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has accelerated; but something happened between 3 and 4 and even more so
at 5—it stopped at a light perhaps? Then ït 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.
That is one way to describe the motion. Another way is by means of a
graph. H we plot the time horizontally and the distance vertically, we obtain a
curve something like that shown in Eig. 8-1. As the tỉme increases, the đistance
Increases, at first very slowly and then more rapidly, and very slowly again for a
little while at 4 minutes; then it increases again for a few minutes and ñnally,
at 9 minutes, appears to have stopped increasing. 'Phese observations can be
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Table 8-2 th
Z 300
f (sec) | s (ft) Ế
0 0 £ 200
1 16 Ờ
2 64 Ê 100
3 144 ễ
5 400 : TIME IN SECONDS ˆ ;
Fig. 8-2. Graph of distance versus time for a falling
made from the graph, without a table. Obviously, for a complete description
one would have to know where the car is at the half-minute marks, too, but we
suppose that the graph means something, that the car has some position at all
the intermediate times.
'The motion of a car is complicated. Eor 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
ở seconds, 14⁄4 feet, and so on; ïf the tabulated numbers are plotted, we get the
nice parabolic curve shown in Fig. 8-2. The formula for this curve can be written
s= 16. (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=ƒ/(0, (8.2)
meaning that s is some quantity depending on ý or, in mathematical phraseology,
ø is a function of . Since we do not know what the function is, there is no way
we can write it in defnite algebraic form.
We have now seen ÿwo examples of motion, adequately described with very
simple ideas, no subtleties. However, there øre subtleties—several of them. In
--- Trang 163 ---
the first place, what do we mean by f£#ne and space? It turns out that these deep
philosophical questions have to be analyzed very carefully in physics, and this
1s not so easy to do. 'Phe theory of relativity shows that our ideas of space and
time are not as simple as one might think at fñrst sight. However, for our present
purposes, for the accuracy that we need at first, we need not be very careful
about defning things precisely. Perhaps you say, “Phat's a terrible thing—I
learned that in seience we have to defñne cuerwthing precisely.” We cannot defne
gmything preciselyl TẾ we attempt to, we get into that paralysis of thought that
comes to philosophers, who sit opposite each other, one saying to the other, “You
don”? know what you are talking about!” “The second one says, “What do you
mean by knou? What do you mean by falking? What do you mean by ow#,”
and so on. In order to be able to talk constructively, we Just have to agree that
we are talking about roughly the same thing. You know as much about time as
we need for the present, but remember that there are some subtleties that have
to be discussed; we shall discuss them later.
Another subtlety involved, and already mentioned, is that ¡t should be possible
to imagine that the moving point we are observing is 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 of atoms, that idea also is false—we
cannot fnd a marker on an atom and watch it move. 'Phat subtlety we shall
have to get around in quantum mechanies. But we are first go¡ing to learn what
the problems are before introducing the complications, and ¿hen we shall be in a
better position to make corrections, in the light of the more recent knowledge
of the subject. We shall, therefore, take a simple point of view about time and
space. We know what these concepts are in a rough way, and those who have
driven a car know what speed means.
8-2 Speed
ven though we know roughly what “speed” means, there are still some
rather deep subtleties; consider that the learned Greeks were never able to
adequately describe problems involving velocity. The subtlety comes when we try
to comprehend exactly what is meant by “speed.” The Greeks got very confused
about this, and a new branch of mathematies had to be discovered beyond the
geometry and algebra of the Greeks, Arabs, and Babylonians. As an illustration
of the dificulty, try to solve this problem by sheer algebra: A balloon is being
infated so that the volume of the balloon is increasing at the rate of 100 em”
--- Trang 164 ---
per second; at what speed is the radius inereasing when the volume is 1000 em”?
'The Greeks were somewhat confused by such problems, being helped, of course,
by some very confusing Greeks. To show that there were difficulties in reasoning
about speed at the time, Zeno produced a large number of paradoxes, of which
we shall mention one to illustrate his point that there are obvious dificulties in
thinking about motion. “Listen,” he says, “to the following argument: Achilles
runs 10 times as fast as a tortoise, nevertheless he can never catch the tortoise.
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 has proceeded 10 meters, having run one-tenth as fast.
NÑow, Achiles has to run another 10 meters to catch up with the tortoise, but on
arriving at the end of that run, he ñnds that the tortoise is still 1 meter ahead
of him; running another meter, he fnds the tortoise 10 centimeters ahead, and
SO On, ød ?nƒfinøtum. Pherefore, at any moment the tortoise is always ahead of
Achilles and Achilles can never catch, up with the tortoise.” What is wrong with
that? It is that a finite amount of time can be divided into an infnite number of
pieces, just as a length of line can be divided into an infnite number of pieces
by dividing repeatedly by bwo. And so, although there are an infnite number
Of sbeps (in the argument) to the point at which Achilles reaches the tortoise,
it doesnt mean that there is an infnite amount of #me. We can see from this
example that there are indeed some subtleties in reasoning about speed.