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So, if you remember last time,
we looked at parametric
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equations -- -- as a way of
describing the motion of a point
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that moves in the plane or in
space as a function of time of
00:00:41.000 --> 00:00:48.000
your favorite parameter that
will tell you how far the motion
00:00:48.000 --> 00:00:54.000
has progressed.
And, I think we did it in
00:00:54.000 --> 00:01:01.000
detail the example of the
cycloid, which is the curve
00:01:01.000 --> 00:01:09.000
traced by a point on a wheel
that's rolling on a flat
00:01:09.000 --> 00:01:14.000
surface.
So, we have this example where
00:01:14.000 --> 00:01:19.000
we have this wheel that's
rolling on the x-axis,
00:01:19.000 --> 00:01:23.000
and we have this point on the
wheel.
00:01:23.000 --> 00:01:31.000
And, as it moves around,
it traces a trajectory that
00:01:31.000 --> 00:01:36.000
moves more or less like this.
OK, so I'm trying a new color.
00:01:36.000 --> 00:01:44.000
Is this visible from the back?
So, no more blue.
00:01:44.000 --> 00:01:52.000
OK, so remember,
in general, we are trying to
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find the position,
so, x of t, y of t,
00:01:58.000 --> 00:02:09.000
maybe z of t if we are in space
-- -- of a moving point along a
00:02:09.000 --> 00:02:17.000
trajectory.
And, one way to think about
00:02:17.000 --> 00:02:25.000
this is in terms of the position
vector.
00:02:25.000 --> 00:02:32.000
So, position vector is just the
vector whose components are
00:02:32.000 --> 00:02:37.000
coordinates of a point,
OK, so if you prefer,
00:02:37.000 --> 00:02:43.000
that's the same thing as a
vector from the origin to the
00:02:43.000 --> 00:02:50.000
moving point.
So, maybe our point is here, P.
00:02:50.000 --> 00:03:02.000
So, this vector here -- This
vector here is vector OP.
00:03:02.000 --> 00:03:12.000
And, that's also the position
vector r of t.
00:03:12.000 --> 00:03:24.000
So, just to give you,
again, that example -- -- if I
00:03:24.000 --> 00:03:34.000
take the cycloid for a wheel of
radius 1,
00:03:34.000 --> 00:03:41.000
and let's say that we are going
at unit speed so that the angle
00:03:41.000 --> 00:03:48.000
that we used as a parameter of
time is the same thing as time
00:03:48.000 --> 00:03:53.000
when the position vector,
in this case,
00:03:53.000 --> 00:04:00.000
we found to be,
just to make sure that they
00:04:00.000 --> 00:04:07.000
have it right,
.
00:04:07.000 --> 00:04:10.000
OK, that's a formula that you
should have in your notes from
00:04:10.000 --> 00:04:13.000
last time, except we had theta
instead of t because we were
00:04:13.000 --> 00:04:16.000
using the angle.
But now I'm saying,
00:04:16.000 --> 00:04:20.000
we are moving at unit speed,
so time and angle are the same
00:04:20.000 --> 00:04:24.000
thing.
So, now, what's interesting
00:04:24.000 --> 00:04:31.000
about this is we can analyze the
motion in more detail.
00:04:31.000 --> 00:04:33.000
OK, so, now that we know the
position of the point as a
00:04:33.000 --> 00:04:37.000
function of time,
we can try to study how it
00:04:37.000 --> 00:04:43.000
varies in particular things like
the speed and acceleration.
00:04:43.000 --> 00:04:48.000
OK, so let's start with speed.
Well, in fact we can do better
00:04:48.000 --> 00:04:51.000
than speed.
Let's not start with speed.
00:04:51.000 --> 00:04:54.000
So, speed is a number.
It tells you how fast you are
00:04:54.000 --> 00:04:58.000
going along your trajectory.
I mean, if you're driving in a
00:04:58.000 --> 00:05:01.000
car, then it tells you how fast
you are going.
00:05:01.000 --> 00:05:03.000
But, unless you have one of
these fancy cars with a GPS,
00:05:03.000 --> 00:05:05.000
it doesn't tell you which
direction you're going.
00:05:05.000 --> 00:05:08.000
And, that's useful information,
too, if you're trying to figure
00:05:08.000 --> 00:05:10.000
out what your trajectory is.
So, in fact,
00:05:10.000 --> 00:05:13.000
there's two aspects to it.
One is how fast you are going,
00:05:13.000 --> 00:05:15.000
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
00:05:19.000 --> 00:05:26.000
about this.
And so, that's called the
00:05:26.000 --> 00:05:32.000
velocity vector.
And, the way we can get it,
00:05:32.000 --> 00:05:37.000
so, it's called usually V,
so, V here stands for velocity
00:05:37.000 --> 00:05:42.000
more than for vector.
And, you just get it by taking
00:05:42.000 --> 00:05:46.000
the derivative of a position
vector with respect to time.
00:05:46.000 --> 00:05:50.000
Now, it's our first time
writing this kind of thing with
00:05:50.000 --> 00:05:52.000
a vector.
So, the basic rule is you can
00:05:52.000 --> 00:05:57.000
take the derivative of a vector
quantity just by taking the
00:05:57.000 --> 00:06:06.000
derivatives of each component.
OK, so that's just dx/dt,
00:06:06.000 --> 00:06:17.000
dy/dt, and if you have z
component, dz/dt.
00:06:17.000 --> 00:06:32.000
So, let me -- OK,
so -- OK, so let's see what it
00:06:32.000 --> 00:06:44.000
is for the cycloid.
So, an example of a cycloid,
00:06:44.000 --> 00:06:54.000
well, so what do we get when we
take the derivatives of this
00:06:54.000 --> 00:07:02.000
formula there?
Well, so, the derivative of t
00:07:02.000 --> 00:07:08.000
is 1- cos(t).
The derivative of 1 is 0.
00:07:08.000 --> 00:07:12.000
The derivative of -cos(t) is
sin(t).
00:07:12.000 --> 00:07:17.000
Very good.
OK, that's at least one thing
00:07:17.000 --> 00:07:20.000
you should remember from single
variable calculus.
00:07:20.000 --> 00:07:24.000
Hopefully you remember even
more than that.
00:07:24.000 --> 00:07:27.000
OK, so that's the velocity
vector.
00:07:27.000 --> 00:07:31.000
It tells us at any time how
fast we are going,
00:07:31.000 --> 00:07:37.000
and in what direction.
So, for example, observe.
00:07:37.000 --> 00:07:40.000
Remember last time at the end
of class we were trying to
00:07:40.000 --> 00:07:43.000
figure out what exactly happens
near the bottom point,
00:07:43.000 --> 00:07:47.000
when we have this motion that
seems to stop and go backwards.
00:07:47.000 --> 00:07:50.000
And, we answered that one way.
But, let's try to understand it
00:07:50.000 --> 00:07:54.000
in terms of velocity.
What if I plug t equals 0 in
00:07:54.000 --> 00:07:57.000
here?
Then, 1- cos(t) is 0,
00:07:57.000 --> 00:08:01.000
sin(t) is 0.
The velocity is 0.
00:08:01.000 --> 00:08:05.000
So, at the time,at that
particular time,
00:08:05.000 --> 00:08:08.000
our point is actually not
moving.
00:08:08.000 --> 00:08:11.000
Of course, it's been moving
just before, and it starts
00:08:11.000 --> 00:08:14.000
moving just afterwards.
It's just the instant,
00:08:14.000 --> 00:08:20.000
at that particular instant,
the speed is zero.
00:08:20.000 --> 00:08:23.000
So, that's especially maybe a
counterintuitive thing,
00:08:23.000 --> 00:08:28.000
but something is moving.
And at that time,
00:08:28.000 --> 00:08:33.000
it's actually stopped.
Now, let's see,
00:08:33.000 --> 00:08:36.000
so that's the vector.
And, it's useful.
00:08:36.000 --> 00:08:39.000
But, if you want just the usual
speed as a number,
00:08:39.000 --> 00:08:43.000
then, what will you do?
Well, you will just take
00:08:43.000 --> 00:08:46.000
exactly the magnitude of this
vector.
00:08:46.000 --> 00:08:56.000
So, speed, which is the scalar
quantity is going to be just the
00:08:56.000 --> 00:09:01.000
magnitude of the vector,
V.
00:09:01.000 --> 00:09:09.000
OK, so, in this case,
while it would be square root
00:09:09.000 --> 00:09:18.000
of (1- cost)^2 sin^2(t),
and if you expand that,
00:09:18.000 --> 00:09:23.000
you will get,
let me take a bit more space,
00:09:23.000 --> 00:09:35.000
it's going to be square root of
1 - 2cos(t) cos^2(t) sin^2(t).
00:09:35.000 --> 00:09:38.000
It seems to simplify a little
bit because we have cos^2 plus
00:09:38.000 --> 00:09:41.000
sin^2.
That's 1.
00:09:41.000 --> 00:09:49.000
So, it's going to be the square
root of 2 - 2cos(t).
00:09:49.000 --> 00:09:52.000
So, at this point,
if I was going to ask you,
00:09:52.000 --> 00:09:55.000
when is the speed the smallest
or the largest?
00:09:55.000 --> 00:09:59.000
You could answer based on that.
See, at t equals 0,
00:09:59.000 --> 00:10:01.000
well, that turns out to be
zero.
00:10:01.000 --> 00:10:04.000
The point is not moving.
At t equals pi,
00:10:04.000 --> 00:10:07.000
that ends up being the square
root of 2 plus 2,
00:10:07.000 --> 00:10:09.000
which is 4.
So, that's 2.
00:10:09.000 --> 00:10:12.000
And, that's when you're truly
at the top of the arch,
00:10:12.000 --> 00:10:15.000
and that's when the point is
moving the fastest.
00:10:15.000 --> 00:10:18.000
In fact, they are spending
twice as fast as the wheel
00:10:18.000 --> 00:10:20.000
because the wheel is moving to
the right at unit speed,
00:10:20.000 --> 00:10:24.000
and the wheel is also rotating.
So, it's moving to the right
00:10:24.000 --> 00:10:29.000
and unit speed relative to the
center so that the two effects
00:10:29.000 --> 00:10:32.000
add up, and give you a speed of
2.
00:10:32.000 --> 00:10:36.000
Anyway, that's a formula we can
get.
00:10:36.000 --> 00:10:48.000
OK, now, what about
acceleration?
00:10:48.000 --> 00:10:53.000
So, here I should warn you that
there is a serious discrepancy
00:10:53.000 --> 00:10:58.000
between the usual intuitive
notion of acceleration,
00:10:58.000 --> 00:11:02.000
the one that you are aware of
when you drive a car and the one
00:11:02.000 --> 00:11:05.000
that we will be using.
So, you might think
00:11:05.000 --> 00:11:08.000
acceleration is just the
directive of speed.
00:11:08.000 --> 00:11:13.000
If my car goes 55 miles an hour
on the highway and it's going a
00:11:13.000 --> 00:11:15.000
constant speed,
it's not accelerating.
00:11:15.000 --> 00:11:18.000
But, let's say that I'm taking
a really tight turn.
00:11:18.000 --> 00:11:19.000
Then, I'm going to feel
something.
00:11:19.000 --> 00:11:21.000
There is some force being
exerted.
00:11:21.000 --> 00:11:24.000
And, in fact,
there is a sideways
00:11:24.000 --> 00:11:28.000
acceleration at that point even
though the speed is not
00:11:28.000 --> 00:11:30.000
changing.
So, the definition will take
00:11:30.000 --> 00:11:34.000
effect.
The acceleration is,
00:11:34.000 --> 00:11:40.000
as a vector,
and the acceleration vector is
00:11:40.000 --> 00:11:47.000
just the derivative of a
velocity vector.
00:11:47.000 --> 00:11:51.000
So, even if the speed is
constant, that means,
00:11:51.000 --> 00:11:55.000
even if a length of the
velocity vector stays the same,
00:11:55.000 --> 00:11:59.000
the velocity vector can still
rotate.
00:11:59.000 --> 00:12:03.000
And, as it rotates,
it uses acceleration.
00:12:03.000 --> 00:12:07.000
OK, and so this is the notion
of acceleration that's relevant
00:12:07.000 --> 00:12:13.000
to physics when you find F=ma;
that's the (a) that you have in
00:12:13.000 --> 00:12:17.000
mind here.
It's a vector.
00:12:17.000 --> 00:12:19.000
Of course, if you are moving in
a straight line,
00:12:19.000 --> 00:12:20.000
then the two notions are the
same.
00:12:20.000 --> 00:12:23.000
I mean, acceleration is also
going to be along the line,
00:12:23.000 --> 00:12:25.000
and it's going to has to do
with the derivative of speed.
00:12:25.000 --> 00:12:30.000
But, in general,
that's not quite the same.
00:12:30.000 --> 00:12:37.000
So, for example,
let's look at the cycloid.
00:12:37.000 --> 00:12:40.000
If we take the example of the
cycloid, well,
00:12:40.000 --> 00:12:44.000
what's the derivative of one
minus cos(t)?
00:12:44.000 --> 00:12:52.000
It's sin(t).
And, what's the derivative of
00:12:52.000 --> 00:12:55.000
sin(t)?
cos(t), OK.
00:12:55.000 --> 00:13:04.000
So, the acceleration vector is
.
00:13:04.000 --> 00:13:09.000
So, in particular,
let's look at what happens at
00:13:09.000 --> 00:13:13.000
time t equals zero when the
point is not moving.
00:13:13.000 --> 00:13:20.000
Well, the acceleration vector
there will be zero from one.
00:13:20.000 --> 00:13:28.000
So, what that means is that if
I look at my trajectory at this
00:13:28.000 --> 00:13:35.000
point, that the acceleration
vector is pointing in that
00:13:35.000 --> 00:13:39.000
direction.
It's the unit vector in the
00:13:39.000 --> 00:13:43.000
vertical direction.
So, my point is not moving at
00:13:43.000 --> 00:13:46.000
that particular time.
But, it's accelerating up.
00:13:46.000 --> 00:13:49.000
So, that means that actually as
it comes down,
00:13:49.000 --> 00:13:53.000
first it's slowing down.
Then it stops here,
00:13:53.000 --> 00:13:56.000
and then it reverses going back
up.
00:13:56.000 --> 00:14:01.000
OK, so that's another way to
understand what we were saying
00:14:01.000 --> 00:14:06.000
last time that the trajectory at
that point has a vertical
00:14:06.000 --> 00:14:11.000
tendency because that's the
direction in which the motion is
00:14:11.000 --> 00:14:16.000
going to occur just before and
just after time zero.
00:14:16.000 --> 00:14:30.000
OK, any questions about that?
No.
00:14:30.000 --> 00:14:36.000
OK, so I should insist maybe on
one thing,
00:14:36.000 --> 00:14:41.000
which is that,
so, we can differentiate
00:14:41.000 --> 00:14:46.000
vectors just component by
component,
00:14:46.000 --> 00:14:50.000
OK, and we can differentiate
vector expressions according to
00:14:50.000 --> 00:14:54.000
certain rules that we'll see in
a moment.
00:14:54.000 --> 00:15:02.000
One thing that we cannot do,
it's not true that the length
00:15:02.000 --> 00:15:12.000
of dr dt, which is the speed,
is equal to the length of dt.
00:15:12.000 --> 00:15:18.000
OK, this is completely false.
And, they are really not the
00:15:18.000 --> 00:15:19.000
same.
So, if you have to
00:15:19.000 --> 00:15:24.000
differentiate the length of a
vector, but basically you are in
00:15:24.000 --> 00:15:25.000
trouble.
If you really,
00:15:25.000 --> 00:15:27.000
really want to do it,
well, the length of the vector
00:15:27.000 --> 00:15:30.000
is the square root of the sums
of the squares of the
00:15:30.000 --> 00:15:32.000
components,
and from that you can use the
00:15:32.000 --> 00:15:34.000
formula for the derivative of
the square root,
00:15:34.000 --> 00:15:36.000
and the chain rule,
and various other things.
00:15:36.000 --> 00:15:39.000
And, you can get there.
But, it will not be a very nice
00:15:39.000 --> 00:15:42.000
expression.
There is no simple formula for
00:15:42.000 --> 00:15:44.000
this kind of thing.
Fortunately,
00:15:44.000 --> 00:15:48.000
we almost never have to compute
this kind of thing because,
00:15:48.000 --> 00:15:51.000
after all, it's not a very
relevant quantity.
00:15:51.000 --> 00:15:53.000
What's more relevant might be
this one.
00:15:53.000 --> 00:15:59.000
This is actually the speed.
This one, I don't know what it
00:15:59.000 --> 00:16:10.000
means.
OK.
00:16:10.000 --> 00:16:14.000
So, let's continue our
exploration.
00:16:14.000 --> 00:16:20.000
So, the next concept that I
want to define is that of arc
00:16:20.000 --> 00:16:23.000
length.
So, arc length is just the
00:16:23.000 --> 00:16:26.000
distance that you have traveled
along the curve,
00:16:26.000 --> 00:16:27.000
OK?
So, if you are in a car,
00:16:27.000 --> 00:16:30.000
you know, it has mileage
counter that tells you how far
00:16:30.000 --> 00:16:33.000
you've gone, how much fuel
you've used if it's a fancy car.
00:16:33.000 --> 00:16:37.000
And, what it does is it
actually integrates the speed of
00:16:37.000 --> 00:16:41.000
the time to give you the arc
length along the trajectory of
00:16:41.000 --> 00:16:45.000
the car.
So, the usual notation that we
00:16:45.000 --> 00:16:51.000
will have is (s) for arc length.
I'm not quite sure how you get
00:16:51.000 --> 00:16:57.000
an (s) out of this,
but it's the usual notation.
00:16:57.000 --> 00:17:14.000
OK, so, (s) is for distance
traveled along the trajectory.
00:17:14.000 --> 00:17:16.000
And, so that makes sense,
of course, we need to fix a
00:17:16.000 --> 00:17:19.000
reference point.
Maybe on the cycloid,
00:17:19.000 --> 00:17:22.000
we'd say it's a distance
starting on the origin.
00:17:22.000 --> 00:17:25.000
In general, maybe you would say
you start at time,
00:17:25.000 --> 00:17:28.000
t equals zero.
But, it's a convention.
00:17:28.000 --> 00:17:31.000
If you knew in advance,
you could have,
00:17:31.000 --> 00:17:35.000
actually, your car's mileage
counter to count backwards from
00:17:35.000 --> 00:17:38.000
the point where the car will die
and start walking.
00:17:38.000 --> 00:17:41.000
I mean, that would be
sneaky-freaky,
00:17:41.000 --> 00:17:45.000
but you could have a negative
arc length that gets closer and
00:17:45.000 --> 00:17:48.000
closer to zero,
and gets to zero at the end of
00:17:48.000 --> 00:17:51.000
a trajectory,
or anything you want.
00:17:51.000 --> 00:17:53.000
I mean, arc length could be
positive or negative.
00:17:53.000 --> 00:17:56.000
Typically it's negative what
you are before the reference
00:17:56.000 --> 00:18:01.000
point, and positive afterwards.
So, now, how does it relate to
00:18:01.000 --> 00:18:08.000
the things we've seen there?
Well, so in particular,
00:18:08.000 --> 00:18:16.000
how do you relate arc length
and time?
00:18:16.000 --> 00:18:22.000
Well, so, there's a simple
relation, which is that the rate
00:18:22.000 --> 00:18:26.000
of change of arc length versus
time,
00:18:26.000 --> 00:18:30.000
well, that's going to be the
speed at which you are moving,
00:18:30.000 --> 00:18:38.000
OK, because the speed as a
scalar quantity tells you how
00:18:38.000 --> 00:18:44.000
much distance you're covering
per unit time.
00:18:44.000 --> 00:18:47.000
OK, and in fact,
to be completely honest,
00:18:47.000 --> 00:18:51.000
I should put an absolute value
here because there is examples
00:18:51.000 --> 00:18:55.000
of curves maybe where your
motion is going back and forth
00:18:55.000 --> 00:18:59.000
along the same curve.
And then, you don't want to
00:18:59.000 --> 00:19:01.000
keep counting arc length all the
time.
00:19:01.000 --> 00:19:04.000
Actually, maybe you want to say
that the arc length increases
00:19:04.000 --> 00:19:05.000
and then decreases along the
curve.
00:19:05.000 --> 00:19:08.000
I mean, you get to choose how
you count it.
00:19:08.000 --> 00:19:10.000
But, in this case,
if you are moving back and
00:19:10.000 --> 00:19:12.000
forth, it would make more sense
to have the arc length first
00:19:12.000 --> 00:19:18.000
increase,
then decrease,
00:19:18.000 --> 00:19:26.000
increase again,
and so on.
00:19:26.000 --> 00:19:34.000
So -- So if you want to know
really what the arc length is,
00:19:34.000 --> 00:19:41.000
then basically the only way to
do it is to integrate speed
00:19:41.000 --> 00:19:45.000
versus time.
So, if you wanted to know how
00:19:45.000 --> 00:19:49.000
long an arch of cycloid is,
you have this nice-looking
00:19:49.000 --> 00:19:51.000
curve;
how long is it?
00:19:51.000 --> 00:19:55.000
Well, you'd have to basically
integrate this quantity from t
00:19:55.000 --> 00:19:57.000
equals zero to 2 pi.
00:20:24.000 --> 00:20:28.000
And, to say the truth,
I don't really know how to
00:20:28.000 --> 00:20:31.000
integrate that.
So, we don't actually have a
00:20:31.000 --> 00:20:34.000
formula for the length at this
point.
00:20:34.000 --> 00:20:41.000
However, we'll see one later
using a cool trick,
00:20:41.000 --> 00:20:47.000
and multi-variable calculus.
So, for now,
00:20:47.000 --> 00:20:52.000
we'll just leave the formula
like that, and we don't know how
00:20:52.000 --> 00:20:55.000
long it is.
Well, you can put that into
00:20:55.000 --> 00:20:57.000
your calculator and get the
numerical value.
00:20:57.000 --> 00:21:07.000
But, that's the best I can
offer.
00:21:07.000 --> 00:21:18.000
Now, another useful notion is
the unit vector to the
00:21:18.000 --> 00:21:25.000
trajectory.
So, the usual notation is T hat.
00:21:25.000 --> 00:21:28.000
It has a hat because it's a
unit vector, and T because it's
00:21:28.000 --> 00:21:32.000
tangent.
Now, how do we get this unit
00:21:32.000 --> 00:21:36.000
vector?
So, maybe I should have pointed
00:21:36.000 --> 00:21:40.000
out before that if you're moving
along some trajectory,
00:21:40.000 --> 00:21:43.000
say you're going in that
direction, then when you're at
00:21:43.000 --> 00:21:47.000
this point,
the velocity vector is going to
00:21:47.000 --> 00:21:53.000
be tangential to the trajectory.
It tells you the direction of
00:21:53.000 --> 00:21:57.000
motion in particular.
So, if you want a unit vector
00:21:57.000 --> 00:22:02.000
that goes in the same direction,
all you have to do is rescale
00:22:02.000 --> 00:22:05.000
it, so, at its length becomes
one.
00:22:05.000 --> 00:22:10.000
So, it's v divided by a
magnitude of v.
00:22:28.000 --> 00:22:33.000
So, it seems like now we have a
lot of different things that
00:22:33.000 --> 00:22:40.000
should be related in some way.
So, let's see what we can say.
00:22:40.000 --> 00:22:50.000
Well, we can say that dr by dt,
so, that's the velocity vector,
00:22:50.000 --> 00:22:59.000
that's the same thing as if I
use the chain rule dr/ds times
00:22:59.000 --> 00:23:06.000
ds/dt.
OK, so, let's think about this
00:23:06.000 --> 00:23:11.000
things.
So, this guy here we've just
00:23:11.000 --> 00:23:17.000
seen.
That's the same as the speed,
00:23:17.000 --> 00:23:21.000
OK?
So, this one here should be v
00:23:21.000 --> 00:23:28.000
divided by its length.
So, that means this actually
00:23:28.000 --> 00:23:34.000
should be the unit vector.
OK, so, let me rewrite that.
00:23:34.000 --> 00:23:40.000
It's T ds/dt.
So, maybe if I actually stated
00:23:40.000 --> 00:23:43.000
directly that way,
see, I'm just saying the
00:23:43.000 --> 00:23:46.000
velocity vector has a length and
a direction.
00:23:46.000 --> 00:23:51.000
The length is the speed.
The direction is tangent to the
00:23:51.000 --> 00:23:51.000
trajectory.
00:24:19.000 --> 00:24:25.000
So, the speed is ds/dt,
and the vector is T hat.
00:24:25.000 --> 00:24:33.000
And, that's how we get this.
So, let's try just to see why
00:24:33.000 --> 00:24:37.000
dr/ds should be T.
Well, let's think of dr/ds.
00:24:37.000 --> 00:24:42.000
dr/ds means position vector r
means you have the origin,
00:24:42.000 --> 00:24:47.000
which is somewhere out there,
and the vector r is here.
00:24:47.000 --> 00:24:51.000
So, dr/ds means we move by a
small amount,
00:24:51.000 --> 00:24:56.000
delta s along the trajectory a
certain distance delta s.
00:24:56.000 --> 00:25:00.000
And, we look at how the
position vector changes.
00:25:00.000 --> 00:25:08.000
Well, we'll have a small change.
Let me call that vector delta r
00:25:08.000 --> 00:25:13.000
corresponding to the size,
corresponding to the length
00:25:13.000 --> 00:25:17.000
delta s.
And now, delta r should be
00:25:17.000 --> 00:25:25.000
essentially roughly equal to,
well, its direction will be
00:25:25.000 --> 00:25:30.000
tangent to the trajectory.
If I take a small enough
00:25:30.000 --> 00:25:33.000
interval,
then the direction will be
00:25:33.000 --> 00:25:37.000
almost tensioned to the
trajectory times the length of
00:25:37.000 --> 00:25:41.000
it will be delta s,
the distance that I have
00:25:41.000 --> 00:25:45.000
traveled.
OK, sorry, maybe I should
00:25:45.000 --> 00:25:50.000
explain that on a separate
board.
00:25:50.000 --> 00:25:56.000
OK, so, let's say that we have
that amount of time,
00:25:56.000 --> 00:26:00.000
delta t.
So, let's zoom into that curve.
00:26:00.000 --> 00:26:12.000
So, we have r at time t.
We have r at time t plus delta
00:26:12.000 --> 00:26:17.000
t.
This vector here I will call
00:26:17.000 --> 00:26:23.000
delta r.
The length of this vector is
00:26:23.000 --> 00:26:28.000
delta s.
And, the direction is
00:26:28.000 --> 00:26:36.000
essentially that of a vector.
OK, so, delta s over delta t,
00:26:36.000 --> 00:26:43.000
that's the distance traveled
divided by the time.
00:26:43.000 --> 00:26:46.000
That's going to be close to the
speed.
00:26:46.000 --> 00:26:57.000
And, delta r is approximately T
times delta s.
00:26:57.000 --> 00:27:04.000
So, now if I divide both sides
by delta t, I get this.
00:27:04.000 --> 00:27:07.000
And, if I take the limit as
delta t turns to zero,
00:27:07.000 --> 00:27:10.000
then I get the same formula
with the derivatives and with an
00:27:10.000 --> 00:27:13.000
equality.
It's an approximation.
00:27:13.000 --> 00:27:15.000
The approximation becomes
better and better if I go to
00:27:15.000 --> 00:27:16.000
smaller intervals.
00:27:38.000 --> 00:27:44.000
OK, are there any questions
about this?
00:27:44.000 --> 00:27:59.000
Yes?
Yes, that's correct.
00:27:59.000 --> 00:28:01.000
OK, so let's be more careful,
actually.
00:28:01.000 --> 00:28:12.000
So, you're asking about whether
the delta r is actually strictly
00:28:12.000 --> 00:28:16.000
tangent to the curve.
Is that -- That's correct.
00:28:16.000 --> 00:28:20.000
Actually, delta r is not
strictly tangent to anything.
00:28:20.000 --> 00:28:23.000
So, maybe I should draw another
picture.
00:28:23.000 --> 00:28:29.000
If I'm going from here to here,
then delta r is going to be
00:28:29.000 --> 00:28:36.000
this arc inside the curve while
the vector will be going in this
00:28:36.000 --> 00:28:39.000
direction, OK?
So, they are not strictly
00:28:39.000 --> 00:28:41.000
parallel to each other.
That's why it's only
00:28:41.000 --> 00:28:44.000
approximately equal.
Similarly, this distance,
00:28:44.000 --> 00:28:48.000
the length of delta r is not
exactly the length along the
00:28:48.000 --> 00:28:50.000
curve.
It's actually a bit shorter.
00:28:50.000 --> 00:28:53.000
But, if we imagine a smaller
and smaller portion of the
00:28:53.000 --> 00:28:56.000
curve,
then this effect of the curve
00:28:56.000 --> 00:29:00.000
being a curve and not a straight
line becomes more and more
00:29:00.000 --> 00:29:02.000
negligible.
If you zoom into the curve
00:29:02.000 --> 00:29:04.000
sufficiently,
then it looks more and more
00:29:04.000 --> 00:29:07.000
like a straight line.
And then, what I said becomes
00:29:07.000 --> 00:29:18.000
true in the limit.
OK? Any other questions?
00:29:18.000 --> 00:29:35.000
No? OK.
So, what happens next?
00:29:35.000 --> 00:29:39.000
OK, so let me show you a nice
example of why we might want to
00:29:39.000 --> 00:29:43.000
use vectors to study parametric
curves because,
00:29:43.000 --> 00:29:46.000
after all, a lot of what's here
you can just do in coordinates.
00:29:46.000 --> 00:29:48.000
And, we don't really need
vectors.
00:29:48.000 --> 00:29:51.000
Well, and truly,
vectors being a language,
00:29:51.000 --> 00:29:54.000
you never strictly need it,
but it's useful to have a
00:29:54.000 --> 00:30:02.000
notion of vectors.
So, I want to tell you a bit
00:30:02.000 --> 00:30:14.000
about Kepler's second law of
celestial mechanics.
00:30:14.000 --> 00:30:20.000
So, that goes back to 1609.
So, that's not exactly recent
00:30:20.000 --> 00:30:24.000
news, OK?
But, still I think it's a very
00:30:24.000 --> 00:30:29.000
interesting example of why you
might want to use vector methods
00:30:29.000 --> 00:30:33.000
to analyze motions.
So, what happened back then was
00:30:33.000 --> 00:30:39.000
Kepler was trying to observe the
motion of planets in the sky,
00:30:39.000 --> 00:30:42.000
and trying to come up with
general explanations of how they
00:30:42.000 --> 00:30:44.000
move.
Before him, people were saying,
00:30:44.000 --> 00:30:46.000
well, they cannot move in a
circle.
00:30:46.000 --> 00:30:48.000
But maybe it's more complicated
than that.
00:30:48.000 --> 00:30:51.000
We need to add smaller circular
motions on top of each other,
00:30:51.000 --> 00:30:53.000
and so on.
They have more and more
00:30:53.000 --> 00:30:56.000
complicated theories.
And then Kepler came with these
00:30:56.000 --> 00:31:00.000
laws that said basically that
planets move in an ellipse
00:31:00.000 --> 00:31:03.000
around the sun,
and that they move in a very
00:31:03.000 --> 00:31:07.000
specific way along that ellipse.
So, there's actually three
00:31:07.000 --> 00:31:11.000
laws, but let me just tell you
about the second one that has a
00:31:11.000 --> 00:31:17.000
very nice vector interpretation.
So, what Kepler's second law
00:31:17.000 --> 00:31:24.000
says is that the motion of
planets is, first of all,
00:31:24.000 --> 00:31:36.000
they move in a plane.
And second, the area swept out
00:31:36.000 --> 00:31:51.000
by the line from the sun to the
planet is swept at constant
00:31:51.000 --> 00:31:57.000
time.
Sorry, is swept at constant
00:31:57.000 --> 00:32:04.000
rate.
From the sun to the planet,
00:32:04.000 --> 00:32:14.000
it is swept out by the line at
a constant rate.
00:32:14.000 --> 00:32:23.000
OK, so that's an interesting
law because it tells you,
00:32:23.000 --> 00:32:27.000
once you know what the orbit of
the planet looks like,
00:32:27.000 --> 00:32:30.000
it tells you how fast it's
going to move on that orbit.
00:33:09.000 --> 00:33:19.000
OK, so let me explain again.
So, this law says maybe the
00:33:19.000 --> 00:33:27.000
sun, let's put the sun here at
the origin, and let's have a
00:33:27.000 --> 00:33:34.000
planet.
Well, the planet orbits around
00:33:34.000 --> 00:33:41.000
the sun -- -- in some
trajectory.
00:33:41.000 --> 00:33:45.000
So, this is supposed to be
light blue.
00:33:45.000 --> 00:33:49.000
Can you see that it's different
from white?
00:33:49.000 --> 00:33:51.000
No?
OK, me neither.
00:33:51.000 --> 00:33:53.000
[LAUGHTER]
OK, it doesn't really matter.
00:33:53.000 --> 00:33:55.000
So, the planet moves on its
orbit.
00:33:55.000 --> 00:34:00.000
And, if you wait for a certain
time, then a bit later it would
00:34:00.000 --> 00:34:04.000
be here, and then here,
and so on.
00:34:04.000 --> 00:34:09.000
Then, you can look at the
amount of area inside this
00:34:09.000 --> 00:34:12.000
triangular wedge.
And, the claim is that the
00:34:12.000 --> 00:34:16.000
amount of area in here is
proportional to the time
00:34:16.000 --> 00:34:18.000
elapsed.
So, in particular,
00:34:18.000 --> 00:34:21.000
if a planet is closer to the
sun, then it has to go faster.
00:34:21.000 --> 00:34:25.000
And, if it's farther away from
the sun, then it has to go
00:34:25.000 --> 00:34:28.000
slower so that the area remains
proportional to time.
00:34:28.000 --> 00:34:32.000
So, it's a very sophisticated
prediction.
00:34:32.000 --> 00:34:36.000
And, I think the way he came to
it was really just by using a
00:34:36.000 --> 00:34:39.000
lot of observations,
and trying to measure what was
00:34:39.000 --> 00:34:44.000
true that wasn't true.
But, let's try to see how we
00:34:44.000 --> 00:34:49.000
can understand that in terms of
all we know today about
00:34:49.000 --> 00:34:52.000
mechanics.
So, in fact,
00:34:52.000 --> 00:34:56.000
what happens is that Newton,
so Newton was quite a bit
00:34:56.000 --> 00:35:04.000
later.
That was the late 17th century
00:35:04.000 --> 00:35:13.000
instead of the beginning of the
17th century.
00:35:13.000 --> 00:35:30.000
So, he was able to explain this
using his laws for gravitational
00:35:30.000 --> 00:35:36.000
attraction.
And, you'll see that if we
00:35:36.000 --> 00:35:41.000
reformulate Kepler's Law in
terms of vectors,
00:35:41.000 --> 00:35:43.000
and if we work a bit with these
vectors,
00:35:43.000 --> 00:35:46.000
we are going to end up with
something that's actually
00:35:46.000 --> 00:35:49.000
completely obvious to us now.
At the time,
00:35:49.000 --> 00:35:52.000
it was very far from obvious,
but to us now to completely
00:35:52.000 --> 00:35:59.000
obvious.
So, let's try to see,
00:35:59.000 --> 00:36:15.000
what does Kepler's law say in
terms of vectors?
00:36:15.000 --> 00:36:24.000
OK, so, let's think of what
kinds of vectors we might want
00:36:24.000 --> 00:36:31.000
to have in here.
Well, it might be good to think
00:36:31.000 --> 00:36:38.000
of, maybe, the position vector,
and maybe its variation.
00:36:38.000 --> 00:36:46.000
So, if we wait a certain amount
of time, we'll have a vector,
00:36:46.000 --> 00:36:53.000
delta r, which is the change in
position vector a various
00:36:53.000 --> 00:36:59.000
interval of time.
OK, so let's start with the
00:36:59.000 --> 00:37:02.000
first step.
What's the most complicated
00:37:02.000 --> 00:37:05.000
thing in here?
It's this area swept out by the
00:37:05.000 --> 00:37:08.000
line.
How do we express that area in
00:37:08.000 --> 00:37:12.000
terms of vectors?
Well, I've almost given the
00:37:12.000 --> 00:37:14.000
answer by drawing this picture,
right?
00:37:14.000 --> 00:37:18.000
If I take a sufficiently small
amount of time,
00:37:18.000 --> 00:37:22.000
this shaded part looks like a
triangle.
00:37:22.000 --> 00:37:25.000
So, we have to find the area of
the triangle.
00:37:25.000 --> 00:37:27.000
Well, we know how to do that
now.
00:37:27.000 --> 00:37:34.000
So, the area is approximately
equal to one half of the area of
00:37:34.000 --> 00:37:40.000
a parallelogram that I could
form from these vectors.
00:37:40.000 --> 00:37:46.000
And, the area of a
parallelogram is given by the
00:37:46.000 --> 00:37:52.000
magnitude of a cross product.
OK, so, I should say,
00:37:52.000 --> 00:37:56.000
this is the area swept in time
delta t.
00:37:56.000 --> 00:38:00.000
You should think of delta t as
relatively small.
00:38:00.000 --> 00:38:05.000
I mean, the scale of a planet
that might still be a few days,
00:38:05.000 --> 00:38:09.000
but small compared to the other
old trajectory.
00:38:09.000 --> 00:38:16.000
So, let's remember that the
amount by which we moved,
00:38:16.000 --> 00:38:20.000
delta r,
is approximately equal to v
00:38:20.000 --> 00:38:25.000
times delta t,
OK, and just using the
00:38:25.000 --> 00:38:36.000
definition of a velocity vector.
So, let's use that.
00:38:36.000 --> 00:38:43.000
Sorry, so it's approximately
equal to r cross v magnitude
00:38:43.000 --> 00:38:48.000
times delta t.
I can take out the delta t,
00:38:48.000 --> 00:38:52.000
which is scalar.
So, now, what does it mean to
00:38:52.000 --> 00:38:55.000
say that area is swept at a
constant rate?
00:38:55.000 --> 00:39:00.000
It means this thing is
proportional to delta t.
00:39:00.000 --> 00:39:05.000
So, that means,
so, the law says,
00:39:05.000 --> 00:39:15.000
in fact, that the length of
this cross product r cross v
00:39:15.000 --> 00:39:25.000
equals a constant.
OK, r cross v has constant
00:39:25.000 --> 00:39:31.000
length.
Any questions about that?
00:39:31.000 --> 00:39:37.000
No? Yes?
Yes, let me try to explain that
00:39:37.000 --> 00:39:40.000
again.
So, what I'm claiming is that
00:39:40.000 --> 00:39:46.000
the length of the cross products
r cross v measures the rate at
00:39:46.000 --> 00:39:50.000
which area is swept by the
position vector.
00:39:50.000 --> 00:39:52.000
I should say,
with a vector of one half of
00:39:52.000 --> 00:39:55.000
this length is the rate at which
area is swept.
00:39:55.000 --> 00:39:58.000
How do we see that?
Well, let's take a small time
00:39:58.000 --> 00:40:01.000
interval, delta t.
In time, delta t,
00:40:01.000 --> 00:40:05.000
our planet moves by v delta t,
OK?
00:40:05.000 --> 00:40:08.000
So, if it moves by v delta t,
it means that this triangle up
00:40:08.000 --> 00:40:12.000
there has two sides.
One is the position vector,
00:40:12.000 --> 00:40:14.000
r.
The other one is v delta t.
00:40:14.000 --> 00:40:18.000
So, its area is given by one
half of the magnitude of a cross
00:40:18.000 --> 00:40:21.000
product.
That's the formula we've seen
00:40:21.000 --> 00:40:24.000
for the area of a triangle in
space.
00:40:24.000 --> 00:40:28.000
So, the area is one half of the
cross product,
00:40:28.000 --> 00:40:33.000
r, and v delta t,
magnitude of the cross product.
00:40:33.000 --> 00:40:37.000
So, to say that the rate at
which area is swept is constant
00:40:37.000 --> 00:40:39.000
means that these two are
proportional.
00:40:39.000 --> 00:40:42.000
Area divided by delta t is
constant at our time.
00:40:42.000 --> 00:40:51.000
And so, this is constant.
OK, now, what about the other
00:40:51.000 --> 00:40:58.000
half of the law?
Well, it says that the motion
00:40:58.000 --> 00:41:04.000
is in a plane,
and so we have a plane in which
00:41:04.000 --> 00:41:09.000
the motion takes place.
And, it contains,
00:41:09.000 --> 00:41:12.000
also, the sun.
And, it contains the
00:41:12.000 --> 00:41:16.000
trajectory.
So, let's think about that
00:41:16.000 --> 00:41:20.000
plane.
Well, I claim that the position
00:41:20.000 --> 00:41:25.000
vector is in the plane.
OK, that's what we are saying.
00:41:25.000 --> 00:41:28.000
But, there is another vector
that I know it is in the plane.
00:41:28.000 --> 00:41:32.000
You could say the position
vector at another time,
00:41:32.000 --> 00:41:34.000
or at any time,
but in fact,
00:41:34.000 --> 00:41:40.000
what's also true is that the
velocity vector is in the plane.
00:41:40.000 --> 00:41:44.000
OK, if I'm moving in the plane,
then position and velocity are
00:41:44.000 --> 00:41:50.000
in there.
So, the plane of motion
00:41:50.000 --> 00:41:59.000
contains r and v.
So, what's the direction of the
00:41:59.000 --> 00:42:08.000
cross product r cross v?
Well, it's the direction that's
00:42:08.000 --> 00:42:19.000
perpendicular to this plane.
So, it's normal to the plane of
00:42:19.000 --> 00:42:24.000
motion.
And, that means, now,
00:42:24.000 --> 00:42:28.000
that actually we've put the two
statements in there into a
00:42:28.000 --> 00:42:33.000
single form because we are
saying r cross v has constant
00:42:33.000 --> 00:42:37.000
length and constant direction.
In fact, in general,
00:42:37.000 --> 00:42:40.000
maybe I should say something
about this.
00:42:40.000 --> 00:42:42.000
So, if you just look at the
position vector,
00:42:42.000 --> 00:42:45.000
and the velocity vector for any
motion at any given time,
00:42:45.000 --> 00:42:48.000
then together,
they determine some plane.
00:42:48.000 --> 00:42:51.000
And, that's the plane that
contains the origin,
00:42:51.000 --> 00:42:54.000
the point, and the velocity
vector.
00:42:54.000 --> 00:42:56.000
If you want,
it's the plane in which the
00:42:56.000 --> 00:42:59.000
motion seems to be going at the
given time.
00:42:59.000 --> 00:43:01.000
Now, of course,
if your motion is not in a
00:43:01.000 --> 00:43:03.000
plane, then that plane will
change.
00:43:03.000 --> 00:43:06.000
It's, however,
instant, if a plane in which
00:43:06.000 --> 00:43:09.000
the motion is taking place at a
given time.
00:43:09.000 --> 00:43:13.000
And, to say that the motion
actually stays in that plane
00:43:13.000 --> 00:43:17.000
forever means that this guy will
not change direction.
00:43:17.000 --> 00:43:25.000
OK, so -- [LAUGHTER]
[APPLAUSE]
00:43:25.000 --> 00:43:42.000
OK, so, Kepler's second law is
actually equivalent to saying
00:43:42.000 --> 00:43:55.000
that r cross v equals a constant
vector, OK?
00:43:55.000 --> 00:44:04.000
That's what the law says.
So, in terms of derivatives,
00:44:04.000 --> 00:44:14.000
it means d by dt of r cross v
is the zero vector.
00:44:14.000 --> 00:44:20.000
OK, now, so there's an
interesting thing to note,
00:44:20.000 --> 00:44:23.000
which is that we can use the
usual product rule for
00:44:23.000 --> 00:44:26.000
derivatives with vector
expressions,
00:44:26.000 --> 00:44:28.000
with dot products or cross
products.
00:44:28.000 --> 00:44:30.000
There's only one catch,
which is that when we
00:44:30.000 --> 00:44:34.000
differentiate a cross product,
we have to be careful that the
00:44:34.000 --> 00:44:36.000
guy on the left stays on the
left.
00:44:36.000 --> 00:44:40.000
The guy on the right stays on
the right.
00:44:40.000 --> 00:44:44.000
OK, so, if you know that uv
prime equals u prime v plus uv
00:44:44.000 --> 00:44:47.000
prime, then you are safe.
If you know it as u prime v
00:44:47.000 --> 00:44:50.000
cross v prime u,
then you are not safe.
00:44:50.000 --> 00:44:52.000
OK, so it's the only thing to
watch for.
00:44:52.000 --> 00:45:05.000
So, product rule is OK for
taking the derivative of a dot
00:45:05.000 --> 00:45:10.000
product.
There, you don't actually even
00:45:10.000 --> 00:45:14.000
need to be very careful about
all the things or the derivative
00:45:14.000 --> 00:45:18.000
of a cross product.
There you just need to be a
00:45:18.000 --> 00:45:27.000
little bit more careful.
OK, so, now that we know that,
00:45:27.000 --> 00:45:39.000
we can write this as dr/dt
cross v plus r cross dv/dt,
00:45:39.000 --> 00:45:42.000
OK?
Well, let's reformulate things
00:45:42.000 --> 00:45:47.000
slightly.
So, dr dt already has a name.
00:45:47.000 --> 00:45:50.000
In fact, that's v.
OK, that's what we call the
00:45:50.000 --> 00:45:55.000
velocity vector.
So, this is v cross v plus r
00:45:55.000 --> 00:46:04.000
cross, what is dv/dt?
That's the acceleration,
00:46:04.000 --> 00:46:11.000
a, equals zero.
OK, so now what's the next step?
00:46:11.000 --> 00:46:15.000
Well, we know what v cross v is
because, remember,
00:46:15.000 --> 00:46:18.000
a vector cross itself is always
zero, OK?
00:46:18.000 --> 00:46:30.000
So, this is the same r cross a
equals zero,
00:46:30.000 --> 00:46:35.000
and that's the same as saying
that the cross product of two
00:46:35.000 --> 00:46:39.000
vectors is zero exactly when the
parallelogram of the form has no
00:46:39.000 --> 00:46:41.000
area.
And, the way in which that
00:46:41.000 --> 00:46:45.000
happens is if they are actually
parallel to each other.
00:46:45.000 --> 00:46:50.000
So, that means the acceleration
is parallel to the position.
00:46:50.000 --> 00:46:55.000
OK, so, in fact,
what Kepler's second law says
00:46:55.000 --> 00:47:02.000
is that the acceleration is
parallel to the position vector.
00:47:02.000 --> 00:47:05.000
And, since we know that
acceleration is caused by a
00:47:05.000 --> 00:47:08.000
force that's equivalent to the
fact that the gravitational
00:47:08.000 --> 00:47:08.000
force --
00:47:13.000 --> 00:47:18.000
-- is parallel to the position
vector, that means,
00:47:18.000 --> 00:47:22.000
well, if you have the sun here
at the origin,
00:47:22.000 --> 00:47:27.000
and if you have your planets,
well, the gravitational force
00:47:27.000 --> 00:47:32.000
caused by the sun should go
along this line.
00:47:32.000 --> 00:47:34.000
In fact, the law doesn't even
say whether it's going towards
00:47:34.000 --> 00:47:37.000
the sun or away from the sun.
Well, what we know now is that,
00:47:37.000 --> 00:47:39.000
of course, the attraction is
towards the sun.
00:47:39.000 --> 00:47:41.000
But, Kepler's law would also be
true, actually,
00:47:41.000 --> 00:47:44.000
if things were going away.
So, in particular,
00:47:44.000 --> 00:47:48.000
say, electric force also has
this property of being towards
00:47:48.000 --> 00:47:50.000
the central charge.
So, actually,
00:47:50.000 --> 00:47:54.000
if you look at motion of
charged particles in an electric
00:47:54.000 --> 00:47:58.000
field caused by a point charged
particle, it also satisfies
00:47:58.000 --> 00:48:01.000
Kepler's law,
satisfies the same law.
00:48:01.000 --> 00:48:03.000
OK, that's the end for today,
thanks.