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So, if you remember last time,
we looked at parametric
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00:00:28 --> 00:00:34
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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00:00:54 --> 00:01:01
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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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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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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00:01:52 --> 00:01:58
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
26
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.
30
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
34
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,
36
00:03:53 --> 00:04:00
we found to be,
just to make sure that they
37
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
39
00:04:10 --> 00:04:13
last time, except we had theta
instead of t because we were
40
00:04:13 --> 00:04:16
using the angle.
But now I'm saying,
41
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
45
00:04:33 --> 00:04:37
function of time,
we can try to study how it
46
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
48
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
55
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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00:05:19 --> 00:05:26
about this.
And so, that's called the
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00:05:26 --> 00:05:32
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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00:05:42 --> 00:05:46
the derivative of a position
vector with respect to time.
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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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00:06:54 --> 00:07:02
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.
88
00:08:01 --> 00:08:05
So, at the time,at that
particular time,
89
00:08:05 --> 00:08:08
our point is actually not
moving.
90
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.
93
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,
95
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.
97
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.
102
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,
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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.
145
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.
148
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)?
156
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
.
159
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
164
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,
169
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.
175
00:14:30 --> 00:14:36
OK, so I should insist maybe on
one thing,
176
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,
178
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.
182
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