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So, if you remember last time, 
we looked at parametric

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equations -- -- as a way of
describing the motion of a point

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that moves in the plane or in
space as a function of time of

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00:00:41 --> 00:00:48
your favorite parameter that
will tell you how far the motion

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00:00:48 --> 00:00:54
has progressed.
And, I think we did it in

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detail the example of the
cycloid, which is the curve

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00:01:01 --> 00:01:09
traced by a point on a wheel
that's rolling on a flat

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surface.
So, we have this example where

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00:01:14 --> 00:01:19
we have this wheel that's
rolling on the x-axis,

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00:01:19 --> 00:01:23
and we have this point on the
wheel.

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00:01:23 --> 00:01:31
And, as it moves around,
it traces a trajectory that

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moves more or less like this.
OK, so I'm trying a new color.

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00:01:36 --> 00:01:44
Is this visible from the back?
So, no more blue.

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00:01:44 --> 00:01:52
OK, so remember,
in general, we are trying to

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find the position,
so, x of t, y of t,

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00:01:58 --> 00:02:09
maybe z of t if we are in space
-- -- of a moving point along a

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00:02:09 --> 00:02:17
trajectory.
And, one way to think about

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00:02:17 --> 00:02:25
this is in terms of the position
vector.

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00:02:25 --> 00:02:32
So, position vector is just the
vector whose components are

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00:02:32 --> 00:02:37
coordinates of a point,
OK, so if you prefer,

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00:02:37 --> 00:02:43
that's the same thing as a
vector from the origin to the

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00:02:43 --> 00:02:50
moving point.
So, maybe our point is here, P.

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00:02:50 --> 00:03:02
So, this vector here -- This
vector here is vector OP.

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00:03:02 --> 00:03:12
And, that's also the position
vector r of t.

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00:03:12 --> 00:03:24
So, just to give you,
again, that example -- -- if I

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00:03:24 --> 00:03:34
take the cycloid for a wheel of
radius 1,

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00:03:34 --> 00:03:41
and let's say that we are going
at unit speed so that the angle

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00:03:41 --> 00:03:48
that we used as a parameter of
time is the same thing as time

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00:03:48 --> 00:03:53
when the position vector,
in this case,

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00:03:53 --> 00:04:00
we found to be,
just to make sure that they

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00:04:00 --> 00:04:07
have it right,
.

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00:04:07 --> 00:04:10
OK, that's a formula that you
should have in your notes from

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00:04:10 --> 00:04:13
last time, except we had theta
instead of t because we were

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00:04:13 --> 00:04:16
using the angle.
But now I'm saying,

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00:04:16 --> 00:04:20
we are moving at unit speed,
so time and angle are the same

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00:04:20 --> 00:04:24
thing.
So, now, what's interesting

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00:04:24 --> 00:04:31
about this is we can analyze the
motion in more detail.

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00:04:31 --> 00:04:33
OK, so, now that we know the
position of the point as a

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00:04:33 --> 00:04:37
function of time,
we can try to study how it

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00:04:37 --> 00:04:43
varies in particular things like
the speed and acceleration.

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00:04:43 --> 00:04:48
OK, so let's start with speed.
Well, in fact we can do better

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00:04:48 --> 00:04:51
than speed.
Let's not start with speed.

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00:04:51 --> 00:04:54
So, speed is a number.
It tells you how fast you are

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00:04:54 --> 00:04:58
going along your trajectory.
I mean, if you're driving in a

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00:04:58 --> 00:05:01
car, then it tells you how fast
you are going.

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00:05:01 --> 00:05:03
But, unless you have one of
these fancy cars with a GPS,

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00:05:03 --> 00:05:05
it doesn't tell you which
direction you're going.

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00:05:05 --> 00:05:08
And, that's useful information,
too, if you're trying to figure

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00:05:08 --> 00:05:10
out what your trajectory is.
So, in fact,

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00:05:10 --> 00:05:13
there's two aspects to it.
One is how fast you are going,

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00:05:13 --> 00:05:15
and the other is in what
direction you're going.

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00:05:15 --> 00:05:19
That means actually we should
use a vector maybe to think

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about this.
And so, that's called the

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velocity vector.
And, the way we can get it,

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so, it's called usually V,
so, V here stands for velocity

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more than for vector.
And, you just get it by taking

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the derivative of a position
vector with respect to time.

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00:05:46 --> 00:05:50
Now, it's our first time
writing this kind of thing with

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a vector.
So, the basic rule is you can

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take the derivative of a vector
quantity just by taking the

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derivatives of each component.
OK, so that's just dx/dt,

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00:06:06 --> 00:06:17
dy/dt, and if you have z
component, dz/dt.

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00:06:17 --> 00:06:32
So, let me -- OK,
so -- OK, so let's see what it

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00:06:32 --> 00:06:44
is for the cycloid.
So, an example of a cycloid,

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well, so what do we get when we
take the derivatives of this

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formula there?
Well, so, the derivative of t

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00:07:02 --> 00:07:08
is 1- cos(t).
The derivative of 1 is 0.

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00:07:08 --> 00:07:12
The derivative of -cos(t) is
sin(t).

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00:07:12 --> 00:07:17
Very good.
OK, that's at least one thing

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00:07:17 --> 00:07:20
you should remember from single
variable calculus.

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00:07:20 --> 00:07:24
Hopefully you remember even
more than that.

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00:07:24 --> 00:07:27
OK, so that's the velocity
vector.

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00:07:27 --> 00:07:31
It tells us at any time how
fast we are going,

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00:07:31 --> 00:07:37
and in what direction.
So, for example, observe.

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00:07:37 --> 00:07:40
Remember last time at the end
of class we were trying to

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00:07:40 --> 00:07:43
figure out what exactly happens
near the bottom point,

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00:07:43 --> 00:07:47
when we have this motion that
seems to stop and go backwards.

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00:07:47 --> 00:07:50
And, we answered that one way.
But, let's try to understand it

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00:07:50 --> 00:07:54
in terms of velocity.
What if I plug t equals 0 in

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00:07:54 --> 00:07:57
here?
Then, 1- cos(t) is 0,

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00:07:57 --> 00:08:01
sin(t) is 0.
The velocity is 0.

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00:08:01 --> 00:08:05
So, at the time,at that
particular time,

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00:08:05 --> 00:08:08
our point is actually not
moving.

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00:08:08 --> 00:08:11
Of course, it's been moving
just before, and it starts

91
00:08:11 --> 00:08:14
moving just afterwards.
It's just the instant,

92
00:08:14 --> 00:08:20
at that particular instant,
the speed is zero.

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00:08:20 --> 00:08:23
So, that's especially maybe a
counterintuitive thing,

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00:08:23 --> 00:08:28
but something is moving.
And at that time,

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00:08:28 --> 00:08:33
it's actually stopped.
Now, let's see,

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00:08:33 --> 00:08:36
so that's the vector.
And, it's useful.

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00:08:36 --> 00:08:39
But, if you want just the usual
speed as a number,

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00:08:39 --> 00:08:43
then, what will you do?
Well, you will just take

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00:08:43 --> 00:08:46
exactly the magnitude of this
vector.

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00:08:46 --> 00:08:56
So, speed, which is the scalar
quantity is going to be just the

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00:08:56 --> 00:09:01
magnitude of the vector,
V.

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00:09:01 --> 00:09:09
OK, so, in this case,
while it would be square root

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00:09:09 --> 00:09:18
of (1- cost)^2 sin^2(t),
and if you expand that,

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00:09:18 --> 00:09:23
you will get,
let me take a bit more space, 

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00:09:23 --> 00:09:35
it's going to be square root of
1 - 2cos(t) cos^2(t) sin^2(t).

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00:09:35 --> 00:09:38
It seems to simplify a little
bit because we have cos^2 plus

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00:09:38 --> 00:09:41
sin^2.
That's 1.

108
00:09:41 --> 00:09:49
So, it's going to be the square
root of 2 - 2cos(t).

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00:09:49 --> 00:09:52
So, at this point,
if I was going to ask you,

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00:09:52 --> 00:09:55
when is the speed the smallest
or the largest?

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00:09:55 --> 00:09:59
You could answer based on that.
See, at t equals 0,

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00:09:59 --> 00:10:01
well, that turns out to be
zero.

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00:10:01 --> 00:10:04
The point is not moving.
At t equals pi,

114
00:10:04 --> 00:10:07
that ends up being the square
root of 2 plus 2,

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00:10:07 --> 00:10:09
which is 4.
So, that's 2.

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00:10:09 --> 00:10:12
And, that's when you're truly
at the top of the arch,

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00:10:12 --> 00:10:15
and that's when the point is
moving the fastest.

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00:10:15 --> 00:10:18
In fact, they are spending
twice as fast as the wheel

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00:10:18 --> 00:10:20
because the wheel is moving to
the right at unit speed,

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00:10:20 --> 00:10:24
and the wheel is also rotating.
So, it's moving to the right

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00:10:24 --> 00:10:29
and unit speed relative to the
center so that the two effects

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00:10:29 --> 00:10:32
add up, and give you a speed of
2.

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00:10:32 --> 00:10:36
Anyway, that's a formula we can
get.

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00:10:36 --> 00:10:48
OK, now, what about
acceleration?

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00:10:48 --> 00:10:53
So, here I should warn you that
there is a serious discrepancy

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00:10:53 --> 00:10:58
between the usual intuitive
notion of acceleration,

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00:10:58 --> 00:11:02
the one that you are aware of
when you drive a car and the one

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00:11:02 --> 00:11:05
that we will be using.
So, you might think

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00:11:05 --> 00:11:08
acceleration is just the
directive of speed.

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00:11:08 --> 00:11:13
If my car goes 55 miles an hour
on the highway and it's going a

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00:11:13 --> 00:11:15
constant speed,
it's not accelerating.

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00:11:15 --> 00:11:18
But, let's say that I'm taking
a really tight turn.

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00:11:18 --> 00:11:19
Then, I'm going to feel
something.

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00:11:19 --> 00:11:21
There is some force being
exerted.

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00:11:21 --> 00:11:24
And, in fact,
there is a sideways

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00:11:24 --> 00:11:28
acceleration at that point even
though the speed is not

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00:11:28 --> 00:11:30
changing.
So, the definition will take

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00:11:30 --> 00:11:34
effect.
The acceleration is,

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00:11:34 --> 00:11:40
as a vector,
and the acceleration vector is

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00:11:40 --> 00:11:47
just the derivative of a
velocity vector.

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00:11:47 --> 00:11:51
So, even if the speed is
constant, that means,

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00:11:51 --> 00:11:55
even if a length of the
velocity vector stays the same,

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00:11:55 --> 00:11:59
the velocity vector can still
rotate.

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00:11:59 --> 00:12:03
And, as it rotates,
it uses acceleration.

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00:12:03 --> 00:12:07
OK, and so this is the notion
of acceleration that's relevant

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00:12:07 --> 00:12:13
to physics when you find F=ma;
that's the (a) that you have in

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00:12:13 --> 00:12:17
mind here.
It's a vector.

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00:12:17 --> 00:12:19
Of course, if you are moving in
a straight line,

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00:12:19 --> 00:12:20
then the two notions are the
same.

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00:12:20 --> 00:12:23
I mean, acceleration is also
going to be along the line,

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00:12:23 --> 00:12:25
and it's going to has to do
with the derivative of speed.

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00:12:25 --> 00:12:30
But, in general,
that's not quite the same.

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00:12:30 --> 00:12:37
So, for example,
let's look at the cycloid.

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00:12:37 --> 00:12:40
If we take the example of the
cycloid, well,

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00:12:40 --> 00:12:44
what's the derivative of one
minus cos(t)?

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00:12:44 --> 00:12:52
It's sin(t).
And, what's the derivative of

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00:12:52 --> 00:12:55
sin(t)?
cos(t), OK.

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00:12:55 --> 00:13:04
So, the acceleration vector is
.

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00:13:04 --> 00:13:09
So, in particular,
let's look at what happens at

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00:13:09 --> 00:13:13
time t equals zero when the
point is not moving.

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00:13:13 --> 00:13:20
Well, the acceleration vector
there will be zero from one.

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00:13:20 --> 00:13:28
So, what that means is that if
I look at my trajectory at this

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00:13:28 --> 00:13:35
point, that the acceleration
vector is pointing in that

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00:13:35 --> 00:13:39
direction.
It's the unit vector in the

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00:13:39 --> 00:13:43
vertical direction.
So, my point is not moving at

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00:13:43 --> 00:13:46
that particular time.
But, it's accelerating up.

167
00:13:46 --> 00:13:49
So, that means that actually as
it comes down,

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00:13:49 --> 00:13:53
first it's slowing down.
Then it stops here,

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00:13:53 --> 00:13:56
and then it reverses going back
up.

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00:13:56 --> 00:14:01
OK, so that's another way to
understand what we were saying

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00:14:01 --> 00:14:06
last time that the trajectory at
that point has a vertical

172
00:14:06 --> 00:14:11
tendency because that's the
direction in which the motion is

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00:14:11 --> 00:14:16
going to occur just before and
just after time zero.

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00:14:16 --> 00:14:30
OK, any questions about that?
No.

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00:14:30 --> 00:14:36
OK, so I should insist maybe on
one thing,

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00:14:36 --> 00:14:41
which is that,
so, we can differentiate

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00:14:41 --> 00:14:46
vectors just component by
component,

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00:14:46 --> 00:14:50
OK, and we can differentiate
vector expressions according to

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00:14:50 --> 00:14:54
certain rules that we'll see in
a moment.

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00:14:54 --> 00:15:02
One thing that we cannot do,
it's not true that the length

181
00:15:02 --> 00:15:12
of dr dt, which is the speed,
is equal to the length of dt.

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00:15:12 --> 00:15:18
OK, this is completely false.
And, they are really not the

183
00:15:18 --> 00:15:19
same.
So, if you have to

184
00:15:19 --> 00:15:24
differentiate the length of a
vector, but basically you are in

185
00:15:24 --> 00:15:25
trouble.
If you really,

186
00:15:25 --> 00:15:27
really want to do it,
well, the length of the vector

187
00:15:27 --> 00:15:30
is the square root of the sums
of the squares of the

188
00:15:30 --> 00:15:32
components,
and from that you can use the

189
00:15:32 --> 00:15:34
formula for the derivative of
the square root,

190
00:15:34 --> 00:15:36
and the chain rule,
and various other things.

191
00:15:36 --> 00:15:39
And, you can get there.
But, it will not be a very nice

192
00:15:39 --> 00:15:42
expression.
There is no simple formula for

193
00:15:42 --> 00:15:44
this kind of thing.
Fortunately,

194
00:15:44 --> 00:15:48
we almost never have to compute
this kind of thing because,

195
00:15:48 --> 00:15:51
after all, it's not a very
relevant quantity.

196
00:15:51 --> 00:15:53
What's more relevant might be
this one.

197
00:15:53 --> 00:15:59
This is actually the speed.
This one, I don't know what it

198
00:15:59 --> 00:16:10
means.
OK.

199
00:16:10 --> 00:16:14
So, let's continue our
exploration.

200
00:16:14 --> 00:16:20
So, the next concept that I
want to define is that of arc

201
00:16:20 --> 00:16:23
length.
So, arc length is just the

202
00:16:23 --> 00:16:26
distance that you have traveled
along the curve,

203
00:16:26 --> 00:16:27
OK?
So, if you are in a car,

204
00:16:27 --> 00:16:30
you know, it has mileage
counter that tells you how far

205
00:16:30 --> 00:16:33
you've gone, how much fuel
you've used if it's a fancy car.

206
00:16:33 --> 00:16:37
And, what it does is it
actually integrates the speed of

207
00:16:37 --> 00:16:41
the time to give you the arc
length along the trajectory of

208
00:16:41 --> 00:16:45
the car.
So, the usual notation that we

209
00:16:45 --> 00:16:51
will have is (s) for arc length.
I'm not quite sure how you get

210
00:16:51 --> 00:16:57
an (s) out of this,
but it's the usual notation.

211
00:16:57 --> 00:17:14
OK, so, (s) is for distance
traveled along the trajectory.

212
00:17:14 --> 00:17:16
And, so that makes sense,
of course, we need to fix a

213
00:17:16 --> 00:17:19
reference point.
Maybe on the cycloid,

214
00:17:19 --> 00:17:22
we'd say it's a distance
starting on the origin.

215
00:17:22 --> 00:17:25
In general, maybe you would say
you start at time,

216
00:17:25 --> 00:17:28
t equals zero.
But, it's a convention.

217
00:17:28 --> 00:17:31
If you knew in advance,
you could have,

218
00:17:31 --> 00:17:35
actually, your car's mileage
counter to count backwards from

219
00:17:35 --> 00:17:38
the point where the car will die
and start walking.

220
00:17:38 --> 00:17:41
I mean, that would be
sneaky-freaky,

221
00:17:41 --> 00:17:45
but you could have a negative
arc length that gets closer and

222
00:17:45 --> 00:17:48
closer to zero,
and gets to zero at the end of

223
00:17:48 --> 00:17:51
a trajectory,
or anything you want.

224
00:17:51 --> 00:17:53
I mean, arc length could be
positive or negative.

225
00:17:53 --> 00:17:56
Typically it's negative what
you are before the reference

226
00:17:56 --> 00:18:01
point, and positive afterwards.
So, now, how does it relate to

227
00:18:01 --> 00:18:08
the things we've seen there?
Well, so in particular,

228
00:18:08 --> 00:18:16
how do you relate arc length
and time?

229
00:18:16 --> 00:18:22
Well, so, there's a simple
relation, which is that the rate

230
00:18:22 --> 00:18:26
of change of arc length versus
time,

231
00:18:26 --> 00:18:30
well, that's going to be the
speed at which you are moving,

232
00:18:30 --> 00:18:38
OK, because the speed as a
scalar quantity tells you how

233
00:18:38 --> 00:18:44
much distance you're covering
per unit time.

234
00:18:44 --> 00:18:47
OK, and in fact,
to be completely honest,

235
00:18:47 --> 00:18:51
I should put an absolute value
here because there is examples

236
00:18:51 --> 00:18:55
of curves maybe where your
motion is going back and forth

237
00:18:55 --> 00:18:59
along the same curve.
And then, you don't want to

238
00:18:59 --> 00:19:01
keep counting arc length all the
time.

239
00:19:01 --> 00:19:04
Actually, maybe you want to say
that the arc length increases

240
00:19:04 --> 00:19:05
and then decreases along the
curve.

241
00:19:05 --> 00:19:08
I mean, you get to choose how
you count it.

242
00:19:08 --> 00:19:10
But, in this case,
if you are moving back and

243
00:19:10 --> 00:19:12
forth, it would make more sense
to have the arc length first

244
00:19:12 --> 00:19:18
increase,
then decrease,

245
00:19:18 --> 00:19:26
increase again,
and so on.

246
00:19:26 --> 00:19:34
So -- So if you want to know
really what the arc length is,

247
00:19:34 --> 00:19:41
then basically the only way to
do it is to integrate speed

248
00:19:41 --> 00:19:45
versus time.
So, if you wanted to know how

249
00:19:45 --> 00:19:49
long an arch of cycloid is,
you have this nice-looking

250
00:19:49 --> 00:19:51
curve;
how long is it?

251
00:19:51 --> 00:19:55
Well, you'd have to basically
integrate this quantity from t

252
00:19:55 --> 00:19:57
equals zero to 2 pi.

253
00:19:57 --> 00:20:24

254
00:20:24 --> 00:20:28
And, to say the truth,
I don't really know how to

255
00:20:28 --> 00:20:31
integrate that.
So, we don't actually have a

256
00:20:31 --> 00:20:34
formula for the length at this
point.

257
00:20:34 --> 00:20:41
However, we'll see one later
using a cool trick,

258
00:20:41 --> 00:20:47
and multi-variable calculus.
So, for now,

259
00:20:47 --> 00:20:52
we'll just leave the formula
like that, and we don't know how

260
00:20:52 --> 00:20:55
long it is.
Well, you can put that into

261
00:20:55 --> 00:20:57
your calculator and get the
numerical value.

262
00:20:57 --> 00:21:07
But, that's the best I can
offer.

263
00:21:07 --> 00:21:18
Now, another useful notion is
the unit vector to the

264
00:21:18 --> 00:21:25
trajectory.
So, the usual notation is T hat.

265
00:21:25 --> 00:21:28
It has a hat because it's a
unit vector, and T because it's

266
00:21:28 --> 00:21:32
tangent.
Now, how do we get this unit

267
00:21:32 --> 00:21:36
vector?
So, maybe I should have pointed

268
00:21:36 --> 00:21:40
out before that if you're moving
along some trajectory,

269
00:21:40 --> 00:21:43
say you're going in that
direction, then when you're at

270
00:21:43 --> 00:21:47
this point,
the velocity vector is going to

271
00:21:47 --> 00:21:53
be tangential to the trajectory.
It tells you the direction of

272
00:21:53 --> 00:21:57
motion in particular.
So, if you want a unit vector

273
00:21:57 --> 00:22:02
that goes in the same direction,
all you have to do is rescale

274
00:22:02 --> 00:22:05
it, so, at its length becomes
one.

275
00:22:05 --> 00:22:10
So, it's v divided by a
magnitude of v.

276
00:22:10 --> 00:22:28
 
 

277
00:22:28 --> 00:22:33
So, it seems like now we have a
lot of different things that

278
00:22:33 --> 00:22:40
should be related in some way.
So, let's see what we can say.

279
00:22:40 --> 00:22:50
Well, we can say that dr by dt,
so, that's the velocity vector,

280
00:22:50 --> 00:22:59
that's the same thing as if I
use the chain rule dr/ds times

281
00:22:59 --> 00:23:06
ds/dt.
OK, so, let's think about this

282
00:23:06 --> 00:23:11
things.
So, this guy here we've just

283
00:23:11 --> 00:23:17
seen.
That's the same as the speed,

284
00:23:17 --> 00:23:21
OK?
So, this one here should be v

285
00:23:21 --> 00:23:28
divided by its length.
So, that means this actually

286
00:23:28 --> 00:23:34
should be the unit vector.
OK, so, let me rewrite that.

287
00:23:34 --> 00:23:40
It's T ds/dt.
So, maybe if I actually stated

288
00:23:40 --> 00:23:43
directly that way,
see, I'm just saying the

289
00:23:43 --> 00:23:46
velocity vector has a length and
a direction.

290
00:23:46 --> 00:23:51
The length is the speed.
The direction is tangent to the

291
00:23:51 --> 00:23:51
trajectory.

292
00:23:51 --> 00:24:19

293
00:24:19 --> 00:24:25
So, the speed is ds/dt,
and the vector is T hat.

294
00:24:25 --> 00:24:33
And, that's how we get this.
So, let's try just to see why

295
00:24:33 --> 00:24:37
dr/ds should be T.
Well, let's think of dr/ds.

296
00:24:37 --> 00:24:42
dr/ds means position vector r
means you have the origin,

297
00:24:42 --> 00:24:47
which is somewhere out there,
and the vector r is here.

298
00:24:47 --> 00:24:51
So, dr/ds means we move by a
small amount,

299
00:24:51 --> 00:24:56
delta s along the trajectory a
certain distance delta s.

300
00:24:56 --> 00:25:00
And, we look at how the
position vector changes.

301
00:25:00 --> 00:25:08
Well, we'll have a small change.
Let me call that vector delta r

302
00:25:08 --> 00:25:13
corresponding to the size,
corresponding to the length

303
00:25:13 --> 00:25:17
delta s.
And now, delta r should be

304
00:25:17 --> 00:25:25
essentially roughly equal to,
well, its direction will be

305
00:25:25 --> 00:25:30
tangent to the trajectory.
If I take a small enough

306
00:25:30 --> 00:25:33
interval,
then the direction will be

307
00:25:33 --> 00:25:37
almost tensioned to the
trajectory times the length of

308
00:25:37 --> 00:25:41
it will be delta s,
the distance that I have

309
00:25:41 --> 00:25:45
traveled.
OK, sorry, maybe I should

310
00:25:45 --> 00:25:50
explain that on a separate
board.

311
00:25:50 --> 00:25:56
OK, so, let's say that we have
that amount of time,

312
00:25:56 --> 00:26:00
delta t.
So, let's zoom into that curve.

313
00:26:00 --> 00:26:12
So, we have r at time t.
We have r at time t plus delta

314
00:26:12 --> 00:26:17
t.
This vector here I will call

315
00:26:17 --> 00:26:23
delta r.
The length of this vector is

316
00:26:23 --> 00:26:28
delta s.
And, the direction is

317
00:26:28 --> 00:26:36
essentially that of a vector.
OK, so, delta s over delta t,

318
00:26:36 --> 00:26:43
that's the distance traveled
divided by the time.

319
00:26:43 --> 00:26:46
That's going to be close to the
speed.

320
00:26:46 --> 00:26:57
And, delta r is approximately T
times delta s.

321
00:26:57 --> 00:27:04
So, now if I divide both sides
by delta t, I get this.

322
00:27:04 --> 00:27:07
And, if I take the limit as
delta t turns to zero,

323
00:27:07 --> 00:27:10
then I get the same formula
with the derivatives and with an

324
00:27:10 --> 00:27:13
equality.
It's an approximation.

325
00:27:13 --> 00:27:15
The approximation becomes
better and better if I go to

326
00:27:15 --> 00:27:16
smaller intervals.

327
00:27:16 --> 00:27:38

328
00:27:38 --> 00:27:44
OK, are there any questions
about this?

329
00:27:44 --> 00:27:59
Yes?
Yes, that's correct.

330
00:27:59 --> 00:28:01
OK, so let's be more careful,
actually.

331
00:28:01 --> 00:28:12
So, you're asking about whether
the delta r is actually strictly

332
00:28:12 --> 00:28:16
tangent to the curve.
Is that -- That's correct.

333
00:28:16 --> 00:28:20
Actually, delta r is not
strictly tangent to anything.

334
00:28:20 --> 00:28:23
So, maybe I should draw another
picture.

335
00:28:23 --> 00:28:29
If I'm going from here to here,
then delta r is going to be

336
00:28:29 --> 00:28:36
this arc inside the curve while
the vector will be going in this

337
00:28:36 --> 00:28:39
direction, OK?
So, they are not strictly

338
00:28:39 --> 00:28:41
parallel to each other.
That's why it's only

339
00:28:41 --> 00:28:44
approximately equal.
Similarly, this distance,

340
00:28:44 --> 00:28:48
the length of delta r is not
exactly the length along the

341
00:28:48 --> 00:28:50
curve.
It's actually a bit shorter.

342
00:28:50 --> 00:28:53
But, if we imagine a smaller
and smaller portion of the

343
00:28:53 --> 00:28:56
curve,
then this effect of the curve

344
00:28:56 --> 00:29:00
being a curve and not a straight
line becomes more and more

345
00:29:00 --> 00:29:02
negligible.
If you zoom into the curve

346
00:29:02 --> 00:29:04
sufficiently,
then it looks more and more

347
00:29:04 --> 00:29:07
like a straight line.
And then, what I said becomes

348
00:29:07 --> 00:29:18
true in the limit.
OK? Any other questions?

349
00:29:18 --> 00:29:35
No? OK.
So, what happens next?

350
00:29:35 --> 00:29:39
OK, so let me show you a nice
example of why we might want to

351
00:29:39 --> 00:29:43
use vectors to study parametric
curves because,

352
00:29:43 --> 00:29:46
after all, a lot of what's here
you can just do in coordinates.

353
00:29:46 --> 00:29:48
And, we don't really need
vectors.

354
00:29:48 --> 00:29:51
Well, and truly,
vectors being a language,

355
00:29:51 --> 00:29:54
you never strictly need it,
but it's useful to have a

356
00:29:54 --> 00:30:02
notion of vectors.
So, I want to tell you a bit

357
00:30:02 --> 00:30:14
about Kepler's second law of
celestial mechanics.

358
00:30:14 --> 00:30:20
So, that goes back to 1609.
So, that's not exactly recent

359
00:30:20 --> 00:30:24
news, OK?
But, still I think it's a very

360
00:30:24 --> 00:30:29
interesting example of why you
might want to use vector methods

361
00:30:29 --> 00:30:33
to analyze motions.
So, what happened back then was

362
00:30:33 --> 00:30:39
Kepler was trying to observe the
motion of planets in the sky,

363
00:30:39 --> 00:30:42
and trying to come up with
general explanations of how they

364
00:30:42 --> 00:30:44
move.
Before him, people were saying,

365
00:30:44 --> 00:30:46
well, they cannot move in a
circle.

366
00:30:46 --> 00:30:48
But maybe it's more complicated
than that.

367
00:30:48 --> 00:30:51
We need to add smaller circular
motions on top of each other,

368
00:30:51 --> 00:30:53
and so on.
They have more and more

369
00:30:53 --> 00:30:56
complicated theories.
And then Kepler came with these

370
00:30:56 --> 00:31:00
laws that said basically that
planets move in an ellipse

371
00:31:00 --> 00:31:03
around the sun,
and that they move in a very

372
00:31:03 --> 00:31:07
specific way along that ellipse.
So, there's actually three

373
00:31:07 --> 00:31:11
laws, but let me just tell you
about the second one that has a

374
00:31:11 --> 00:31:17
very nice vector interpretation.
So, what Kepler's second law

375
00:31:17 --> 00:31:24
says is that the motion of
planets is, first of all,

376
00:31:24 --> 00:31:36
they move in a plane.
And second, the area swept out

377
00:31:36 --> 00:31:51
by the line from the sun to the
planet is swept at constant

378
00:31:51 --> 00:31:57
time.
Sorry, is swept at constant

379
00:31:57 --> 00:32:04
rate.
From the sun to the planet,

380
00:32:04 --> 00:32:14
it is swept out by the line at
a constant rate.

381
00:32:14 --> 00:32:23
OK, so that's an interesting
law because it tells you,

382
00:32:23 --> 00:32:27
once you know what the orbit of
the planet looks like,

383
00:32:27 --> 00:32:30
it tells you how fast it's
going to move on that orbit.

384
00:32:30 --> 00:33:09
 
 

385
00:33:09 --> 00:33:19
OK, so let me explain again.
So, this law says maybe the

386
00:33:19 --> 00:33:27
sun, let's put the sun here at
the origin, and let's have a

387
00:33:27 --> 00:33:34
planet.
Well, the planet orbits around

388
00:33:34 --> 00:33:41
the sun -- -- in some
trajectory.

389
00:33:41 --> 00:33:45
So, this is supposed to be
light blue.

390
00:33:45 --> 00:33:49
Can you see that it's different
from white?

391
00:33:49 --> 00:33:51
No?
OK, me neither.

392
00:33:51 --> 00:33:53
[LAUGHTER]
OK, it doesn't really matter.

393
00:33:53 --> 00:33:55
So, the planet moves on its
orbit.

394
00:33:55 --> 00:34:00
And, if you wait for a certain
time, then a bit later it would

395
00:34:00 --> 00:34:04
be here, and then here,
and so on.

396
00:34:04 --> 00:34:09
Then, you can look at the
amount of area inside this

397
00:34:09 --> 00:34:12
triangular wedge.
And, the claim is that the

398
00:34:12 --> 00:34:16
amount of area in here is
proportional to the time

399
00:34:16 --> 00:34:18
elapsed.
So, in particular,

400
00:34:18 --> 00:34:21
if a planet is closer to the
sun, then it has to go faster.

401
00:34:21 --> 00:34:25
And, if it's farther away from
the sun, then it has to go

402
00:34:25 --> 00:34:28
slower so that the area remains
proportional to time.

403
00:34:28 --> 00:34:32
So, it's a very sophisticated
prediction.

404
00:34:32 --> 00:34:36
And, I think the way he came to
it was really just by using a

405
00:34:36 --> 00:34:39
lot of observations,
and trying to measure what was

406
00:34:39 --> 00:34:44
true that wasn't true.
But, let's try to see how we

407
00:34:44 --> 00:34:49
can understand that in terms of
all we know today about

408
00:34:49 --> 00:34:52
mechanics.
So, in fact,

409
00:34:52 --> 00:34:56
what happens is that Newton,
so Newton was quite a bit

410
00:34:56 --> 00:35:04
later.
That was the late 17th century

411
00:35:04 --> 00:35:13
instead of the beginning of the
17th century.

412
00:35:13 --> 00:35:30
So, he was able to explain this
using his laws for gravitational

413
00:35:30 --> 00:35:36
attraction.
And, you'll see that if we

414
00:35:36 --> 00:35:41
reformulate Kepler's Law in
terms of vectors,

415
00:35:41 --> 00:35:43
and if we work a bit with these
vectors,

416
00:35:43 --> 00:35:46
we are going to end up with
something that's actually

417
00:35:46 --> 00:35:49
completely obvious to us now.
At the time,

418
00:35:49 --> 00:35:52
it was very far from obvious,
but to us now to completely

419
00:35:52 --> 00:35:59
obvious.
So, let's try to see,

420
00:35:59 --> 00:36:15
what does Kepler's law say in
terms of vectors?

421
00:36:15 --> 00:36:24
OK, so, let's think of what
kinds of vectors we might want

422
00:36:24 --> 00:36:31
to have in here.
Well, it might be good to think

423
00:36:31 --> 00:36:38
of, maybe, the position vector,
and maybe its variation.

424
00:36:38 --> 00:36:46
So, if we wait a certain amount
of time, we'll have a vector,

425
00:36:46 --> 00:36:53
delta r, which is the change in
position vector a various

426
00:36:53 --> 00:36:59
interval of time.
OK, so let's start with the

427
00:36:59 --> 00:37:02
first step.
What's the most complicated

428
00:37:02 --> 00:37:05
thing in here?
It's this area swept out by the

429
00:37:05 --> 00:37:08
line.
How do we express that area in

430
00:37:08 --> 00:37:12
terms of vectors?
Well, I've almost given the

431
00:37:12 --> 00:37:14
answer by drawing this picture,
right?

432
00:37:14 --> 00:37:18
If I take a sufficiently small
amount of time,

433
00:37:18 --> 00:37:22
this shaded part looks like a
triangle.

434
00:37:22 --> 00:37:25
So, we have to find the area of
the triangle.

435
00:37:25 --> 00:37:27
Well, we know how to do that
now.

436
00:37:27 --> 00:37:34
So, the area is approximately
equal to one half of the area of

437
00:37:34 --> 00:37:40
a parallelogram that I could
form from these vectors.

438
00:37:40 --> 00:37:46
And, the area of a
parallelogram is given by the

439
00:37:46 --> 00:37:52
magnitude of a cross product.
OK, so, I should say,

440
00:37:52 --> 00:37:56
this is the area swept in time
delta t.

441
00:37:56 --> 00:38:00
You should think of delta t as
relatively small.

442
00:38:00 --> 00:38:05
I mean, the scale of a planet
that might still be a few days,

443
00:38:05 --> 00:38:09
but small compared to the other
old trajectory.

444
00:38:09 --> 00:38:16
So, let's remember that the
amount by which we moved,

445
00:38:16 --> 00:38:20
delta r,
is approximately equal to v

446
00:38:20 --> 00:38:25
times delta t,
OK, and just using the

447
00:38:25 --> 00:38:36
definition of a velocity vector.
So, let's use that.

448
00:38:36 --> 00:38:43
Sorry, so it's approximately
equal to r cross v magnitude

449
00:38:43 --> 00:38:48
times delta t.
I can take out the delta t,

450
00:38:48 --> 00:38:52
which is scalar.
So, now, what does it mean to

451
00:38:52 --> 00:38:55
say that area is swept at a
constant rate?

452
00:38:55 --> 00:39:00
It means this thing is
proportional to delta t.

453
00:39:00 --> 00:39:05
So, that means,
so, the law says,

454
00:39:05 --> 00:39:15
in fact, that the length of
this cross product r cross v

455
00:39:15 --> 00:39:25
equals a constant.
OK, r cross v has constant

456
00:39:25 --> 00:39:31
length.
Any questions about that?

457
00:39:31 --> 00:39:37
No? Yes?
Yes, let me try to explain that

458
00:39:37 --> 00:39:40
again.
So, what I'm claiming is that

459
00:39:40 --> 00:39:46
the length of the cross products
r cross v measures the rate at

460
00:39:46 --> 00:39:50
which area is swept by the
position vector.

461
00:39:50 --> 00:39:52
I should say,
with a vector of one half of

462
00:39:52 --> 00:39:55
this length is the rate at which
area is swept.

463
00:39:55 --> 00:39:58
How do we see that?
Well, let's take a small time

464
00:39:58 --> 00:40:01
interval, delta t.
In time, delta t,

465
00:40:01 --> 00:40:05
our planet moves by v delta t,
OK?

466
00:40:05 --> 00:40:08
So, if it moves by v delta t,
it means that this triangle up

467
00:40:08 --> 00:40:12
there has two sides.
One is the position vector,

468
00:40:12 --> 00:40:14
r.
The other one is v delta t.

469
00:40:14 --> 00:40:18
So, its area is given by one
half of the magnitude of a cross

470
00:40:18 --> 00:40:21
product.
That's the formula we've seen

471
00:40:21 --> 00:40:24
for the area of a triangle in
space.

472
00:40:24 --> 00:40:28
So, the area is one half of the
cross product,

473
00:40:28 --> 00:40:33
r, and v delta t,
magnitude of the cross product.

474
00:40:33 --> 00:40:37
So, to say that the rate at
which area is swept is constant

475
00:40:37 --> 00:40:39
means that these two are
proportional.

476
00:40:39 --> 00:40:42
Area divided by delta t is
constant at our time.

477
00:40:42 --> 00:40:51
And so, this is constant.
OK, now, what about the other

478
00:40:51 --> 00:40:58
half of the law?
Well, it says that the motion

479
00:40:58 --> 00:41:04
is in a plane,
and so we have a plane in which

480
00:41:04 --> 00:41:09
the motion takes place.
And, it contains,

481
00:41:09 --> 00:41:12
also, the sun.
And, it contains the

482
00:41:12 --> 00:41:16
trajectory.
So, let's think about that

483
00:41:16 --> 00:41:20
plane.
Well, I claim that the position

484
00:41:20 --> 00:41:25
vector is in the plane.
OK, that's what we are saying.

485
00:41:25 --> 00:41:28
But, there is another vector
that I know it is in the plane.

486
00:41:28 --> 00:41:32
You could say the position
vector at another time,

487
00:41:32 --> 00:41:34
or at any time,
but in fact,

488
00:41:34 --> 00:41:40
what's also true is that the
velocity vector is in the plane.

489
00:41:40 --> 00:41:44
OK, if I'm moving in the plane,
then position and velocity are

490
00:41:44 --> 00:41:50
in there.
So, the plane of motion

491
00:41:50 --> 00:41:59
contains r and v.
So, what's the direction of the

492
00:41:59 --> 00:42:08
cross product r cross v?
Well, it's the direction that's

493
00:42:08 --> 00:42:19
perpendicular to this plane.
So, it's normal to the plane of

494
00:42:19 --> 00:42:24
motion.
And, that means, now, 

495
00:42:24 --> 00:42:28
that actually we've put the two
statements in there into a

496
00:42:28 --> 00:42:33
single form because we are
saying r cross v has constant

497
00:42:33 --> 00:42:37
length and constant direction.
In fact, in general,

498
00:42:37 --> 00:42:40
maybe I should say something
about this.

499
00:42:40 --> 00:42:42
So, if you just look at the
position vector,

500
00:42:42 --> 00:42:45
and the velocity vector for any
motion at any given time,

501
00:42:45 --> 00:42:48
then together,
they determine some plane.

502
00:42:48 --> 00:42:51
And, that's the plane that
contains the origin,

503
00:42:51 --> 00:42:54
the point, and the velocity
vector.

504
00:42:54 --> 00:42:56
If you want,
it's the plane in which the

505
00:42:56 --> 00:42:59
motion seems to be going at the
given time.

506
00:42:59 --> 00:43:01
Now, of course,
if your motion is not in a

507
00:43:01 --> 00:43:03
plane, then that plane will
change.

508
00:43:03 --> 00:43:06
It's, however,
instant, if a plane in which

509
00:43:06 --> 00:43:09
the motion is taking place at a
given time.

510
00:43:09 --> 00:43:13
And, to say that the motion
actually stays in that plane

511
00:43:13 --> 00:43:17
forever means that this guy will
not change direction.

512
00:43:17 --> 00:43:25
OK, so -- [LAUGHTER]
[APPLAUSE]

513
00:43:25 --> 00:43:42
OK, so, Kepler's second law is
actually equivalent to saying

514
00:43:42 --> 00:43:55
that r cross v equals a constant
vector, OK?

515
00:43:55 --> 00:44:04
That's what the law says.
So, in terms of derivatives,

516
00:44:04 --> 00:44:14
it means d by dt of r cross v
is the zero vector.

517
00:44:14 --> 00:44:20
OK, now, so there's an
interesting thing to note,

518
00:44:20 --> 00:44:23
which is that we can use the
usual product rule for

519
00:44:23 --> 00:44:26
derivatives with vector
expressions,

520
00:44:26 --> 00:44:28
with dot products or cross
products.

521
00:44:28 --> 00:44:30
There's only one catch,
which is that when we

522
00:44:30 --> 00:44:34
differentiate a cross product,
we have to be careful that the

523
00:44:34 --> 00:44:36
guy on the left stays on the
left.

524
00:44:36 --> 00:44:40
The guy on the right stays on
the right.

525
00:44:40 --> 00:44:44
OK, so, if you know that uv
prime equals u prime v plus uv

526
00:44:44 --> 00:44:47
prime, then you are safe.
If you know it as u prime v

527
00:44:47 --> 00:44:50
cross v prime u,
then you are not safe.

528
00:44:50 --> 00:44:52
OK, so it's the only thing to
watch for.

529
00:44:52 --> 00:45:05
So, product rule is OK for
taking the derivative of a dot

530
00:45:05 --> 00:45:10
product.
There, you don't actually even

531
00:45:10 --> 00:45:14
need to be very careful about
all the things or the derivative

532
00:45:14 --> 00:45:18
of a cross product.
There you just need to be a

533
00:45:18 --> 00:45:27
little bit more careful.
OK, so, now that we know that,

534
00:45:27 --> 00:45:39
we can write this as dr/dt
cross v plus r cross dv/dt,

535
00:45:39 --> 00:45:42
OK?
Well, let's reformulate things

536
00:45:42 --> 00:45:47
slightly.
So, dr dt already has a name.

537
00:45:47 --> 00:45:50
In fact, that's v.
OK, that's what we call the

538
00:45:50 --> 00:45:55
velocity vector.
So, this is v cross v plus r

539
00:45:55 --> 00:46:04
cross, what is dv/dt?
That's the acceleration,

540
00:46:04 --> 00:46:11
a, equals zero.
OK, so now what's the next step?

541
00:46:11 --> 00:46:15
Well, we know what v cross v is
because, remember,

542
00:46:15 --> 00:46:18
a vector cross itself is always
zero, OK?

543
00:46:18 --> 00:46:30
So, this is the same r cross a
equals zero,

544
00:46:30 --> 00:46:35
and that's the same as saying
that the cross product of two

545
00:46:35 --> 00:46:39
vectors is zero exactly when the
parallelogram of the form has no

546
00:46:39 --> 00:46:41
area.
And, the way in which that

547
00:46:41 --> 00:46:45
happens is if they are actually
parallel to each other.

548
00:46:45 --> 00:46:50
So, that means the acceleration
is parallel to the position.

549
00:46:50 --> 00:46:55
OK, so, in fact,
what Kepler's second law says

550
00:46:55 --> 00:47:02
is that the acceleration is
parallel to the position vector.

551
00:47:02 --> 00:47:05
And, since we know that
acceleration is caused by a

552
00:47:05 --> 00:47:08
force that's equivalent to the
fact that the gravitational

553
00:47:08 --> 00:47:08
force --

554
00:47:08 --> 00:47:13

555
00:47:13 --> 00:47:18
-- is parallel to the position
vector, that means,

556
00:47:18 --> 00:47:22
well, if you have the sun here
at the origin,

557
00:47:22 --> 00:47:27
and if you have your planets,
well, the gravitational force

558
00:47:27 --> 00:47:32
caused by the sun should go
along this line.

559
00:47:32 --> 00:47:34
In fact, the law doesn't even
say whether it's going towards

560
00:47:34 --> 00:47:37
the sun or away from the sun.
Well, what we know now is that,

561
00:47:37 --> 00:47:39
of course, the attraction is
towards the sun.

562
00:47:39 --> 00:47:41
But, Kepler's law would also be
true, actually,

563
00:47:41 --> 00:47:44
if things were going away.
So, in particular,

564
00:47:44 --> 00:47:48
say, electric force also has
this property of being towards

565
00:47:48 --> 00:47:50
the central charge.
So, actually,

566
00:47:50 --> 00:47:54
if you look at motion of
charged particles in an electric

567
00:47:54 --> 00:47:58
field caused by a point charged
particle, it also satisfies

568
00:47:58 --> 00:48:01
Kepler's law,
satisfies the same law.

569
00:48:01 --> 00:48:03
OK, that's the end for today,
thanks.

570
00:48:03 --> 00:48:04