Feynman_Lectures_on_Physics_Volume_1_Chapter_05.pdf

Time and Distance

5-1 Motion

In this chapter we shall consider some aspects of the concepts of time and

distance. It has been emphasized earlier that physics, as do all the sciences, de-

pends on observation. One might also say that the development of the physical

sciences to their present form has depended to a large extent on the emphasis

which has been placed on the making of quantitative observations. Only with

quantitative observations can one arrive at quantitative relationships, which are

the heart of physics.

Many people would like to place the beginnings of physics with the work

done 350 years ago by Galileo, and to call him the first physicist. Until that time,

the study of motion had been a philosophical one based on arguments that could

be thought up in one's head. Most of the arguments had been presented by

Aristotle and other Greek philosophers, and were taken as "proven." Galileo

was skeptical, and did an experiment on motion which was essentially this: He

allowed a ball to roll down an inclined trough and observed the motion. He did

not, however, just look; he measured how far the ball went in how long a time.

The way to measure a distance was well known long before Galileo, but there

were no accurate ways of measuring time, particularly short times. Although he

later devised more satisfactory clocks (though not like the ones we know), Galileo's

first experiments on motion were done by using his pulse to count off equal in-

tervals of time. Let us do the same.

We may count off beats of a pulse as the ball rolls down the track: "one .. .

two ... three .. . four .. . five ... six ... seven . . . eight..." We ask a friend to

make a small mark at the location of the ball at each count; we can then measure

the distance the ball travelled from the point of release in one, or two, or three,

etc., equal intervals of time. Galileo expressed the result of his observations in

this way: if the location of the ball is marked at 1, 2, 3, 4,... units of time from

the instant of Its release, those marks are distant from the starting point in propor-

tion to the numbers 1, 4, 9, 16, ... Today we would say the distance is propor-

tional to the square of the time:

5-1 Motion

5-2 Time

5-3 Short times

5-4 Long times

5-5 Units and standards of time

5-6 Large distances

5-7 Short distances

Fig. 5-1. A ball rolls down an

dined track.

The study of motion, which is basic to all of physics, treats with the questions:

where? and when?

5-2 Time

Let us consider first what we mean by time. What is time? It would be nice

if we could find a good definition of time. Webster defines "a time" as "a period,"

and the latter as "a time," which doesn't seem to be very useful. Perhaps we should

say: "Time is what happens when nothing else happens." Which also doesn't

get us very far. Maybe it is just as well if we face the fact that time is one of the

things we probably cannot define (in the dictionary sense), and just say that it

is what we already know it to be: it is how long we wait!

What really matters anyway is not how we define time, but how we measure

it. One way of measuring time is to utilize something which happens over and

over again in a regular fashion—something which is periodic. For example, a

day. A day seems to happen over and over again. But when you begin to think

5-1

about it, you might well ask: "Are days periodic; are they regular? Are all days

the same length?" One certainly has the impression that days in summer are longer

than days in winter. Of course, some of the days in winter seem to get awfully

long if one is very bored. You have certainly heard someone say, "My, but this

has been a long day!"

It does seem, however, that days are about the same length on the average.

Is there any way we can test whether the days are the same length—either from

one day to the next, or at least on the average? One way is to make a comparison

with some other periodic phenomenon. Let us see how such a comparison might

be made with an hour glass. With an hour glass, we can "create" a periodic

occurrence if we have someone standing by it day and night to turn it over when-

ever the last grain of sand runs out.

We could then count the turnings of the glass from each morning to the next.

We would find, this time, that the number of "hours" (i.e., turnings of the glass)

was not the same each "day." We should distrust the sun, or the glass, or both.

After some thought, it might occur to us to count the "hours" from noon to noon.

(Noon is here defined not as 12:00 o'clock, but that instant when the sun is at its

highest point.) We would find, this time, that the number of "hours" each day

is the same.

We now have some confidence that both the "hour" and the "day" have a

regular periodicity, i.e., mark off successive equal intervals of time, although we

have not proved that either one is "really" periodic. Someone might question

whether there might not be some omnipotent being who would slow down the

flow of sand every night and speed it up during the day. Our experiment does not,

of course, give us an answer to this sort of question. All we can say is that we find

that a regularity of one kind fits together with a regularity of another kind. We

can just say that we base our definition of time on the repetition of some apparently

periodic event.

5-3 Short times

We should now notice that in the process of checking on the reproducibility

of the day, we have received an important by-product. We have found a way of

measuring, more accurately, fractions of a day. We have found a way of counting

time in smaller pieces. Can we carry the process further, and learn to measure

even smaller intervals of time?

Galileo decided that a given pendulum always swings back and forth in equal

intervals of time so long as the size of the swing is kept small. A test comparing

the number of swings of a pendulum in one "hour" shows that such is indeed the

case. We can in this way mark fractions of an hour. If we use a mechanical device

to count the swings—and to keep them going—we have the pendulum clock of

our grandfathers.

Let us agree that if our pendulum oscillates 3600 times in one hour (and if

there are 24 such hours in a day), we shall call each period of the pendulum one

"second." We have then divided our original unit of time into approximately

10 5 parts. We can apply the same principles to divide the second into smaller and

smaller intervals. It is, you will realize, not practical to make mechanical pen-

dulums which go arbitrarily fast, but we can now make electrical pendulums,

called oscillators, which can provide a periodic occurrence with a very short

period of swing. In these electronic oscillators it is an electrical current which

swings to and fro, in a manner analogous to the swinging of the bob of the pendulum.

We can make a series of such electronic oscillators, each with a period 10

times shorter than the previous one. We may "calibrate" each oscillator against

the next slower one by counting the number of swings it makes for one swing of

the slower oscillator. When the period of oscillation of our clock is shorter than

a fraction of a second, we cannot count the oscillations without the help of some

device which extends our powers of observation. One such device is the electron-

beam oscilloscope, which acts as a sort of microscope for short times. This device

plots on a fluorescent screen a graph of electrical current (or voltage) versus time.

5-2

By connecting the oscilloscope to two of our oscillators in sequence, so that it

plots a graph first of the current in one of our oscillators and then of the current

in the other, we get two graphs like those shown in Fig. 5-2. We can readily

determine the number of periods of the faster oscillator in one period of the

slower oscillator.

With modern electronic techniques, oscillators have been built with periods

as short as about 10~ 12 second, and they have been calibrated (by comparison

methods such as we have described) in terms of our standard unit of time, the

second. With the invention and perfection of the "laser," or light amplifier, in

the past few years, it has become possible to make oscillators with even shorter

periods than 10~ 12 second, but it has not yet been possible to calibrate them by

the methods which have been described, although it will no doubt soon be possible.

Times shorter than 10~ 12 second have been measured, but by a different tech-

nique. In effect, a different definition of "time" has been used. One way has been

to observe the distance between two happenings on a moving object. If, for

example, the headlights of a moving automobile are turned on and then off,

we can figure out how long the lights were on if we know where they were turned

on and off and how fast the car was moving. The time is the distance over which

the lights were on divided by the speed.

Within the past few years, just such a technique was used to measure the

lifetime of the pð°-meson. By observing in a microscope the minute tracks left in

a photographic emulsion in which p°-mesons had been created one saw that a

p°-meson (known to be travelling at a certain speed nearly that of light) went a

distance of about 10 - 7 meter, on the average, before disintegrating. It lived for

only about 10~ 16 sec. It should be emphasized that we have here used a some-

what different definition of "time" than before. So long as there are no inconsist-

encies in our understanding, however, we feel fairly confident that our definitions

are sufficiently equivalent.

By extending our techniques—and if necessary our definitions—still further

we can infer the time duration of still faster physical events. We can speak of the

period of a nuclear vibration. We can speak of the lifetime of the newly discovered

strange resonances (particles) mentioned in Chapter 2. Their complete life occupies

a time span of only 10 - 24 second, approximately the time it would take light

(which moves at the fastest known speed) to cross the nucleus of hydrogen (the

smallest known object).

What about still smaller times? Does "time" exist on a still smaller scale?

Does it make any sense to speak of smaller times if we cannot measure—or

perhaps even think sensibly about—something which happens in a shorter time?

Perhaps not. These are some of the open questions which you will be asking and

perhaps answering in the next twenty or thirty years.

Fig. 5-2. Two views of an oscilloscope

screen. In (a) the oscilloscope is connected

to one oscillator, in (b) it is connected to an

oscillator with a period one-tenth as long.

5-4 Long times

Let us now consider times longer than one day. Measurement of longer times

is easy; we just count the days—so long as there is someone around to do the-

counting. First we find that there is another natural periodicity: the year, about

365 days. We have also discovered that nature has sometimes provided a counter

for the years, in the form of tree rings or river-bottom sediments. In some cases

we can use these natural time markers to determine the time which has passed

since some early event.

When we cannot count the years for the measurement of long times, we must

look for other ways to measure. One of the most successful is the use of radio-

active material as a "clock." In this case we do not have a periodic occurrence,

as for the day or the pendulum, but a new kind of "regularity." We find that the

radioactivity of a particular sample of material decreases by the same fraction

for successive equal increases in its age. If we plot a graph of the radioactivity

observed as a function of time (say in days), we obtain a curve like that shown in

Fig. 5-3. We observe that if the radioactivity decreases to one-half in T days

(called the "half-life"), then it decreases to one-quarter in another T days, and so

5-3

Fig. 5-3. The decrease with time of

radioactivity. The activity decreases by

one-half in each "half-life," T.

on. In an arbitrary time interval t there are t/T "half-lives," and the fraction left

after this time t is ^') tlT .

If we knew that a piece of material, say a piece of wood, had contained an

amount A of radioactive material when it was formed, and we found out by a direct

measurement that it now contains the amount B, we could compute the age of

the object, t, by solving the equation

There are, fortunately, cases in which we can know the amount of radioactivity

that was in an object when it was formed. We know, for example, that the carbon

dioxide in the air contains a certain small fraction of the radioactive carbon

isotope C 14 (replenished continuously by the action of cosmic rays). If we measure

the total carbon content of an object, we know that a certain fraction of that amount

was originally the radioactive C 14 ; we know, therefore, the starting amount A

to use in the formula above. Carbon-14 has a half-life of 5000 years. By careful

measurements we can measure the amount left after 20 half-lives or so and can

therefore "date" organic objects which grew as long as 100,000 years ago.

We would like to know, and we think we do know, the life of still older things.

Much of our knowledge is based on the measurements of other radioactive iso-

topes which have different half-lives. If we make measurements with an isotope

with a longer half-life, then we are able to measure longer times. Uranium, for

example, has an isotope whose half-life is about 10 9 years, so that if some material

was formed with uranium in it 10 9 years ago, only half the uranium would remain

today. When the uranium disintegrates, it changes into lead. Consider a piece

of rock which was formed a long time ago in some chemical process. Lead, being

of a chemical nature different from uranium, would appear in one part of the rock

and uranium would appear in another part of the rock. The uranium and lead

5-4

would be separate. If we look at that piece of rock today, where there should only

be uranium we will how find a certain fraction of uranium and a certain fraction

of lead. By comparing these fractions, we can tell what percent of the uranium

disappeared and changed into lead. By this method, the age of certain rocks has

been determined to be several billion years. An extension of this method, not

using particular rocks but looking at the uranium and lead in the oceans and using

averages over the earth, has been used to determine (within the past few years)

that the age of the earth itself is approximately 5.5 billion years.

It is encouraging that the age of the earth is found to be the same as the age

of the meteorites which land on the earth, as determined by the uranium method.

It appears that the earth was formed out of rocks floating in space, and that the

meteorites are, quite likely, some of that material left over. At some time more than

five billion years ago, the universe started. It is now believed that at least our part

of the universe had its beginning about ten or twelve billion years ago. We do

not know what happened before then. In fact, we may well ask again: Does the

question make any sense? Does an earlier time have any meaning?

5-5 Units and standards of time

We have implied that it is convenient if we start with some standard unit of

time, say a day or a second, and refer all other times to some multiple or fraction

of this unit. What shall we take as our basic standard of time? Shall we take the

human pulse? If we compare pulses, we find that they seem to vary a lot. On

comparing two clocks, one finds they do not vary so much. You might then say,

well, let us take a clock. But whose clock? There is a story of a Swiss boy who

wanted all of the clocks in his town to ring noon at the same time. So he went

around trying to convince everyone of the value of this. Everyone thought it was

a marvelous idea so long as all of the other clocks rang noon when his did! It is

rather difficult to decide whose clock we should take as a standard. Fortunately,

we all share one clock—the earth. For a long time the rotational period of the

earth has been taken as the basic standard of time. As measurements have been

made more and more precise, however, it has been found that the rotation of the

earth is not exactly periodic, when measured in terms of the best clocks. These

"best" clocks are those which we have reason to believe are accurate because they

agree with each other. We now believe that, for various reasons, some days are

longer than others, some days are shorter, and on the average the period of the

earth becomes a little longer as the centuries pass.

Until very recently we had found nothing much better than the earth's

period, so all clocks have been related to the length of the day, and the second

has been defined as 1/86400 of an average day. Recently we have been gaining

experience with some natural oscillators which we now believe would provide a

more constant time reference than the earth, and which are also based on a natural

phenomenon available to everyone. These are the so-called "atomic clocks."

Their basic internal period is that of an atomic vibration which is very insensitive

to the temperature or any other external effects. These clocks keep time to an

accuracy of one part in 10 9 or better. Within the past two years an improved

atomic clock which operates on the vibration of the hydrogen atom has been de-

signed and built by Professor Norman Ramsey at Harvard University. He believes

that this clock might be 100 times more accurate still. Measurements now in

progress will show whether this is true or not.

We may expect that since it has been possible to build clocks much more

accurate than astronomical time, there will soon be an agreement among scientists

to define the unit of time in terms of one of the atomic clock standards.

5-6 Large distances

Let us now turn to the question of distance. How far, or how big, are things?

Everybody knows that the way you measure distance is to start with a stick and

count. Or start with a thumb and count. You begin with a unit and count. How

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