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So, if you remember last time,
we looked at parametric
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00:00:28,000 --> 00:00:34,000
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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00:00:41,000 --> 00:00:48,000
your favorite parameter that
will tell you how far the motion
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00:00:48,000 --> 00:00:54,000
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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00:01:01,000 --> 00:01:09,000
traced by a point on a wheel
that's rolling on a flat
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00:01:09,000 --> 00:01:14,000
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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00:01:19,000 --> 00:01:23,000
and we have this point on the
wheel.
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00:01:23,000 --> 00:01:31,000
And, as it moves around,
it traces a trajectory that
18
00:01:31,000 --> 00:01:36,000
moves more or less like this.
OK, so I'm trying a new color.
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00:01:36,000 --> 00:01:44,000
Is this visible from the back?
So, no more blue.
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00:01:44,000 --> 00:01:52,000
OK, so remember,
in general, we are trying to
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00:01:52,000 --> 00:01:58,000
find the position,
so, x of t, y of t,
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00:01:58,000 --> 00:02:09,000
maybe z of t if we are in space
-- -- of a moving point along a
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00:02:09,000 --> 00:02:17,000
trajectory.
And, one way to think about
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this is in terms of the position
vector.
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00:02:25,000 --> 00:02:32,000
So, position vector is just the
vector whose components are
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00:02:32,000 --> 00:02:37,000
coordinates of a point,
OK, so if you prefer,
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00:02:37,000 --> 00:02:43,000
that's the same thing as a
vector from the origin to the
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00:02:43,000 --> 00:02:50,000
moving point.
So, maybe our point is here, P.
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00:02:50,000 --> 00:03:02,000
So, this vector here -- This
vector here is vector OP.
30
00:03:02,000 --> 00:03:12,000
And, that's also the position
vector r of t.
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00:03:12,000 --> 00:03:24,000
So, just to give you,
again, that example -- -- if I
32
00:03:24,000 --> 00:03:34,000
take the cycloid for a wheel of
radius 1,
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00:03:34,000 --> 00:03:41,000
and let's say that we are going
at unit speed so that the angle
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00:03:41,000 --> 00:03:48,000
that we used as a parameter of
time is the same thing as time
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00:03:48,000 --> 00:03:53,000
when the position vector,
in this case,
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00:03:53,000 --> 00:04:00,000
we found to be,
just to make sure that they
37
00:04:00,000 --> 00:04:07,000
have it right,
.
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00:04:07,000 --> 00:04:10,000
OK, that's a formula that you
should have in your notes from
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00:04:10,000 --> 00:04:13,000
last time, except we had theta
instead of t because we were
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00:04:13,000 --> 00:04:16,000
using the angle.
But now I'm saying,
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00:04:16,000 --> 00:04:20,000
we are moving at unit speed,
so time and angle are the same
42
00:04:20,000 --> 00:04:24,000
thing.
So, now, what's interesting
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00:04:24,000 --> 00:04:31,000
about this is we can analyze the
motion in more detail.
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00:04:31,000 --> 00:04:33,000
OK, so, now that we know the
position of the point as a
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00:04:33,000 --> 00:04:37,000
function of time,
we can try to study how it
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00:04:37,000 --> 00:04:43,000
varies in particular things like
the speed and acceleration.
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00:04:43,000 --> 00:04:48,000
OK, so let's start with speed.
Well, in fact we can do better
48
00:04:48,000 --> 00:04:51,000
than speed.
Let's not start with speed.
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00:04:51,000 --> 00:04:54,000
So, speed is a number.
It tells you how fast you are
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00:04:54,000 --> 00:04:58,000
going along your trajectory.
I mean, if you're driving in a
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00:04:58,000 --> 00:05:01,000
car, then it tells you how fast
you are going.
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00:05:01,000 --> 00:05:03,000
But, unless you have one of
these fancy cars with a GPS,
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00:05:03,000 --> 00:05:05,000
it doesn't tell you which
direction you're going.
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00:05:05,000 --> 00:05:08,000
And, that's useful information,
too, if you're trying to figure
55
00:05:08,000 --> 00:05:10,000
out what your trajectory is.
So, in fact,
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00:05:10,000 --> 00:05:13,000
there's two aspects to it.
One is how fast you are going,
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and the other is in what
direction you're going.
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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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00:05:37,000 --> 00:05:42,000
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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00:05:46,000 --> 00:05:50,000
Now, it's our first time
writing this kind of thing with
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00:05:50,000 --> 00:05:52,000
a vector.
So, the basic rule is you can
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00:05:52,000 --> 00:05:57,000
take the derivative of a vector
quantity just by taking the
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00:05:57,000 --> 00:06:06,000
derivatives of each component.
OK, so that's just dx/dt,
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00:06:06,000 --> 00:06:17,000
dy/dt, and if you have z
component, dz/dt.
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00:06:17,000 --> 00:06:32,000
So, let me -- OK,
so -- OK, so let's see what it
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is for the cycloid.
So, an example of a cycloid,
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00:06:44,000 --> 00:06:54,000
well, so what do we get when we
take the derivatives of this
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00:06:54,000 --> 00:07:02,000
formula there?
Well, so, the derivative of t
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is 1- cos(t).
The derivative of 1 is 0.
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00:07:08,000 --> 00:07:12,000
The derivative of -cos(t) is
sin(t).
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00:07:12,000 --> 00:07:17,000
Very good.
OK, that's at least one thing
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00:07:17,000 --> 00:07:20,000
you should remember from single
variable calculus.
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Hopefully you remember even
more than that.
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00:07:24,000 --> 00:07:27,000
OK, so that's the velocity
vector.
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00:07:27,000 --> 00:07:31,000
It tells us at any time how
fast we are going,
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00:07:31,000 --> 00:07:37,000
and in what direction.
So, for example, observe.
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00:07:37,000 --> 00:07:40,000
Remember last time at the end
of class we were trying to
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00:07:40,000 --> 00:07:43,000
figure out what exactly happens
near the bottom point,
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00:07:43,000 --> 00:07:47,000
when we have this motion that
seems to stop and go backwards.
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00:07:47,000 --> 00:07:50,000
And, we answered that one way.
But, let's try to understand it
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00:07:50,000 --> 00:07:54,000
in terms of velocity.
What if I plug t equals 0 in
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00:07:54,000 --> 00:07:57,000
here?
Then, 1- cos(t) is 0,
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00:07:57,000 --> 00:08:01,000
sin(t) is 0.
The velocity is 0.
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00:08:01,000 --> 00:08:05,000
So, at the time,at that
particular time,
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00:08:05,000 --> 00:08:08,000
our point is actually not
moving.
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00:08:08,000 --> 00:08:11,000
Of course, it's been moving
just before, and it starts
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00:08:11,000 --> 00:08:14,000
moving just afterwards.
It's just the instant,
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00:08:14,000 --> 00:08:20,000
at that particular instant,
the speed is zero.
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So, that's especially maybe a
counterintuitive thing,
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but something is moving.
And at that time,
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it's actually stopped.
Now, let's see,
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00:08:33,000 --> 00:08:36,000
so that's the vector.
And, it's useful.
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00:08:36,000 --> 00:08:39,000
But, if you want just the usual
speed as a number,
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00:08:39,000 --> 00:08:43,000
then, what will you do?
Well, you will just take
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00:08:43,000 --> 00:08:46,000
exactly the magnitude of this
vector.
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So, speed, which is the scalar
quantity is going to be just the
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magnitude of the vector,
V.
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00:09:01,000 --> 00:09:09,000
OK, so, in this case,
while it would be square root
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00:09:09,000 --> 00:09:18,000
of (1- cost)^2 sin^2(t),
and if you expand that,
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00:09:18,000 --> 00:09:23,000
you will get,
let me take a bit more space,
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it's going to be square root of
1 - 2cos(t) cos^2(t) sin^2(t).
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It seems to simplify a little
bit because we have cos^2 plus
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sin^2.
That's 1.
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So, it's going to be the square
root of 2 - 2cos(t).
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00:09:49,000 --> 00:09:52,000
So, at this point,
if I was going to ask you,
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00:09:52,000 --> 00:09:55,000
when is the speed the smallest
or the largest?
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00:09:55,000 --> 00:09:59,000
You could answer based on that.
See, at t equals 0,
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00:09:59,000 --> 00:10:01,000
well, that turns out to be
zero.
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00:10:01,000 --> 00:10:04,000
The point is not moving.
At t equals pi,
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00:10:04,000 --> 00:10:07,000
that ends up being the square
root of 2 plus 2,
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00:10:07,000 --> 00:10:09,000
which is 4.
So, that's 2.
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00:10:09,000 --> 00:10:12,000
And, that's when you're truly
at the top of the arch,
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00:10:12,000 --> 00:10:15,000
and that's when the point is
moving the fastest.
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00:10:15,000 --> 00:10:18,000
In fact, they are spending
twice as fast as the wheel
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00:10:18,000 --> 00:10:20,000
because the wheel is moving to
the right at unit speed,
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00:10:20,000 --> 00:10:24,000
and the wheel is also rotating.
So, it's moving to the right
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00:10:24,000 --> 00:10:29,000
and unit speed relative to the
center so that the two effects
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00:10:29,000 --> 00:10:32,000
add up, and give you a speed of
2.
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00:10:32,000 --> 00:10:36,000
Anyway, that's a formula we can
get.
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00:10:36,000 --> 00:10:48,000
OK, now, what about
acceleration?
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00:10:48,000 --> 00:10:53,000
So, here I should warn you that
there is a serious discrepancy
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00:10:53,000 --> 00:10:58,000
between the usual intuitive
notion of acceleration,
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00:10:58,000 --> 00:11:02,000
the one that you are aware of
when you drive a car and the one
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00:11:02,000 --> 00:11:05,000
that we will be using.
So, you might think
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00:11:05,000 --> 00:11:08,000
acceleration is just the
directive of speed.
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00:11:08,000 --> 00:11:13,000
If my car goes 55 miles an hour
on the highway and it's going a
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00:11:13,000 --> 00:11:15,000
constant speed,
it's not accelerating.
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00:11:15,000 --> 00:11:18,000
But, let's say that I'm taking
a really tight turn.
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00:11:18,000 --> 00:11:19,000
Then, I'm going to feel
something.
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00:11:19,000 --> 00:11:21,000
There is some force being
exerted.
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00:11:21,000 --> 00:11:24,000
And, in fact,
there is a sideways
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00:11:24,000 --> 00:11:28,000
acceleration at that point even
though the speed is not
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00:11:28,000 --> 00:11:30,000
changing.
So, the definition will take
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00:11:30,000 --> 00:11:34,000
effect.
The acceleration is,
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00:11:34,000 --> 00:11:40,000
as a vector,
and the acceleration vector is
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00:11:40,000 --> 00:11:47,000
just the derivative of a
velocity vector.
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00:11:47,000 --> 00:11:51,000
So, even if the speed is
constant, that means,
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00:11:51,000 --> 00:11:55,000
even if a length of the
velocity vector stays the same,
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00:11:55,000 --> 00:11:59,000
the velocity vector can still
rotate.
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00:11:59,000 --> 00:12:03,000
And, as it rotates,
it uses acceleration.
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00:12:03,000 --> 00:12:07,000
OK, and so this is the notion
of acceleration that's relevant
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00:12:07,000 --> 00:12:13,000
to physics when you find F=ma;
that's the (a) that you have in
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00:12:13,000 --> 00:12:17,000
mind here.
It's a vector.
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00:12:17,000 --> 00:12:19,000
Of course, if you are moving in
a straight line,
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00:12:19,000 --> 00:12:20,000
then the two notions are the
same.
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00:12:20,000 --> 00:12:23,000
I mean, acceleration is also
going to be along the line,
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00:12:23,000 --> 00:12:25,000
and it's going to has to do
with the derivative of speed.
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00:12:25,000 --> 00:12:30,000
But, in general,
that's not quite the same.
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00:12:30,000 --> 00:12:37,000
So, for example,
let's look at the cycloid.
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00:12:37,000 --> 00:12:40,000
If we take the example of the
cycloid, well,
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00:12:40,000 --> 00:12:44,000
what's the derivative of one
minus cos(t)?
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00:12:44,000 --> 00:12:52,000
It's sin(t).
And, what's the derivative of
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00:12:52,000 --> 00:12:55,000
sin(t)?
cos(t), OK.
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00:12:55,000 --> 00:13:04,000
So, the acceleration vector is
.
159
00:13:04,000 --> 00:13:09,000
So, in particular,
let's look at what happens at
160
00:13:09,000 --> 00:13:13,000
time t equals zero when the
point is not moving.
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00:13:13,000 --> 00:13:20,000
Well, the acceleration vector
there will be zero from one.
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00:13:20,000 --> 00:13:28,000
So, what that means is that if
I look at my trajectory at this
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00:13:28,000 --> 00:13:35,000
point, that the acceleration
vector is pointing in that
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00:13:35,000 --> 00:13:39,000
direction.
It's the unit vector in the
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00:13:39,000 --> 00:13:43,000
vertical direction.
So, my point is not moving at
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00:13:43,000 --> 00:13:46,000
that particular time.
But, it's accelerating up.
167
00:13:46,000 --> 00:13:49,000
So, that means that actually as
it comes down,
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00:13:49,000 --> 00:13:53,000
first it's slowing down.
Then it stops here,
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00:13:53,000 --> 00:13:56,000
and then it reverses going back
up.
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00:13:56,000 --> 00:14:01,000
OK, so that's another way to
understand what we were saying
171
00:14:01,000 --> 00:14:06,000
last time that the trajectory at
that point has a vertical
172
00:14:06,000 --> 00:14:11,000
tendency because that's the
direction in which the motion is
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00:14:11,000 --> 00:14:16,000
going to occur just before and
just after time zero.
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00:14:16,000 --> 00:14:30,000
OK, any questions about that?
No.
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00:14:30,000 --> 00:14:36,000
OK, so I should insist maybe on
one thing,
176
00:14:36,000 --> 00:14:41,000
which is that,
so, we can differentiate
177
00:14:41,000 --> 00:14:46,000
vectors just component by
component,
178
00:14:46,000 --> 00:14:50,000
OK, and we can differentiate
vector expressions according to
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00:14:50,000 --> 00:14:54,000
certain rules that we'll see in
a moment.
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00:14:54,000 --> 00:15:02,000
One thing that we cannot do,
it's not true that the length
181
00:15:02,000 --> 00:15:12,000
of dr dt, which is the speed,
is equal to the length of dt.
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00:15:12,000 --> 00:15:18,000
OK, this is completely false.
And, they are really not the
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00:15:18,000 --> 00:15:19,000
same.
So, if you have to
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00:15:19,000 --> 00:15:24,000
differentiate the length of a
vector, but basically you are in
185
00:15:24,000 --> 00:15:25,000
trouble.
If you really,
186
00:15:25,000 --> 00:15:27,000
really want to do it,
well, the length of the vector
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00:15:27,000 --> 00:15:30,000
is the square root of the sums
of the squares of the
188
00:15:30,000 --> 00:15:32,000
components,
and from that you can use the
189
00:15:32,000 --> 00:15:34,000
formula for the derivative of
the square root,
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00:15:34,000 --> 00:15:36,000
and the chain rule,
and various other things.
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00:15:36,000 --> 00:15:39,000
And, you can get there.
But, it will not be a very nice
192
00:15:39,000 --> 00:15:42,000
expression.
There is no simple formula for
193
00:15:42,000 --> 00:15:44,000
this kind of thing.
Fortunately,
194
00:15:44,000 --> 00:15:48,000
we almost never have to compute
this kind of thing because,
195
00:15:48,000 --> 00:15:51,000
after all, it's not a very
relevant quantity.
196
00:15:51,000 --> 00:15:53,000
What's more relevant might be
this one.
197
00:15:53,000 --> 00:15:59,000
This is actually the speed.
This one, I don't know what it
198
00:15:59,000 --> 00:16:10,000
means.
OK.
199
00:16:10,000 --> 00:16:14,000
So, let's continue our
exploration.
200
00:16:14,000 --> 00:16:20,000
So, the next concept that I
want to define is that of arc
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00:16:20,000 --> 00:16:23,000
length.
So, arc length is just the
202
00:16:23,000 --> 00:16:26,000
distance that you have traveled
along the curve,
203
00:16:26,000 --> 00:16:27,000
OK?
So, if you are in a car,
204
00:16:27,000 --> 00:16:30,000
you know, it has mileage
counter that tells you how far
205
00:16:30,000 --> 00:16:33,000
you've gone, how much fuel
you've used if it's a fancy car.
206
00:16:33,000 --> 00:16:37,000
And, what it does is it
actually integrates the speed of
207
00:16:37,000 --> 00:16:41,000
the time to give you the arc
length along the trajectory of
208
00:16:41,000 --> 00:16:45,000
the car.
So, the usual notation that we
209
00:16:45,000 --> 00:16:51,000
will have is (s) for arc length.
I'm not quite sure how you get
210
00:16:51,000 --> 00:16:57,000
an (s) out of this,
but it's the usual notation.
211
00:16:57,000 --> 00:17:14,000
OK, so, (s) is for distance
traveled along the trajectory.
212
00:17:14,000 --> 00:17:16,000
And, so that makes sense,
of course, we need to fix a
213
00:17:16,000 --> 00:17:19,000
reference point.
Maybe on the cycloid,
214
00:17:19,000 --> 00:17:22,000
we'd say it's a distance
starting on the origin.
215
00:17:22,000 --> 00:17:25,000
In general, maybe you would say
you start at time,
216
00:17:25,000 --> 00:17:28,000
t equals zero.
But, it's a convention.
217
00:17:28,000 --> 00:17:31,000
If you knew in advance,
you could have,
218
00:17:31,000 --> 00:17:35,000
actually, your car's mileage
counter to count backwards from
219
00:17:35,000 --> 00:17:38,000
the point where the car will die
and start walking.
220
00:17:38,000 --> 00:17:41,000
I mean, that would be
sneaky-freaky,
221
00:17:41,000 --> 00:17:45,000
but you could have a negative
arc length that gets closer and
222
00:17:45,000 --> 00:17:48,000
closer to zero,
and gets to zero at the end of
223
00:17:48,000 --> 00:17:51,000
a trajectory,
or anything you want.
224
00:17:51,000 --> 00:17:53,000
I mean, arc length could be
positive or negative.
225
00:17:53,000 --> 00:17:56,000
Typically it's negative what
you are before the reference
226
00:17:56,000 --> 00:18:01,000
point, and positive afterwards.
So, now, how does it relate to
227
00:18:01,000 --> 00:18:08,000
the things we've seen there?
Well, so in particular,
228
00:18:08,000 --> 00:18:16,000
how do you relate arc length
and time?
229
00:18:16,000 --> 00:18:22,000
Well, so, there's a simple
relation, which is that the rate
230
00:18:22,000 --> 00:18:26,000
of change of arc length versus
time,
231
00:18:26,000 --> 00:18:30,000
well, that's going to be the
speed at which you are moving,
232
00:18:30,000 --> 00:18:38,000
OK, because the speed as a
scalar quantity tells you how
233
00:18:38,000 --> 00:18:44,000
much distance you're covering
per unit time.
234
00:18:44,000 --> 00:18:47,000
OK, and in fact,
to be completely honest,
235
00:18:47,000 --> 00:18:51,000
I should put an absolute value
here because there is examples
236
00:18:51,000 --> 00:18:55,000
of curves maybe where your
motion is going back and forth
237
00:18:55,000 --> 00:18:59,000
along the same curve.
And then, you don't want to
238
00:18:59,000 --> 00:19:01,000
keep counting arc length all the
time.
239
00:19:01,000 --> 00:19:04,000
Actually, maybe you want to say
that the arc length increases
240
00:19:04,000 --> 00:19:05,000
and then decreases along the
curve.
241
00:19:05,000 --> 00:19:08,000
I mean, you get to choose how
you count it.
242
00:19:08,000 --> 00:19:10,000
But, in this case,
if you are moving back and
243
00:19:10,000 --> 00:19:12,000
forth, it would make more sense
to have the arc length first
244
00:19:12,000 --> 00:19:18,000
increase,
then decrease,
245
00:19:18,000 --> 00:19:26,000
increase again,
and so on.
246
00:19:26,000 --> 00:19:34,000
So -- So if you want to know
really what the arc length is,
247
00:19:34,000 --> 00:19:41,000
then basically the only way to
do it is to integrate speed
248
00:19:41,000 --> 00:19:45,000
versus time.
So, if you wanted to know how
249
00:19:45,000 --> 00:19:49,000
long an arch of cycloid is,
you have this nice-looking
250
00:19:49,000 --> 00:19:51,000
curve;
how long is it?
251
00:19:51,000 --> 00:19:55,000
Well, you'd have to basically
integrate this quantity from t
252
00:19:55,000 --> 00:19:57,000
equals zero to 2 pi.
253
00:20:24,000 --> 00:20:28,000
And, to say the truth,
I don't really know how to
254
00:20:28,000 --> 00:20:31,000
integrate that.
So, we don't actually have a
255
00:20:31,000 --> 00:20:34,000
formula for the length at this
point.
256
00:20:34,000 --> 00:20:41,000
However, we'll see one later
using a cool trick,
257
00:20:41,000 --> 00:20:47,000
and multi-variable calculus.
So, for now,
258
00:20:47,000 --> 00:20:52,000
we'll just leave the formula
like that, and we don't know how
259
00:20:52,000 --> 00:20:55,000
long it is.
Well, you can put that into
260
00:20:55,000 --> 00:20:57,000
your calculator and get the
numerical value.
261
00:20:57,000 --> 00:21:07,000
But, that's the best I can
offer.
262
00:21:07,000 --> 00:21:18,000
Now, another useful notion is
the unit vector to the
263
00:21:18,000 --> 00:21:25,000
trajectory.
So, the usual notation is T hat.
264
00:21:25,000 --> 00:21:28,000
It has a hat because it's a
unit vector, and T because it's
265
00:21:28,000 --> 00:21:32,000
tangent.
Now, how do we get this unit
266
00:21:32,000 --> 00:21:36,000
vector?
So, maybe I should have pointed
267
00:21:36,000 --> 00:21:40,000
out before that if you're moving
along some trajectory,
268
00:21:40,000 --> 00:21:43,000
say you're going in that
direction, then when you're at
269
00:21:43,000 --> 00:21:47,000
this point,
the velocity vector is going to
270
00:21:47,000 --> 00:21:53,000
be tangential to the trajectory.
It tells you the direction of
271
00:21:53,000 --> 00:21:57,000
motion in particular.
So, if you want a unit vector
272
00:21:57,000 --> 00:22:02,000
that goes in the same direction,
all you have to do is rescale
273
00:22:02,000 --> 00:22:05,000
it, so, at its length becomes
one.
274
00:22:05,000 --> 00:22:10,000
So, it's v divided by a
magnitude of v.
275
00:22:28,000 --> 00:22:33,000
So, it seems like now we have a
lot of different things that
276
00:22:33,000 --> 00:22:40,000
should be related in some way.
So, let's see what we can say.
277
00:22:40,000 --> 00:22:50,000
Well, we can say that dr by dt,
so, that's the velocity vector,
278
00:22:50,000 --> 00:22:59,000
that's the same thing as if I
use the chain rule dr/ds times
279
00:22:59,000 --> 00:23:06,000
ds/dt.
OK, so, let's think about this
280
00:23:06,000 --> 00:23:11,000
things.
So, this guy here we've just
281
00:23:11,000 --> 00:23:17,000
seen.
That's the same as the speed,
282
00:23:17,000 --> 00:23:21,000
OK?
So, this one here should be v
283
00:23:21,000 --> 00:23:28,000
divided by its length.
So, that means this actually
284
00:23:28,000 --> 00:23:34,000
should be the unit vector.
OK, so, let me rewrite that.
285
00:23:34,000 --> 00:23:40,000
It's T ds/dt.
So, maybe if I actually stated
286
00:23:40,000 --> 00:23:43,000
directly that way,
see, I'm just saying the
287
00:23:43,000 --> 00:23:46,000
velocity vector has a length and
a direction.
288
00:23:46,000 --> 00:23:51,000
The length is the speed.
The direction is tangent to the
289
00:23:51,000 --> 00:23:51,000
trajectory.
290
00:24:19,000 --> 00:24:25,000
So, the speed is ds/dt,
and the vector is T hat.
291
00:24:25,000 --> 00:24:33,000
And, that's how we get this.
So, let's try just to see why
292
00:24:33,000 --> 00:24:37,000
dr/ds should be T.
Well, let's think of dr/ds.
293
00:24:37,000 --> 00:24:42,000
dr/ds means position vector r
means you have the origin,
294
00:24:42,000 --> 00:24:47,000
which is somewhere out there,
and the vector r is here.
295
00:24:47,000 --> 00:24:51,000
So, dr/ds means we move by a
small amount,
296
00:24:51,000 --> 00:24:56,000
delta s along the trajectory a
certain distance delta s.
297
00:24:56,000 --> 00:25:00,000
And, we look at how the
position vector changes.
298
00:25:00,000 --> 00:25:08,000
Well, we'll have a small change.
Let me call that vector delta r
299
00:25:08,000 --> 00:25:13,000
corresponding to the size,
corresponding to the length
300
00:25:13,000 --> 00:25:17,000
delta s.
And now, delta r should be
301
00:25:17,000 --> 00:25:25,000
essentially roughly equal to,
well, its direction will be
302
00:25:25,000 --> 00:25:30,000
tangent to the trajectory.
If I take a small enough
303
00:25:30,000 --> 00:25:33,000
interval,
then the direction will be
304
00:25:33,000 --> 00:25:37,000
almost tensioned to the
trajectory times the length of
305
00:25:37,000 --> 00:25:41,000
it will be delta s,
the distance that I have
306
00:25:41,000 --> 00:25:45,000
traveled.
OK, sorry, maybe I should
307
00:25:45,000 --> 00:25:50,000
explain that on a separate
board.
308
00:25:50,000 --> 00:25:56,000
OK, so, let's say that we have
that amount of time,
309
00:25:56,000 --> 00:26:00,000
delta t.
So, let's zoom into that curve.
310
00:26:00,000 --> 00:26:12,000
So, we have r at time t.
We have r at time t plus delta
311
00:26:12,000 --> 00:26:17,000
t.
This vector here I will call
312
00:26:17,000 --> 00:26:23,000
delta r.
The length of this vector is
313
00:26:23,000 --> 00:26:28,000
delta s.
And, the direction is
314
00:26:28,000 --> 00:26:36,000
essentially that of a vector.
OK, so, delta s over delta t,
315
00:26:36,000 --> 00:26:43,000
that's the distance traveled
divided by the time.
316
00:26:43,000 --> 00:26:46,000
That's going to be close to the
speed.
317
00:26:46,000 --> 00:26:57,000
And, delta r is approximately T
times delta s.
318
00:26:57,000 --> 00:27:04,000
So, now if I divide both sides
by delta t, I get this.
319
00:27:04,000 --> 00:27:07,000
And, if I take the limit as
delta t turns to zero,
320
00:27:07,000 --> 00:27:10,000
then I get the same formula
with the derivatives and with an
321
00:27:10,000 --> 00:27:13,000
equality.
It's an approximation.
322
00:27:13,000 --> 00:27:15,000
The approximation becomes
better and better if I go to
323
00:27:15,000 --> 00:27:16,000
smaller intervals.
324
00:27:38,000 --> 00:27:44,000
OK, are there any questions
about this?
325
00:27:44,000 --> 00:27:59,000
Yes?
Yes, that's correct.
326
00:27:59,000 --> 00:28:01,000
OK, so let's be more careful,
actually.
327
00:28:01,000 --> 00:28:12,000
So, you're asking about whether
the delta r is actually strictly
328
00:28:12,000 --> 00:28:16,000
tangent to the curve.
Is that -- That's correct.
329
00:28:16,000 --> 00:28:20,000
Actually, delta r is not
strictly tangent to anything.
330
00:28:20,000 --> 00:28:23,000
So, maybe I should draw another
picture.
331
00:28:23,000 --> 00:28:29,000
If I'm going from here to here,
then delta r is going to be
332
00:28:29,000 --> 00:28:36,000
this arc inside the curve while
the vector will be going in this
333
00:28:36,000 --> 00:28:39,000
direction, OK?
So, they are not strictly
334
00:28:39,000 --> 00:28:41,000
parallel to each other.
That's why it's only
335
00:28:41,000 --> 00:28:44,000
approximately equal.
Similarly, this distance,
336
00:28:44,000 --> 00:28:48,000
the length of delta r is not
exactly the length along the
337
00:28:48,000 --> 00:28:50,000
curve.
It's actually a bit shorter.
338
00:28:50,000 --> 00:28:53,000
But, if we imagine a smaller
and smaller portion of the
339
00:28:53,000 --> 00:28:56,000
curve,
then this effect of the curve
340
00:28:56,000 --> 00:29:00,000
being a curve and not a straight
line becomes more and more
341
00:29:00,000 --> 00:29:02,000
negligible.
If you zoom into the curve
342
00:29:02,000 --> 00:29:04,000
sufficiently,
then it looks more and more
343
00:29:04,000 --> 00:29:07,000
like a straight line.
And then, what I said becomes
344
00:29:07,000 --> 00:29:18,000
true in the limit.
OK? Any other questions?
345
00:29:18,000 --> 00:29:35,000
No? OK.
So, what happens next?
346
00:29:35,000 --> 00:29:39,000
OK, so let me show you a nice
example of why we might want to
347
00:29:39,000 --> 00:29:43,000
use vectors to study parametric
curves because,
348
00:29:43,000 --> 00:29:46,000
after all, a lot of what's here
you can just do in coordinates.
349
00:29:46,000 --> 00:29:48,000
And, we don't really need
vectors.
350
00:29:48,000 --> 00:29:51,000
Well, and truly,
vectors being a language,
351
00:29:51,000 --> 00:29:54,000
you never strictly need it,
but it's useful to have a
352
00:29:54,000 --> 00:30:02,000
notion of vectors.
So, I want to tell you a bit
353
00:30:02,000 --> 00:30:14,000
about Kepler's second law of
celestial mechanics.
354
00:30:14,000 --> 00:30:20,000
So, that goes back to 1609.
So, that's not exactly recent
355
00:30:20,000 --> 00:30:24,000
news, OK?
But, still I think it's a very
356
00:30:24,000 --> 00:30:29,000
interesting example of why you
might want to use vector methods
357
00:30:29,000 --> 00:30:33,000
to analyze motions.
So, what happened back then was
358
00:30:33,000 --> 00:30:39,000
Kepler was trying to observe the
motion of planets in the sky,
359
00:30:39,000 --> 00:30:42,000
and trying to come up with
general explanations of how they
360
00:30:42,000 --> 00:30:44,000
move.
Before him, people were saying,
361
00:30:44,000 --> 00:30:46,000
well, they cannot move in a
circle.
362
00:30:46,000 --> 00:30:48,000
But maybe it's more complicated
than that.
363
00:30:48,000 --> 00:30:51,000
We need to add smaller circular
motions on top of each other,
364
00:30:51,000 --> 00:30:53,000
and so on.
They have more and more
365
00:30:53,000 --> 00:30:56,000
complicated theories.
And then Kepler came with these
366
00:30:56,000 --> 00:31:00,000
laws that said basically that
planets move in an ellipse
367
00:31:00,000 --> 00:31:03,000
around the sun,
and that they move in a very
368
00:31:03,000 --> 00:31:07,000
specific way along that ellipse.
So, there's actually three
369
00:31:07,000 --> 00:31:11,000
laws, but let me just tell you
about the second one that has a
370
00:31:11,000 --> 00:31:17,000
very nice vector interpretation.
So, what Kepler's second law
371
00:31:17,000 --> 00:31:24,000
says is that the motion of
planets is, first of all,
372
00:31:24,000 --> 00:31:36,000
they move in a plane.
And second, the area swept out
373
00:31:36,000 --> 00:31:51,000
by the line from the sun to the
planet is swept at constant
374
00:31:51,000 --> 00:31:57,000
time.
Sorry, is swept at constant
375
00:31:57,000 --> 00:32:04,000
rate.
From the sun to the planet,
376
00:32:04,000 --> 00:32:14,000
it is swept out by the line at
a constant rate.
377
00:32:14,000 --> 00:32:23,000
OK, so that's an interesting
law because it tells you,
378
00:32:23,000 --> 00:32:27,000
once you know what the orbit of
the planet looks like,
379
00:32:27,000 --> 00:32:30,000
it tells you how fast it's
going to move on that orbit.
380
00:33:09,000 --> 00:33:19,000
OK, so let me explain again.
So, this law says maybe the
381
00:33:19,000 --> 00:33:27,000
sun, let's put the sun here at
the origin, and let's have a
382
00:33:27,000 --> 00:33:34,000
planet.
Well, the planet orbits around
383
00:33:34,000 --> 00:33:41,000
the sun -- -- in some
trajectory.
384
00:33:41,000 --> 00:33:45,000
So, this is supposed to be
light blue.
385
00:33:45,000 --> 00:33:49,000
Can you see that it's different
from white?
386
00:33:49,000 --> 00:33:51,000
No?
OK, me neither.
387
00:33:51,000 --> 00:33:53,000
[LAUGHTER]
OK, it doesn't really matter.
388
00:33:53,000 --> 00:33:55,000
So, the planet moves on its
orbit.
389
00:33:55,000 --> 00:34:00,000
And, if you wait for a certain
time, then a bit later it would
390
00:34:00,000 --> 00:34:04,000
be here, and then here,
and so on.
391
00:34:04,000 --> 00:34:09,000
Then, you can look at the
amount of area inside this
392
00:34:09,000 --> 00:34:12,000
triangular wedge.
And, the claim is that the
393
00:34:12,000 --> 00:34:16,000
amount of area in here is
proportional to the time
394
00:34:16,000 --> 00:34:18,000
elapsed.
So, in particular,
395
00:34:18,000 --> 00:34:21,000
if a planet is closer to the
sun, then it has to go faster.
396
00:34:21,000 --> 00:34:25,000
And, if it's farther away from
the sun, then it has to go
397
00:34:25,000 --> 00:34:28,000
slower so that the area remains
proportional to time.
398
00:34:28,000 --> 00:34:32,000
So, it's a very sophisticated
prediction.
399
00:34:32,000 --> 00:34:36,000
And, I think the way he came to
it was really just by using a
400
00:34:36,000 --> 00:34:39,000
lot of observations,
and trying to measure what was
401
00:34:39,000 --> 00:34:44,000
true that wasn't true.
But, let's try to see how we
402
00:34:44,000 --> 00:34:49,000
can understand that in terms of
all we know today about
403
00:34:49,000 --> 00:34:52,000
mechanics.
So, in fact,
404
00:34:52,000 --> 00:34:56,000
what happens is that Newton,
so Newton was quite a bit
405
00:34:56,000 --> 00:35:04,000
later.
That was the late 17th century
406
00:35:04,000 --> 00:35:13,000
instead of the beginning of the
17th century.
407
00:35:13,000 --> 00:35:30,000
So, he was able to explain this
using his laws for gravitational
408
00:35:30,000 --> 00:35:36,000
attraction.
And, you'll see that if we
409
00:35:36,000 --> 00:35:41,000
reformulate Kepler's Law in
terms of vectors,
410
00:35:41,000 --> 00:35:43,000
and if we work a bit with these
vectors,
411
00:35:43,000 --> 00:35:46,000
we are going to end up with
something that's actually
412
00:35:46,000 --> 00:35:49,000
completely obvious to us now.
At the time,
413
00:35:49,000 --> 00:35:52,000
it was very far from obvious,
but to us now to completely
414
00:35:52,000 --> 00:35:59,000
obvious.
So, let's try to see,
415
00:35:59,000 --> 00:36:15,000
what does Kepler's law say in
terms of vectors?
416
00:36:15,000 --> 00:36:24,000
OK, so, let's think of what
kinds of vectors we might want
417
00:36:24,000 --> 00:36:31,000
to have in here.
Well, it might be good to think
418
00:36:31,000 --> 00:36:38,000
of, maybe, the position vector,
and maybe its variation.
419
00:36:38,000 --> 00:36:46,000
So, if we wait a certain amount
of time, we'll have a vector,
420
00:36:46,000 --> 00:36:53,000
delta r, which is the change in
position vector a various
421
00:36:53,000 --> 00:36:59,000
interval of time.
OK, so let's start with the
422
00:36:59,000 --> 00:37:02,000
first step.
What's the most complicated
423
00:37:02,000 --> 00:37:05,000
thing in here?
It's this area swept out by the
424
00:37:05,000 --> 00:37:08,000
line.
How do we express that area in
425
00:37:08,000 --> 00:37:12,000
terms of vectors?
Well, I've almost given the
426
00:37:12,000 --> 00:37:14,000
answer by drawing this picture,
right?
427
00:37:14,000 --> 00:37:18,000
If I take a sufficiently small
amount of time,
428
00:37:18,000 --> 00:37:22,000
this shaded part looks like a
triangle.
429
00:37:22,000 --> 00:37:25,000
So, we have to find the area of
the triangle.
430
00:37:25,000 --> 00:37:27,000
Well, we know how to do that
now.
431
00:37:27,000 --> 00:37:34,000
So, the area is approximately
equal to one half of the area of
432
00:37:34,000 --> 00:37:40,000
a parallelogram that I could
form from these vectors.
433
00:37:40,000 --> 00:37:46,000
And, the area of a
parallelogram is given by the
434
00:37:46,000 --> 00:37:52,000
magnitude of a cross product.
OK, so, I should say,
435
00:37:52,000 --> 00:37:56,000
this is the area swept in time
delta t.
436
00:37:56,000 --> 00:38:00,000
You should think of delta t as
relatively small.
437
00:38:00,000 --> 00:38:05,000
I mean, the scale of a planet
that might still be a few days,
438
00:38:05,000 --> 00:38:09,000
but small compared to the other
old trajectory.
439
00:38:09,000 --> 00:38:16,000
So, let's remember that the
amount by which we moved,
440
00:38:16,000 --> 00:38:20,000
delta r,
is approximately equal to v
441
00:38:20,000 --> 00:38:25,000
times delta t,
OK, and just using the
442
00:38:25,000 --> 00:38:36,000
definition of a velocity vector.
So, let's use that.
443
00:38:36,000 --> 00:38:43,000
Sorry, so it's approximately
equal to r cross v magnitude
444
00:38:43,000 --> 00:38:48,000
times delta t.
I can take out the delta t,
445
00:38:48,000 --> 00:38:52,000
which is scalar.
So, now, what does it mean to
446
00:38:52,000 --> 00:38:55,000
say that area is swept at a
constant rate?
447
00:38:55,000 --> 00:39:00,000
It means this thing is
proportional to delta t.
448
00:39:00,000 --> 00:39:05,000
So, that means,
so, the law says,
449
00:39:05,000 --> 00:39:15,000
in fact, that the length of
this cross product r cross v
450
00:39:15,000 --> 00:39:25,000
equals a constant.
OK, r cross v has constant
451
00:39:25,000 --> 00:39:31,000
length.
Any questions about that?
452
00:39:31,000 --> 00:39:37,000
No? Yes?
Yes, let me try to explain that
453
00:39:37,000 --> 00:39:40,000
again.
So, what I'm claiming is that
454
00:39:40,000 --> 00:39:46,000
the length of the cross products
r cross v measures the rate at
455
00:39:46,000 --> 00:39:50,000
which area is swept by the
position vector.
456
00:39:50,000 --> 00:39:52,000
I should say,
with a vector of one half of
457
00:39:52,000 --> 00:39:55,000
this length is the rate at which
area is swept.
458
00:39:55,000 --> 00:39:58,000
How do we see that?
Well, let's take a small time
459
00:39:58,000 --> 00:40:01,000
interval, delta t.
In time, delta t,
460
00:40:01,000 --> 00:40:05,000
our planet moves by v delta t,
OK?
461
00:40:05,000 --> 00:40:08,000
So, if it moves by v delta t,
it means that this triangle up
462
00:40:08,000 --> 00:40:12,000
there has two sides.
One is the position vector,
463
00:40:12,000 --> 00:40:14,000
r.
The other one is v delta t.
464
00:40:14,000 --> 00:40:18,000
So, its area is given by one
half of the magnitude of a cross
465
00:40:18,000 --> 00:40:21,000
product.
That's the formula we've seen
466
00:40:21,000 --> 00:40:24,000
for the area of a triangle in
space.
467
00:40:24,000 --> 00:40:28,000
So, the area is one half of the
cross product,
468
00:40:28,000 --> 00:40:33,000
r, and v delta t,
magnitude of the cross product.
469
00:40:33,000 --> 00:40:37,000
So, to say that the rate at
which area is swept is constant
470
00:40:37,000 --> 00:40:39,000
means that these two are
proportional.
471
00:40:39,000 --> 00:40:42,000
Area divided by delta t is
constant at our time.
472
00:40:42,000 --> 00:40:51,000
And so, this is constant.
OK, now, what about the other
473
00:40:51,000 --> 00:40:58,000
half of the law?
Well, it says that the motion
474
00:40:58,000 --> 00:41:04,000
is in a plane,
and so we have a plane in which
475
00:41:04,000 --> 00:41:09,000
the motion takes place.
And, it contains,
476
00:41:09,000 --> 00:41:12,000
also, the sun.
And, it contains the
477
00:41:12,000 --> 00:41:16,000
trajectory.
So, let's think about that
478
00:41:16,000 --> 00:41:20,000
plane.
Well, I claim that the position
479
00:41:20,000 --> 00:41:25,000
vector is in the plane.
OK, that's what we are saying.
480
00:41:25,000 --> 00:41:28,000
But, there is another vector
that I know it is in the plane.
481
00:41:28,000 --> 00:41:32,000
You could say the position
vector at another time,
482
00:41:32,000 --> 00:41:34,000
or at any time,
but in fact,
483
00:41:34,000 --> 00:41:40,000
what's also true is that the
velocity vector is in the plane.
484
00:41:40,000 --> 00:41:44,000
OK, if I'm moving in the plane,
then position and velocity are
485
00:41:44,000 --> 00:41:50,000
in there.
So, the plane of motion
486
00:41:50,000 --> 00:41:59,000
contains r and v.
So, what's the direction of the
487
00:41:59,000 --> 00:42:08,000
cross product r cross v?
Well, it's the direction that's
488
00:42:08,000 --> 00:42:19,000
perpendicular to this plane.
So, it's normal to the plane of
489
00:42:19,000 --> 00:42:24,000
motion.
And, that means, now,
490
00:42:24,000 --> 00:42:28,000
that actually we've put the two
statements in there into a
491
00:42:28,000 --> 00:42:33,000
single form because we are
saying r cross v has constant
492
00:42:33,000 --> 00:42:37,000
length and constant direction.
In fact, in general,
493
00:42:37,000 --> 00:42:40,000
maybe I should say something
about this.
494
00:42:40,000 --> 00:42:42,000
So, if you just look at the
position vector,
495
00:42:42,000 --> 00:42:45,000
and the velocity vector for any
motion at any given time,
496
00:42:45,000 --> 00:42:48,000
then together,
they determine some plane.
497
00:42:48,000 --> 00:42:51,000
And, that's the plane that
contains the origin,
498
00:42:51,000 --> 00:42:54,000
the point, and the velocity
vector.
499
00:42:54,000 --> 00:42:56,000
If you want,
it's the plane in which the
500
00:42:56,000 --> 00:42:59,000
motion seems to be going at the
given time.
501
00:42:59,000 --> 00:43:01,000
Now, of course,
if your motion is not in a
502
00:43:01,000 --> 00:43:03,000
plane, then that plane will
change.
503
00:43:03,000 --> 00:43:06,000
It's, however,
instant, if a plane in which
504
00:43:06,000 --> 00:43:09,000
the motion is taking place at a
given time.
505
00:43:09,000 --> 00:43:13,000
And, to say that the motion
actually stays in that plane
506
00:43:13,000 --> 00:43:17,000
forever means that this guy will
not change direction.
507
00:43:17,000 --> 00:43:25,000
OK, so -- [LAUGHTER]
[APPLAUSE]
508
00:43:25,000 --> 00:43:42,000
OK, so, Kepler's second law is
actually equivalent to saying
509
00:43:42,000 --> 00:43:55,000
that r cross v equals a constant
vector, OK?
510
00:43:55,000 --> 00:44:04,000
That's what the law says.
So, in terms of derivatives,
511
00:44:04,000 --> 00:44:14,000
it means d by dt of r cross v
is the zero vector.
512
00:44:14,000 --> 00:44:20,000
OK, now, so there's an
interesting thing to note,
513
00:44:20,000 --> 00:44:23,000
which is that we can use the
usual product rule for
514
00:44:23,000 --> 00:44:26,000
derivatives with vector
expressions,
515
00:44:26,000 --> 00:44:28,000
with dot products or cross
products.
516
00:44:28,000 --> 00:44:30,000
There's only one catch,
which is that when we
517
00:44:30,000 --> 00:44:34,000
differentiate a cross product,
we have to be careful that the
518
00:44:34,000 --> 00:44:36,000
guy on the left stays on the
left.
519
00:44:36,000 --> 00:44:40,000
The guy on the right stays on
the right.
520
00:44:40,000 --> 00:44:44,000
OK, so, if you know that uv
prime equals u prime v plus uv
521
00:44:44,000 --> 00:44:47,000
prime, then you are safe.
If you know it as u prime v
522
00:44:47,000 --> 00:44:50,000
cross v prime u,
then you are not safe.
523
00:44:50,000 --> 00:44:52,000
OK, so it's the only thing to
watch for.
524
00:44:52,000 --> 00:45:05,000
So, product rule is OK for
taking the derivative of a dot
525
00:45:05,000 --> 00:45:10,000
product.
There, you don't actually even
526
00:45:10,000 --> 00:45:14,000
need to be very careful about
all the things or the derivative
527
00:45:14,000 --> 00:45:18,000
of a cross product.
There you just need to be a
528
00:45:18,000 --> 00:45:27,000
little bit more careful.
OK, so, now that we know that,
529
00:45:27,000 --> 00:45:39,000
we can write this as dr/dt
cross v plus r cross dv/dt,
530
00:45:39,000 --> 00:45:42,000
OK?
Well, let's reformulate things
531
00:45:42,000 --> 00:45:47,000
slightly.
So, dr dt already has a name.
532
00:45:47,000 --> 00:45:50,000
In fact, that's v.
OK, that's what we call the
533
00:45:50,000 --> 00:45:55,000
velocity vector.
So, this is v cross v plus r
534
00:45:55,000 --> 00:46:04,000
cross, what is dv/dt?
That's the acceleration,
535
00:46:04,000 --> 00:46:11,000
a, equals zero.
OK, so now what's the next step?
536
00:46:11,000 --> 00:46:15,000
Well, we know what v cross v is
because, remember,
537
00:46:15,000 --> 00:46:18,000
a vector cross itself is always
zero, OK?
538
00:46:18,000 --> 00:46:30,000
So, this is the same r cross a
equals zero,
539
00:46:30,000 --> 00:46:35,000
and that's the same as saying
that the cross product of two
540
00:46:35,000 --> 00:46:39,000
vectors is zero exactly when the
parallelogram of the form has no
541
00:46:39,000 --> 00:46:41,000
area.
And, the way in which that
542
00:46:41,000 --> 00:46:45,000
happens is if they are actually
parallel to each other.
543
00:46:45,000 --> 00:46:50,000
So, that means the acceleration
is parallel to the position.
544
00:46:50,000 --> 00:46:55,000
OK, so, in fact,
what Kepler's second law says
545
00:46:55,000 --> 00:47:02,000
is that the acceleration is
parallel to the position vector.
546
00:47:02,000 --> 00:47:05,000
And, since we know that
acceleration is caused by a
547
00:47:05,000 --> 00:47:08,000
force that's equivalent to the
fact that the gravitational
548
00:47:08,000 --> 00:47:08,000
force --
549
00:47:13,000 --> 00:47:18,000
-- is parallel to the position
vector, that means,
550
00:47:18,000 --> 00:47:22,000
well, if you have the sun here
at the origin,
551
00:47:22,000 --> 00:47:27,000
and if you have your planets,
well, the gravitational force
552
00:47:27,000 --> 00:47:32,000
caused by the sun should go
along this line.
553
00:47:32,000 --> 00:47:34,000
In fact, the law doesn't even
say whether it's going towards
554
00:47:34,000 --> 00:47:37,000
the sun or away from the sun.
Well, what we know now is that,
555
00:47:37,000 --> 00:47:39,000
of course, the attraction is
towards the sun.
556
00:47:39,000 --> 00:47:41,000
But, Kepler's law would also be
true, actually,
557
00:47:41,000 --> 00:47:44,000
if things were going away.
So, in particular,
558
00:47:44,000 --> 00:47:48,000
say, electric force also has
this property of being towards
559
00:47:48,000 --> 00:47:50,000
the central charge.
So, actually,
560
00:47:50,000 --> 00:47:54,000
if you look at motion of
charged particles in an electric
561
00:47:54,000 --> 00:47:58,000
field caused by a point charged
particle, it also satisfies
562
00:47:58,000 --> 00:48:01,000
Kepler's law,
satisfies the same law.
563
00:48:01,000 --> 00:48:03,000
OK, that's the end for today,
thanks.