you of a joke which you surely must have heard. At the pointwhere the lady in the car is caught by a cop, the cop comes up to her and says,“Lady, you were going 60 miles an hour!” She says, “That’s impossible, sir, I was travellingfor only seven minutes. It is ridiculous—how can I go 60 miles an hour when I wasn’t going an hour?” How would you answerher if you were the cop? Of course, if you were really the cop, then no subtletiesare involved; it is very simple: you say, “Tell that to the judge!” But let ussuppose that we do not have that escape and we make a more honest, intellectualattack on the problem, and try to explain to this lady what we mean by the ideathat she was going 60 miles an hour. Just what do we mean? We say, “What wemean, lady, is this: if you kept on going the same way as you are going now, inthe next hour you would go 60 miles.” She could say, “Well, my foot was off the acceleratorand the car was slowing down, so if I kept on going that way it would not go 60 miles.” Or consider the falling ball and suppose we want to knowits speed at the time three seconds if the ball kept on going the way it isgoing. What does that mean—kept on accelerating, going faster? No—kepton going with the same velocity. But that is what we are trying todefine! For if the ball keeps on going the way it is going, it will just keepon going the way it is going. Thus we need to define the velocity better. Whathas to be kept the same? The lady can also argue this way: “If I kept on goingthe way I’m going for one more hour, I would run into that wall at the end ofthe street!” It is not so easy to say what we mean.
为了以更清晰的方式，理解微妙，我们给你讲一个笑话，你肯定听过。一位女士开车，被警察拦住，警察对她说：“女士，你的车速是1小时60英里！”她说：“不可能，先生，我只开了7分钟，很可笑，我还没开到1小时，怎么可能在1小时内走了60英里？”如果你是那个警察，你会如何回答呢？当然，如果你真是那个警察，那么，不会牵扯到任何微妙，你会简单地说：“去跟法官讲吧！”但是，假设我们没有这个借口，而是要更诚实地、更理智地直面这个问题，尝试给这位女士解释，通过1小时60英里这个概念，我们的意思是什么？仅仅是：我们的意思是什么？我们说：“女士，我们的意思是这样，如果你以现在的方式继续走下去，那么，在下面1小时内，你将会走60英里。”她会说：“其实，我的脚已经不在加速器上了，这个汽车正在减速，如果我继续走的话，它也不会走60英里。”或者，考虑那个下落球的情况，假设球一直按它的方式在走，我们想知道，在下落三秒时，它的速度。那意思是什么呢--继续加速，或走的更快？不，是继续走，且是用同样的速度。但这正是我们尝试去定义的！因为，如果这个球，是按它正在走的方式继续走的话，那么，它当将只是按这种方式，继续走。这样，我们就需要更好地去定义速度。需要被保持为同样的，是什么呢？这个女士，也可以这样辩解：“如果我以现在正在走的方式，再走1小时，那么，我就会撞到街尽头的墙上了！”所以，要把你的意思说清楚，并不是那么容易。

Many physicists think that measurement isthe only definition of anything. Obviously, then, we should use the instrumentthat measures the speed—the speedometer—and say, “Look, lady, your speedometerreads 60 .” So she says, “My speedometer is broken and didn’t read at all.” Doesthat mean the car is standing still? We believe that there is something to measurebefore we build the speedometer. Only then can we say, for example, “The speedometerisn’t working right,” or “the speedometer is broken.” That would be ameaningless sentence if the velocity had no meaning independent of thespeedometer. So we have in our minds, obviously, an idea that is independent ofthe speedometer, and the speedometer is meant only to measure this idea. So letus see if we can get a better definition of the idea. We say, “Yes, of course,before you went an hour, you would hit that wall, but if you went one second, youwould go 88 feet; lady, you were going 88 feet per second, and if you kept on going, the next second itwould be 88 feet, and the wall down there is farther away than that.” Shesays, “Yes, but there’s no law against going 88 feet per second! There is only a law against going 60 miles an hour.” “But,” we reply, “it’s the same thing.” If it isthe same thing, it should not be necessary to go into this circumlocution about88 feet per second. In fact, the falling ball could not keep goingthe same way even one second because it would be changing speed, and we shallhave to define speed somehow.
很多物理学家认为，任何事物的唯一定义方式，就是测量。因此，很显然，我们应该使用测量速度的仪器--速度计，并且说：“看，女士，你的速度计是60。”于是她说：“我的速度计坏了，根本无法读数”。这是否意味着她的车原地没动？我们相信，在我们建造速度计之前，是有某种东西，可用于测量的。只有那样，我们才可以说，例如：“速度计工作的不正常,”，或者，“速度计坏了。”如果速度，在离开速度计时，没有任何意义，那么，这句话将没有意义。于是，很显然，在我们心中，有一个想法，它是独立于速度计的，而速度计，只是用来测量这个想法的。让我们看看，我们是否能够为此想法，得到一个更好的定义。我们说：“是的，当然，在你走了一个小时之前，你就可能将撞上那堵墙，但是，如果你走一秒钟，你将走88英尺；女士，你是按88英尺每秒走的，如果你继续这样走，下一秒钟又是88英尺，而那面墙，相比之下，还是很远。”她说：“是的，但是，没有任何法律，是反对走88英尺每秒的！只有一条法律，反对60英里1小时。”“但是”，我们回答说：“这是一样的”。如果是一样的，那么，就没有必要婉转曲折地说88英尺每秒。事实上，下落的球，无法始终保持同样的方式，即使一秒也不行，因为，它会不断地改变速度，我们必须以某种方式，来定义速度。

The foregoing definition involves a newidea, an idea that was not available to the Greeks in a general form. That ideawas to take an infinitesimal distance and the corresponding infinitesimaltime, form the ratio, and watch what happens to that ratio as the time thatwe use gets smaller and smaller and smaller. In other words, take a limit ofthe distance travelled divided by the time required, as the time taken getssmaller and smaller, ad infinitum. This idea was invented by Newton andby Leibniz, independently, and is the beginning of a new branch of mathematics,called the differential calculus. Calculus was invented in order todescribe motion, and its first application was to the problem of defining whatis meant by going “60 miles an hour.”
上述定义，包含了一个新的想法，希腊人没有办法以一种普遍的形式，来得到这个想法。这个想法就是，用无限小的距离，及相应的无限小的时间，来形成比率，然后看，随着时间变得越来越小，越来越小，这个比率会发生什么变化。换句话说，取所走距离的极限，让它除以所要求的时间，让时间变得越来越小，直到无限小。这个想法，由牛顿和莱布尼兹，分别独立发明，是一门新的数学分支的开始，被称为微积分。微积分的发明，是为了描述运动，其第一个应用，就是如何定义：走“60英里1小时”是什么意思。

Let us try to define velocity a littlebetter. Suppose that in a short time, ϵ , the car or other body goes a short distance x ; then the velocity, v , is defined as
现在让我们看看，能不能把速度，定义的更好一点。假设在一个短的时间ϵ内，汽车或一个物体，走了短的距离x，则速度v就可定义为：
v=x/ϵ,
an approximation that becomes better and better as the ϵ is taken smaller and smaller. If a mathematical expression is desired,we can say that the velocity equals the limit as the ϵ is made to go smaller and smaller in the expression x/ϵ, or
这是一个近似，随着ϵ取得越来越小，它就可变得越来越好。如果我们期望一个数学表达式的话，我们可以说，速度，就是ϵ的取值，越来越小时，表达式x/ϵ的值，或者：

We cannot do the same thing with the lady in the car, because thetable is incomplete. We know only where she was at intervals of one minute; we canget a rough idea that she was going 5000 ft/min during the 7 th minute, but we do not know, at exactly the moment 7 minutes, whether she had been speeding up and the speed was 4900 ft/min at the beginning of the 6 th minute, and is now 5100 ft/min, or something else, because we do not have the exactdetails in between. So only if the table were completed with an infinite numberof entries could we really calculate the velocity from such a table. On the otherhand, when we have a complete mathematical formula, as in the case of a fallingbody (Eq. 8.1),then it is possible to calculate the velocity, because we can calculate theposition at any time whatsoever.
对于汽车女士情况，我们不能做同样的事情，因为表不完整。我们只知道，在每一分钟的间隔处，她在那里；我们只能得到一个粗略的想法：在那7分钟，她的速度是5000英尺/分钟，但是我们不知道，在7分钟的时间过程中，她在哪里加速了，在第6分钟的开始，速度是4900英尺/分钟，而现在是5100英尺/分钟，或者其他，因为我们不知道中间的准确细节。所以，只有当这个表是完整的，且拥有无限的记录条目时，我们才能从这张表出发，准确地计算速度。另一方面，当我们有一个完整的数学公式，例如下落球的那种情况（公式8.1），这时，才可能计算速度，因为，任何时间，球的位置，我们都能计算。

Let us take as an example the problem of determiningthe velocity of the falling ball at the particular time 5 seconds. One way to do this is to see from Table 8–2 what itdid in the 5 th second; it went 400−256=144 ft, so it is going 144 ft/sec; however, that is wrong, because the speed is changing; onthe average it is 144 ft/sec during this interval, but the ball is speeding up and isreally going faster than 144 ft/sec. We want to find out exactly how fast. Thetechnique involved in this process is the following: We know where the ball wasat 5 sec. At 5.1 sec, the distance that it has gone all together is 16(5.1)2=416.16 ft (see Eq. 8.1).At 5 sec it had already fallen 400 ft; in the last tenth of a second it fell 416.16−400=16.16 ft. Since 16.16 ft in 0.1 sec is the same as 161.6 ft/sec, that is the speed more or less, but it is not exactlycorrect. Is that the speed at 5 , or at 5.1 , or halfway between at 5.05 sec, or when is that the speed? Never mind—the problemwas to find the speed at 5 seconds, and we do not have exactly that; we have to do abetter job. So, we take one-thousandth of a second more than 5 sec, or 5.001 sec, and calculate the total fall as
s=16(5.001)2=16(25.010001)=400.160016 ft.
In the last 0.001 sec the ball fell 0.160016 ft, and if we divide this number by 0.001 sec we obtain the speed as 160.016 ft/sec. That is closer, very close, but it is still not exact.It should now be evident what we must do to find the speed exactly. To performthe mathematics we state the problem a little more abstractly: to find thevelocity at a special time, t0 , which in the original problem was 5 sec. Now the distance at t0 , which we call s0 , is 16t02 , or 400 ft in this case. In order to find the velocity, we ask, “At thetime t0+(a little bit) , or t0+ϵ , where is the body?” The new position is 16(t0+ϵ)2=16t02+32t0ϵ+16ϵ2. So it is farther along than it was before, because before it wasonly 16t02 . This distance we shall call s0+(a little bitmore) , or s0+x (if x is the extra bit). Now if we subtract the distance at t0from the distance at t0+ϵ , we get x , the extra distance gone, as x=32t0⋅ϵ+16ϵ2 . Our first approximation to the velocity is
v=x/ϵ=32t0+16ϵ.(8.4)
下落的球，在5秒钟时，速度为何，现在，我们就把这个问题，作为一个例子。做此事的一个方法，就是从表8-2，看球在第5秒时，做了什么；它走了400−256=144 ft, 于是，其速度为144 ft/sec；然而，这是错的，因为速度在改变；在此间隔，平均速度是144 ft/sec，但是球在加速，确实要比144 ft/sec更快了。这个过程所包含的技术如下：我们知道，5秒时，球在哪里。 在5.1秒，球总共走的距离是16(5.1)2=416.16 ft (见公式 8.1).在5秒时，它下落了400 ft；在最后的这十分之一秒，它下降了 416.16−400=16.16 ft。由于16.16 ft/0.1 sec，与161.6 ft/sec一样，也就是说，速度或多或少改变了，但这并不完全正确。这是5秒时的速度，还是5.1秒时的速度，或是到5.05秒之间的速度吗？或者，这是什么时候的速度？不要担心，问题是要找出5秒时的速度，而我们并没有它的准确值；所以我们必须做的更好才行。于是，我们在5秒的基础上，再多取千分之一秒，或者是5.001 sec, 然后计算总的下落距离：
s=16(5.001)2=16(25.010001)=400.160016 ft.
在最后的0.001 sec，球下降了0.160016 ft，如果我们用0.001 sec，除以这个数，得到速度为 160.016 ft/sec。这很接近了，非常接近，但仍不准确。要准确地找出速度，应该做什么，现在已经很明显了。为了执行数学，我们稍微抽象地来陈述这个问题：找出在时间t0处的速度，t0在原问题中是5 sec。现在，在t0处的距离，我们称为s0，在这里，它是16t02, or 400 ft。为了找出速度，我们问：“在时间 t0+(一个小量),或 t0+ϵ，物体在哪里？”新的位置是：16(t0+ϵ)2=16t02+32t0ϵ+16ϵ2。所以，它比以前要长了，以前只是16t02。这个距离，我们将称为s0+(一个小量) ，或 s0+x(如果 x 是额外小量的话)。现在，如果我们从t0+ϵ处的距离中，减去t0处的距离去，我们就会得到 x ，额外走的距离就是： x=32t0⋅ϵ+16ϵ2 。我们对速度的第一个近似就是：
v=x/ϵ=32t0+16ϵ.(8.4)

The true velocity is the value of this ratio, x/ϵ , when ϵ becomes vanishingly small. In other words, after forming the ratio, wetake the limit as ϵ gets smaller and smaller, that is, approaches 0 . The equation reduces to,
v(attime t0)=32t0.
In our problem, t0=5 sec, so the solution is v= 32×5= 160 ft/sec. A few lines above, where we took ϵ as 0.1 and 0.001 sec successively, the value we got for v was a little more than this, but now we see that the actual velocityis precisely 160 ft/sec.
真正的速度，就是当ϵ趋于消失时，比率x/ϵ的值。换句话说，在形成这个比率后，我们取极限，让ϵ越来越小，也就是说，趋近于零。这个方程可简化为：
v(在时间 t0)=32t0.
在我们的问题中，t0=5 sec，于是，答案就是v= 32×5= 160 ft/sec。在上面的几行中，我们取ϵ为0.1和0.001 sec，非常成功，我们为得到v值，比此值稍微大一点，但现在我们看到，实际速度就是精确的160 ft/sec。

8–3Speed as a derivative 8-3 速度作为一个导出项
The procedure we have just carried out isperformed so often in mathematics that for convenience special notations havebeen assigned to our quantities ϵ and x . In this notation, the ϵ used above becomes Δt and x becomes Δs . This Δt means “an extra bit of t ,” and carries an implication that it can be made smaller. Theprefix Δ is not a multiplier, any more than sinθ means s⋅i⋅n⋅θ —it simply defines a time increment, and reminds us of its specialcharacter. Δs has an analogous meaning for the distance s . Since Δ is not a factor, it cannot be cancelled in the ratio Δs/Δtto give s/t , any more than the ratio sinθ/sin2θ can be reduced to 1/2 by cancellation. In this notation, velocity is equal to the limitof Δs/Δt when Δt gets smaller, or
我们上面刚讲的过程，在数学中，经常被运用，以至于为了方便，给我们的量ϵ和x，都安排了特殊的符号。在这套符号中，上面用的ϵ变成了Δt，x变成了Δs。Δt意味着“一个极小的时间t”，暗示着它可以变得更小。前缀Δ不是一个乘数，正如sinθ 并不意味着 s⋅i⋅n⋅θ，它只定义了一个时间的增量，提醒我们其独特性，对于距离，Δs有类似的意义。由于Δ不是一个因子，所以，在比率 Δs/Δt中，它不能被取消掉，以给出s/t，正如比例sinθ/sin2θ，不能通过简化，而给出1/2一样。在这套符号中，速度等于：当Δt越来越小时，Δs/Δt的极限，或：
(8.5)
This is really the same as our previous expression (8.3)with ϵ and x , but it has the advantage of showing that something is changing, andit keeps track of what is changing.
对于ϵ和 x，这个公式，与前面的表达式（8.3），是一样的，但是，它有一个优势，即可以指出：有某物在改变，且它可以追踪：那个正在改变的某物。