Understanding Particle Motion and Forces

Toward the earth is less than the number that are

coming from the other side because here they have no opposition from no Sun there

And it's easy to see after some mental effort

That the further the Sun is away the less in Proportion of all of the particles

are being taken out of the possible directions in which particles can come I mean

the solar size

appears smaller and in fact inversely is the square of the distance So there will

therefore be an impulse toward the Sun on the earth That's inversely is the square

of the distance and is the result of large numbers of very simple operations.

Just just hit

one after the other from all directions and therefore the strangeness of the

mathematical relation will be very much reduced because the fundamental operation

is much simpler than Calculating the inverse of the

square of the distance this machine does the calculating these particles bouncing

The only trouble with it is that it doesn't work

For other reasons every theory that

you make up has to be Analyzed against all the possible consequences and to see if

it predicts anything else and this predicts something else

If the earth is moving this way More particles will hit it from the front than from

the back If you're running in

the rain more rain hits you from the front of the face than in the back of the head

Because you're running into the rain and so

as the earth is moving in this direction It's running into the particles rather and

running away from the ones that are chasing it from behind So that more particles

hit it from the front than from the back and there would be a force also Sideways

whenever there was any motion This sideways

force would slow the earth up in the orbit and certainly would not have lasted The

at least three four billion years that has been going around the Sun.

So that's the end of that theory

Well, you say that was a good one though,

it got rid of the mathematics for a while maybe

Maybe I can invent a better one and

Maybe you can because nobody knows the ultimate

But sit up to today from the time of Newton.

No one has invented another theoretical description of the

mathematical machinery behind this law

which does anything else but

say the same thing over again or make the mathematics harder and

At the same time doesn't produce some wrong phenomenon

I mean they have like this model does it but it produces something which isn't true

So there is no model of the theory of gravitation today

other than the mathematical form If this were the only law of this character,

it would not be it would be interesting and rather annoying But what turns out to

be true is that the more we investigate and the more laws we find and the deeper we

penetrate nature This disease that every one of our laws is a purely mathematical

statement in rather complex and abstruse mathematics This is relatively

simple mathematics.

It gets more and

more abstruse and more and more difficult as mathematics and why I

Haven't the slightest idea why

it is only my purpose in this lecture to tell you about this fact In other words,

it's my purpose in this lecture to explain really why I cannot satisfy

you if you do not

understand mathematics Too well in trying to explain nature in any other way.

That is the burden of this lecture In fact to just tell you the fact

That is impossible to answer the really the

honestly the challenge of explaining in a way

that a person can feel The beauties of the laws

of nature without having some deep understanding of mathematics.

I'm sorry seems to be the case

You might say alright then there's no explanation of the law at

least tell me what the law is Why not tell me

in words instead of in the symbols?

Mathematics is just a language and I

want to be able to translate the language and in fact I can and with patience I

think I partly did I could go a little further and explain more detail that

this means that if it's twice as far away The force is one fourth as much and so on

and can convert all these into words I would be in other words kind to the layman

as they all sit Hopefully that you will explain them and various different people

Get different reputations for their skill at explaining to the

layman in layman's language these difficult and abstruse subjects

The layman then searches for book after book with the hope

That he will avoid

the complexity so it's ultimately set in even by the best expositor of this

type He reads the things hoping yet one after

he finds as he reads a generally increased confusion one complicated statement

after the other one difficult to understand thing after the other all

apparently disconnected from one another and It becomes a little obscure and he

hopes that maybe in some other book There's some explanation which avoids which I

mean the man almost made it you see Maybe another fellow

makes it right and I don't think it's possible because there's another

feature mathematics is not just a language

Mathematics is a language plus reasoning.

It's like a language plus logic

It's a mathematics is a tool for reasoning.

It's in fact a big collection of

the results of some person's careful thought and reasoning

By mathematics it is possible to connect one statement to another For instance,

I can say that the force is directed toward the Sun.

I can also tell you as I did before that

the planet moves so said if I draw a line from the Sun to the planet and

draw another one and Some definite period like three weeks later the area

That's swung out by the planet Is exactly the same as it will be in the next

three weeks in the next three weeks and so on as it goes around the

Sun Now I can explain both of those statements to you carefully,

but I cannot explain why they're both the same

So that if you don't appreciate the mathematics and you cannot see

That the great variety

of facts the enormous apparent Complexity of nature with all its funny laws and

rules each of which have been carefully explained to you are really very Closely

interwoven that logic permits you to go from one or to the other

It may be unbelievable that I can demonstrate that equal times will

be swept out if the forces are directed toward the Sun and Just to

if I may try I will show you one demonstration to show you that those two things

really are equivalent and

So that you can appreciate that there's more to merely the statement of the two

laws That the two laws are connected such that reasoning alone will bring you from

one to the other and the mathematics is disorganized reasoning And that

it's good to know how to do that because they will appreciate the beauty of the

relationship of the statement So I'm going to prove if I may the relationship

That if the forces are directed

Toward the Sun that the equal areas are swept out in equal time.

So we thought here's the Sun and

We imagine at a certain time.

Let's say the planet is here

And is moving in such a way that

let's say one second later or one hour pick any time Let's say one second later.

It's moved into such a manner that it has gotten to the position, too

Now If the Sun

did not exert any force on it then by Galileo's principle It would keep right on

going in a straight line So in the same interval of

time Later the next second it'll move exactly the same distance in

the same straight line to the position three were there no force All right.

Now first we're

going to show that if there's no force equal areas are swept out in equal time.

I Is the vertical distance to the base and

That if the out if the triangle is sort of cocky, there's a name for it,

which I forget up to up to

Then the altitude is this vertical height here

See, I know about the triangles.

I just don't know the name

Now let us draw

the lines to these two points in the case that there was no motion whatsoever

The question this doesn't draw very well I'm not accurate,

but these two distances are equal remember question is are these two areas equal?

Well consider the triangle

Made from the Sun and the two points one and two.

It's this one

What's its area it's this space multiplied by this height

and What about the other triangle?

Which is the triangle in motion from two to three

It's this base times the same altitude the two triangles have the same altitude and

as I indicated the same base and have the same Area so so far so good If there were

no force from the Sun equal areas would be swept out in

equal times the two triangles have equal area but there is a force from the Sun and

During this interval one to two to three the Sun is

pulling and changing the motion in various directions this way this way that way

to get a good approximation to

that will take the central position or average position here and say that the whole

effect during this interval was To change the motion by some

amount in this direction toward the Sun

That means that although the particles were moving this way and would have