Centripetal Acceleration (part 2)

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:01.450
Welcome back.

00:00:01.450 --> 00:00:04.582
So where I left off, I was
hopefully trying to give you

00:00:04.582 --> 00:00:07.650
an intuition of this inward
acceleration.

00:00:07.650 --> 00:00:09.820
In this case, I have a rock
spinning around, and I'm

00:00:09.820 --> 00:00:11.020
holding it on a string.

00:00:11.020 --> 00:00:15.100
And I want to think about what
the inward acceleration is

00:00:15.100 --> 00:00:16.440
dependent on.

00:00:16.440 --> 00:00:19.570
And so we talked a little bit
about its velocity in these

00:00:19.570 --> 00:00:21.250
multiple red arrows.

00:00:21.250 --> 00:00:22.740
These are potential
velocity vectors.

00:00:22.740 --> 00:00:25.760
So the point I was trying to
make is, if the velocity

00:00:25.760 --> 00:00:30.070
vector is bigger, to change its
direction requires even

00:00:30.070 --> 00:00:31.550
more acceleration.

00:00:31.550 --> 00:00:34.230
Because the change in the x
direction is bigger, and then

00:00:34.230 --> 00:00:35.780
the change in the y direction
is bigger.

00:00:35.780 --> 00:00:39.580
Because for example, right here,
it's y-- I'll draw it in

00:00:39.580 --> 00:00:41.540
a different color-- it's
y velocity vector.

00:00:41.540 --> 00:00:44.350
And I'll do it for
the big velocity.

00:00:44.350 --> 00:00:46.100
Its y velocity vector
was here.

00:00:46.100 --> 00:00:49.720
And then now its y velocity
vector is almost essentially

00:00:49.720 --> 00:00:52.150
the whole vector.

00:00:52.150 --> 00:00:54.490
So when the velocity increases,
and let's say this

00:00:54.490 --> 00:00:55.940
is the case of the big arrow.

00:00:55.940 --> 00:00:59.480
If the velocity was less, like
the small arrow, its velocity

00:00:59.480 --> 00:01:05.890
of vectors was here, and it
would have to go to here.

00:01:05.890 --> 00:01:08.480
So I don't know if it's
completely obvious, but the

00:01:08.480 --> 00:01:11.220
change in velocity, for example
in the y direction

00:01:11.220 --> 00:01:14.770
over this course of time,
has to be more.

00:01:14.770 --> 00:01:17.140
And likewise, the change in x
direction over this, and this

00:01:17.140 --> 00:01:20.500
might be more obvious, because
here this is the x velocity

00:01:20.500 --> 00:01:22.810
vector if the velocity
was less.

00:01:22.810 --> 00:01:25.840
And this is the x velocity
vector if it was more.

00:01:25.840 --> 00:01:27.020
And that's going to 0.

00:01:27.020 --> 00:01:29.060
Between when the dot goes
from here to here.

00:01:29.060 --> 00:01:31.920
Here it has no x component.

00:01:31.920 --> 00:01:34.672
So clearly, I would have had
to decelerate in the x

00:01:34.672 --> 00:01:39.420
direction more, if we have
a larger velocity vector.

00:01:39.420 --> 00:01:41.315
I'm either giving you an
intuition, or I'm confusing

00:01:41.315 --> 00:01:43.450
you, but I'll keep going.

00:01:43.450 --> 00:01:48.220
So velocity clearly-- the
greater the velocity, the more

00:01:48.220 --> 00:01:51.290
I'm going to have to pull
in on this one.

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The more the inward acceleration
has to be to keep

00:01:53.800 --> 00:01:55.730
this thing going
in the circle.

00:01:55.730 --> 00:01:57.840
And not only does the
acceleration have to be bigger

00:01:57.840 --> 00:02:00.295
just to get that velocity vector
down, but you've got to

00:02:00.295 --> 00:02:05.490
realize when, between this
point and this point, the

00:02:05.490 --> 00:02:08.360
faster I go, the less time it's
going to take me to get

00:02:08.360 --> 00:02:10.340
from here to here.

00:02:10.340 --> 00:02:13.980
So not only am I going to have
to change the velocity more, I

00:02:13.980 --> 00:02:16.670
have to do that in less time.

00:02:16.670 --> 00:02:20.310
So the velocity is affecting
this acceleration that I need

00:02:20.310 --> 00:02:21.390
to pull in two ways.

00:02:21.390 --> 00:02:24.650
So the higher the velocity, I
have to change more of the

00:02:24.650 --> 00:02:27.290
velocity, and I have to
do it in less time.

00:02:27.290 --> 00:02:30.170
And it actually turns out
that the acceleration is

00:02:30.170 --> 00:02:34.410
proportional to the
velocity squared.

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And then there's another
term, and I won't

00:02:35.680 --> 00:02:36.360
prove it in this video.

00:02:36.360 --> 00:02:38.240
I'm going to prove it
in another video.

00:02:38.240 --> 00:02:40.360
And that might be out of your
scope just now, because it

00:02:40.360 --> 00:02:42.120
requires some calculus.

00:02:42.120 --> 00:02:43.420
But I want to give
you an intuition.

00:02:43.420 --> 00:02:45.670
So we know that the acceleration
is dependent on

00:02:45.670 --> 00:02:46.880
the velocity squared.

00:02:46.880 --> 00:02:49.800
And I want to make that point of
why it's not just velocity,

00:02:49.800 --> 00:02:55.390
because there's two ways that
the velocity is affecting how

00:02:55.390 --> 00:02:56.980
much of an acceleration
I need.

00:02:56.980 --> 00:02:58.880
One, is just the magnitude
of the velocity.

00:02:58.880 --> 00:03:02.170
The more that is, the more I
have to accelerate to change

00:03:02.170 --> 00:03:03.070
this direction.

00:03:03.070 --> 00:03:06.800
And the second is, between this
point and this point, I

00:03:06.800 --> 00:03:09.190
have less time to change
that velocity.

00:03:09.190 --> 00:03:11.470
So the velocity affects
it in two ways.

00:03:11.470 --> 00:03:14.370
And that's where you get an
intuition for the v squared.

00:03:14.370 --> 00:03:17.780
The other thing that matters
is actually the radius.

00:03:17.780 --> 00:03:20.960
So let's say this
is the radius.

00:03:20.960 --> 00:03:22.990
And the acceleration
is inversely

00:03:22.990 --> 00:03:24.850
proportional to the radius.

00:03:24.850 --> 00:03:26.180
And why is that?

00:03:26.180 --> 00:03:28.700
Well, it's kind of like the
second, I guess we can call

00:03:28.700 --> 00:03:32.940
it, velocity argument I gave.
The bigger the radius.

00:03:32.940 --> 00:03:35.260
So let's say that, let's look at
this rate and let's look at

00:03:35.260 --> 00:03:37.480
another radius that is bigger.

00:03:37.480 --> 00:03:39.350
Let's say that this
is another radius.

00:03:39.350 --> 00:03:41.040
Let's say if the string
was longer and

00:03:41.040 --> 00:03:42.290
this was another case.

00:03:44.860 --> 00:03:46.830
Let's says this was the case.

00:03:46.830 --> 00:03:48.830
But the velocity was the same.

00:03:48.830 --> 00:03:53.110
So this is the case with a
larger-- so let's say it's the

00:03:53.110 --> 00:03:54.930
same velocity vector.

00:03:54.930 --> 00:03:56.890
So we have the same thing
at play, to go from

00:03:56.890 --> 00:03:58.600
there to say, there.

00:04:01.190 --> 00:04:05.320
We have to change the velocity
the same magnitude.

00:04:05.320 --> 00:04:10.250
As we had to change the velocity
from here to here.

00:04:13.050 --> 00:04:14.920
But one thing changes.

00:04:14.920 --> 00:04:18.099
This distance is longer.

00:04:18.099 --> 00:04:20.430
So if this velocity is the same
as this velocity, but

00:04:20.430 --> 00:04:23.830
this distance is longer,
you have more time

00:04:23.830 --> 00:04:25.550
to change the velocity.

00:04:25.550 --> 00:04:29.140
So the larger the radius, the
more time you have to change

00:04:29.140 --> 00:04:30.240
the velocity.

00:04:30.240 --> 00:04:32.780
And so the less acceleration
you need.

00:04:32.780 --> 00:04:35.590
Hopefully that makes sense.

00:04:35.590 --> 00:04:37.170
Let me repeat it.

00:04:37.170 --> 00:04:40.280
Because I think it's nice
to have the intuition.

00:04:40.280 --> 00:04:43.730
Between this point and this
point, unless you have a

00:04:43.730 --> 00:04:48.040
smaller radius, if I want to
change the acceleration vector

00:04:48.040 --> 00:04:55.380
from that to that, I have
however long it takes to go

00:04:55.380 --> 00:04:56.030
this distance.

00:04:56.030 --> 00:05:00.660
I have that long to change
its velocity vector.

00:05:00.660 --> 00:05:02.990
But if the radius was
a little longer.

00:05:02.990 --> 00:05:05.840
And I had the same velocity, so
here the velocity is that.

00:05:05.840 --> 00:05:08.680
And then here, the
velocity is that.

00:05:08.680 --> 00:05:12.180
I have a little bit more time
to change its velocity.

00:05:12.180 --> 00:05:15.470
And remember, acceleration is
just change in velocity over

00:05:15.470 --> 00:05:16.890
change in time.

00:05:16.890 --> 00:05:21.600
And so if I have the same change
in velocity, but I have

00:05:21.600 --> 00:05:23.250
more time to do it with.

00:05:23.250 --> 00:05:25.170
This is a bigger number.

00:05:25.170 --> 00:05:28.370
The required acceleration
is less.

00:05:28.370 --> 00:05:30.330
So maybe I've given
you an intuition.

00:05:30.330 --> 00:05:31.630
And I will prove
this for sure.

00:05:31.630 --> 00:05:38.150
Because this is kind of a very
touchy, very soft way, and not

00:05:38.150 --> 00:05:42.000
very rigorous way of proving
that the required acceleration

00:05:42.000 --> 00:05:44.490
to make something going in a
circle is v squared over r.

00:05:44.490 --> 00:05:47.270
But accept that as a bit of
faith, and I'll prove it.

00:05:47.270 --> 00:05:48.960
Especially once you learn your
calculus, and you can watch

00:05:48.960 --> 00:05:49.810
that video.

00:05:49.810 --> 00:05:53.890
And we could do a couple of
problems. And this is probably

00:05:53.890 --> 00:05:54.930
what you care about anyway.

00:05:54.930 --> 00:05:58.100
But I think it's nice to
have that intuition.

00:05:58.100 --> 00:06:05.880
So once again, let's say
we're in deep space.

00:06:08.700 --> 00:06:11.280
Let's do a little
orbit problem.

00:06:11.280 --> 00:06:12.290
Let say that this is Earth.

00:06:12.290 --> 00:06:15.420
I will do Earth in blue.

00:06:15.420 --> 00:06:18.480
Let's say that is Earth.

00:06:18.480 --> 00:06:23.610
And I have a satellite that
is going around Earth.

00:06:23.610 --> 00:06:29.040
And I'm not going to calculate
its-- this is a satellite

00:06:29.040 --> 00:06:31.370
that's going around Earth.

00:06:31.370 --> 00:06:33.730
And I'm going to do a whole
other video on actually

00:06:33.730 --> 00:06:37.300
figuring out the gravitational
force depending on how far

00:06:37.300 --> 00:06:38.530
away you are from an object.

00:06:38.530 --> 00:06:43.010
But let's just assume that its
gravitational force is similar

00:06:43.010 --> 00:06:44.800
to what it is on the
surface of Earth.

00:06:44.800 --> 00:06:47.540
And that is a big assumption,
or let's assume this is a

00:06:47.540 --> 00:06:50.170
different planet and I'm going
to give you the gravitational

00:06:50.170 --> 00:06:51.300
acceleration.

00:06:51.300 --> 00:06:56.590
Let's say that the planet, and
maybe this isn't Earth, is

00:06:56.590 --> 00:06:57.520
pulling inwards.

00:06:57.520 --> 00:06:59.340
It always pulls towards
its center.

00:06:59.340 --> 00:07:02.220
And I'll do that in
a different color.

00:07:02.220 --> 00:07:07.700
That it's pulling inwards
at 30 feet per second.

00:07:07.700 --> 00:07:13.200
My question is, what does the
velocity of the satellite have

00:07:13.200 --> 00:07:16.890
to be for the object
to stay in orbit?

00:07:16.890 --> 00:07:19.670
Well, we could just apply the
formula and then maybe we can

00:07:19.670 --> 00:07:21.380
have a little intuition.

00:07:21.380 --> 00:07:24.040
So acceleration is equal--
oh, I'm sorry.

00:07:24.040 --> 00:07:38.810
Let's say that the object is
6,000 feet from the center.

00:07:38.810 --> 00:07:41.970
And when we learn gravity, we'll
learn actually that it's

00:07:41.970 --> 00:07:44.670
the distance from the radius
that you're going around is

00:07:44.670 --> 00:07:45.640
actually relative
to the center.

00:07:45.640 --> 00:07:47.260
Well, actually it's always
going to be,

00:07:47.260 --> 00:07:49.900
because this is a circle.

00:07:49.900 --> 00:07:50.560
So, the radius.

00:07:50.560 --> 00:07:51.800
This is the center
of that circle.

00:07:51.800 --> 00:07:52.760
You don't have to worry
about that.

00:07:52.760 --> 00:07:54.330
It has nothing to
do with gravity.

00:07:54.330 --> 00:07:57.990
OK, so the acceleration
is v squared over r.

00:07:57.990 --> 00:08:02.590
Acceleration, we know is
30 feet per second.

00:08:02.590 --> 00:08:05.200
We want to figure out the
velocity, and then the radius,

00:08:05.200 --> 00:08:09.330
I said, is 6,000 feet.

00:08:09.330 --> 00:08:10.820
So let's ignore the
units for now.

00:08:10.820 --> 00:08:13.530
This could make things
complex.

00:08:13.530 --> 00:08:16.260
Multiply both sides of this
equation times 6,000, so 3

00:08:16.260 --> 00:08:18.080
times 6 is 18.

00:08:18.080 --> 00:08:18.980
You'll have four 0's.

00:08:18.980 --> 00:08:21.690
2, 3, 4.

00:08:21.690 --> 00:08:24.200
So v squared is equal
to 18,000.

00:08:24.200 --> 00:08:26.120
Everything's in feet
per second.

00:08:26.120 --> 00:08:27.110
And so what's v?

00:08:27.110 --> 00:08:30.922
So v is equal to the square
root of 180,000.

00:08:30.922 --> 00:08:34.159
Let me get the calculator out,
because I haven't memorized my

00:08:34.159 --> 00:08:37.470
perfect squares that high.

00:08:37.470 --> 00:08:39.100
1, 2, 3.

00:08:39.100 --> 00:08:42.770
Take the square root, so 424.

00:08:42.770 --> 00:08:48.530
So the required velocity
is 424 feet per second.

00:08:48.530 --> 00:08:51.660
So if the satellite is at that
velocity, it will go in

00:08:51.660 --> 00:08:54.970
perfect orbit around
this planet.

00:08:54.970 --> 00:08:56.860
Now what happens if
this object's

00:08:56.860 --> 00:09:00.560
velocity is a little slower.

00:09:00.560 --> 00:09:02.280
What happens-- let me do it in
a different color so you know

00:09:02.280 --> 00:09:03.680
what I'm talking about.

00:09:03.680 --> 00:09:05.790
What happens if its velocity
is a little

00:09:05.790 --> 00:09:06.400
bit less than that?

00:09:06.400 --> 00:09:11.820
Let's say its is 300
feet per second.

00:09:11.820 --> 00:09:14.720
Well, then it's not going to be
able to travel far enough

00:09:14.720 --> 00:09:20.180
tangent to the circle
to-- well, it'll

00:09:20.180 --> 00:09:21.630
essentially just do this.

00:09:21.630 --> 00:09:24.480
It'll always be getting
a little bit closer.

00:09:24.480 --> 00:09:28.520
And it'll spiral in and
it'll hit the planet.

00:09:28.520 --> 00:09:32.510
And another way to think about
what the required velocity is,

00:09:32.510 --> 00:09:36.520
you have to go fast enough so
that you're always falling in.

00:09:36.520 --> 00:09:39.180
But every time you fall in, you
want to go in this case,

00:09:39.180 --> 00:09:42.440
the right enough so that you're
an equal distant--

00:09:42.440 --> 00:09:44.540
you're still the same distance
away from the center.

00:09:44.540 --> 00:09:46.630
So that your distance from the
center never changes if you're

00:09:46.630 --> 00:09:47.950
going fast enough.

00:09:47.950 --> 00:09:50.950
And then what happens if its
velocity is too fast?

00:09:50.950 --> 00:09:53.530
What happens in the case
where its velocity is

00:09:53.530 --> 00:09:56.800
600 feet per second?

00:09:56.800 --> 00:10:01.010
Well, in that case, the object
will have enough velocity to

00:10:01.010 --> 00:10:04.020
actually escape the orbit
of the planet.

00:10:04.020 --> 00:10:06.020
But anyway, that's enough
about that.

00:10:06.020 --> 00:10:08.390
I'm actually out of
time, so I'll see

00:10:08.390 --> 00:10:09.910
you in the next video.


Introduction to centripetal acceleration (part 1)

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:03.960
Let's learn a little bit about
centripetal acceleration.

00:00:03.960 --> 00:00:05.690
So let's say I have a dot.

00:00:05.690 --> 00:00:06.550
It could be anything.

00:00:06.550 --> 00:00:08.160
And we're in deep space, so
we're not going to think about

00:00:08.160 --> 00:00:10.080
gravity and all these
things yet.

00:00:10.080 --> 00:00:13.540
Let's say I have some object
that's floating through space

00:00:13.540 --> 00:00:14.370
with some velocity.

00:00:14.370 --> 00:00:15.210
Let me draw the object.

00:00:15.210 --> 00:00:17.800
It's this dot.

00:00:17.800 --> 00:00:19.250
And it has some velocity.

00:00:19.250 --> 00:00:23.660
And I will draw its
velocity vector.

00:00:23.660 --> 00:00:27.200
Let's say that this is-- I'll
draw this really big.

00:00:27.200 --> 00:00:28.020
That's not big.

00:00:28.020 --> 00:00:29.270
Let me make a thicker line.

00:00:31.510 --> 00:00:32.140
There you go.

00:00:32.140 --> 00:00:34.000
Thicker line.

00:00:34.000 --> 00:00:35.950
If you can see that.

00:00:35.950 --> 00:00:38.550
So let's say that's the
dot's velocity vector.

00:00:41.560 --> 00:00:43.130
And I have a question for you.

00:00:43.130 --> 00:00:45.190
So this is what the dot's doing
at exactly this moment.

00:00:45.190 --> 00:00:48.170
Let's say a little bit later,
I want the dot to be doing

00:00:48.170 --> 00:00:49.700
something like this.

00:00:49.700 --> 00:00:55.010
I want its velocity vector
to look like-- i want its

00:00:55.010 --> 00:00:58.240
velocity factor to
look like this.

00:00:58.240 --> 00:00:59.490
A little bit later.

00:01:01.900 --> 00:01:05.630
And its color changes
to yellow.

00:01:05.630 --> 00:01:06.450
That's out of the scope.

00:01:06.450 --> 00:01:08.270
Color changing is out of the
scope of this lecture.

00:01:08.270 --> 00:01:11.620
But let's say a little bit
later I want its velocity

00:01:11.620 --> 00:01:16.880
vector to look something
like that.

00:01:16.880 --> 00:01:18.720
So I was a little careful,
because I want it to have the

00:01:18.720 --> 00:01:21.950
same magnitude, but just a
little different direction.

00:01:21.950 --> 00:01:23.980
I want it to tilt into the
right a little bit.

00:01:23.980 --> 00:01:27.650
Let me draw that.

00:01:27.650 --> 00:01:30.780
And my question is, what had to
be the acceleration acting,

00:01:30.780 --> 00:01:33.630
when this was its velocity
vector, in order to get its

00:01:33.630 --> 00:01:37.070
velocity vector to look like
this a little bit later?

00:01:37.070 --> 00:01:38.840
Well, right here, its
velocity vector is

00:01:38.840 --> 00:01:41.840
completely in the x direction.

00:01:41.840 --> 00:01:43.510
And there's no y component.

00:01:43.510 --> 00:01:45.140
Let's break down this
velocity vector

00:01:45.140 --> 00:01:47.240
into the x and y component.

00:01:47.240 --> 00:01:50.818
So its x component looks
something like this.

00:01:50.818 --> 00:01:54.725
And its y component looks
something like this.

00:01:57.530 --> 00:02:01.140
So in order to change its
velocity, something would have

00:02:01.140 --> 00:02:04.710
to-- Because clearly here
its y component was 0.

00:02:04.710 --> 00:02:06.260
There's no y component.

00:02:06.260 --> 00:02:07.570
And now there is
a y component.

00:02:07.570 --> 00:02:10.310
So the acceleration that you
would have to apply to this

00:02:10.310 --> 00:02:13.720
dot would have to have
some y component.

00:02:13.720 --> 00:02:17.090
The acceleration would have
to have some y component.

00:02:17.090 --> 00:02:18.700
Let's say that.

00:02:18.700 --> 00:02:20.660
This is acceleration, that's
why it's not the same size.

00:02:20.660 --> 00:02:21.593
Actually let me draw
acceleration

00:02:21.593 --> 00:02:23.370
in a different color.

00:02:23.370 --> 00:02:25.160
And I'll try to do this in a
bunch of different ways, so

00:02:25.160 --> 00:02:28.036
that you get an intuition
for it.

00:02:28.036 --> 00:02:30.690
I'm doing it in a slightly
different blue.

00:02:30.690 --> 00:02:33.430
So you have to accelerate in the
y direction a bit, and we

00:02:33.430 --> 00:02:36.210
don't know how long, but long
enough to get its velocity in

00:02:36.210 --> 00:02:38.360
the y direction this big.

00:02:38.360 --> 00:02:39.370
And then what would you have to

00:02:39.370 --> 00:02:41.460
accelerate in the x direction?

00:02:41.460 --> 00:02:44.760
Well, your x component
shrinks a little bit.

00:02:44.760 --> 00:02:47.170
This was its x component, and
now since you're tilted down,

00:02:47.170 --> 00:02:49.510
you get more y and less x.

00:02:49.510 --> 00:02:53.770
So what direction does
acceleration have to go in to

00:02:53.770 --> 00:02:59.280
shrink the x component?

00:02:59.280 --> 00:03:00.690
You have to go opposite.

00:03:00.690 --> 00:03:04.020
So your acceleration is going
to go that way a little bit.

00:03:04.020 --> 00:03:06.415
And so if you were to add these
two vectors, you would

00:03:06.415 --> 00:03:08.620
have to accelerate something--
the acceleration vector would

00:03:08.620 --> 00:03:09.870
look something like this.

00:03:17.350 --> 00:03:20.740
So I'm-- And this might be a
little confusing to you, but

00:03:20.740 --> 00:03:25.385
all I'm showing is that, I want
you to think about-- Say

00:03:25.385 --> 00:03:28.060
an object is traveling through
space at a constant velocity,

00:03:28.060 --> 00:03:30.540
what does the acceleration
vector have to look like in

00:03:30.540 --> 00:03:33.310
order for that object to
curve a little bit?

00:03:33.310 --> 00:03:36.670
So for example, if this was
the-- let me draw, let's say

00:03:36.670 --> 00:03:39.400
this object was here.

00:03:39.400 --> 00:03:42.530
And if we had no force on it,
it would just keep moving in

00:03:42.530 --> 00:03:43.930
this direction.

00:03:43.930 --> 00:03:46.220
Its path would look like this.

00:03:46.220 --> 00:03:47.260
I'll draw the path in brown.

00:03:47.260 --> 00:03:48.210
The path would look like this.

00:03:48.210 --> 00:03:49.330
It would be here, and then
it would be here

00:03:49.330 --> 00:03:50.400
a few seconds later.

00:03:50.400 --> 00:03:52.220
It would just keep going in the
direction that it's at.

00:03:52.220 --> 00:03:54.430
And we know that from Newton's
law, an object in motion tends

00:03:54.430 --> 00:03:55.730
to stay in motion.

00:03:55.730 --> 00:03:59.120
And the only way you can have a
change in velocity is if you

00:03:59.120 --> 00:04:01.840
have some net force
and some net

00:04:01.840 --> 00:04:04.160
acceleration on the object.

00:04:04.160 --> 00:04:06.410
So in order for the object to
curve, in order for the

00:04:06.410 --> 00:04:10.080
object's path to look something
like this-- so

00:04:10.080 --> 00:04:11.010
here's the other path.

00:04:11.010 --> 00:04:15.600
Let's say the object's
path is like this.

00:04:15.600 --> 00:04:17.480
And let's say it
keeps curving.

00:04:17.480 --> 00:04:18.620
The object's path
is like that.

00:04:18.620 --> 00:04:19.940
It keeps curving.

00:04:19.940 --> 00:04:21.070
Goes there, to there.

00:04:21.070 --> 00:04:22.670
It's going in a curved motion.

00:04:22.670 --> 00:04:24.720
What has to keep happening?

00:04:24.720 --> 00:04:27.240
Well, in order for the object's
velocity to go from

00:04:27.240 --> 00:04:31.330
this direction, in the x
direction, in order for its

00:04:31.330 --> 00:04:33.750
velocity to go from this,
to go from this.

00:04:33.750 --> 00:04:36.130
You have to accelerate
a little bit inwards.

00:04:36.130 --> 00:04:36.990
We drew that right here.

00:04:36.990 --> 00:04:39.450
You have to have an inward
acceleration.

00:04:39.450 --> 00:04:41.800
And now its velocity-- and now
if you did nothing else, and

00:04:41.800 --> 00:04:44.730
the object will just keep
going in this direction.

00:04:44.730 --> 00:04:47.560
So in order to make it curve
little more, you've got to

00:04:47.560 --> 00:04:48.580
pull in a little bit again.

00:04:48.580 --> 00:04:51.770
So you have to have-- and then
if you don't pull in again,

00:04:51.770 --> 00:04:53.620
it'll just keep going
in this direction.

00:04:53.620 --> 00:04:55.560
So you got to keep pulling
inwards on it.

00:04:58.530 --> 00:05:01.250
And I think you're starting
to get a sense of what I'm

00:05:01.250 --> 00:05:04.930
saying, so if you want the
object to go in a circle, for

00:05:04.930 --> 00:05:09.210
example, there has to be
constant inward force always

00:05:09.210 --> 00:05:10.560
pulling on the object.

00:05:10.560 --> 00:05:15.190
Because if there isn't that
constant inward-- so this is

00:05:15.190 --> 00:05:17.270
this off white color, this
is when there's a

00:05:17.270 --> 00:05:18.310
constant inward force.

00:05:18.310 --> 00:05:20.030
The object's going to travel
in a circular path.

00:05:23.910 --> 00:05:26.600
Well, what if that inward
force doesn't exist?

00:05:26.600 --> 00:05:28.710
Let's say that inward force is
there, and then all of a

00:05:28.710 --> 00:05:31.460
sudden when the object's here,
the inward force disappears.

00:05:31.460 --> 00:05:34.100
Then the object's just going
to go in a straight line.

00:05:34.100 --> 00:05:37.280
Tangent to the circular path.

00:05:37.280 --> 00:05:37.830
And that makes more sense.

00:05:37.830 --> 00:05:38.980
And I'll go into it
in a little bit

00:05:38.980 --> 00:05:39.720
more detail in a second.

00:05:39.720 --> 00:05:42.170
But if you are spinning
an object around in

00:05:42.170 --> 00:05:43.490
a circle on a string.

00:05:43.490 --> 00:05:46.490
And the string is providing the
inward force, and as soon

00:05:46.490 --> 00:05:49.730
as you let go of the string,
the object goes off in a

00:05:49.730 --> 00:05:50.500
straight line.

00:05:50.500 --> 00:05:53.320
And so that that's what
we're talking about.

00:05:53.320 --> 00:05:56.215
So what does this inward
acceleration has-- actually,

00:05:56.215 --> 00:05:58.170
before I go into actually
calculating what that inward

00:05:58.170 --> 00:06:00.940
acceleration has to be, let's
think about what happens when

00:06:00.940 --> 00:06:02.860
that inwards acceleration
isn't enough

00:06:02.860 --> 00:06:04.110
or if it's too much?

00:06:09.650 --> 00:06:12.290
So I just said, if I have an--
let's say this is like the

00:06:12.290 --> 00:06:13.510
center of our rotation.

00:06:13.510 --> 00:06:16.160
Let's say this is where the
inward force is coming from.

00:06:16.160 --> 00:06:17.140
Right there.

00:06:17.140 --> 00:06:18.770
And let's say this
is my object.

00:06:18.770 --> 00:06:21.070
Let say it's a spaceship.

00:06:21.070 --> 00:06:22.320
That's my spaceship.

00:06:25.110 --> 00:06:27.440
Fire coming out of it.

00:06:27.440 --> 00:06:29.410
And actually the fire would
only go on an impulse.

00:06:29.410 --> 00:06:31.300
Because once you have a little
fire coming out in space,

00:06:31.300 --> 00:06:33.480
there's no wind resistance,
so it'll keep just going.

00:06:33.480 --> 00:06:35.850
You don't have to keep-- so the
science fiction movies,

00:06:35.850 --> 00:06:38.310
where the fire just
keeps going, that

00:06:38.310 --> 00:06:39.010
doesn't make sense.

00:06:39.010 --> 00:06:41.400
You would just have an impulse
of fire that would accelerate

00:06:41.400 --> 00:06:43.690
you, and then you wouldn't have
to keep fueling the fire

00:06:43.690 --> 00:06:44.820
in the back of the spaceship.

00:06:44.820 --> 00:06:47.455
But anyway, maybe you realize
it already, or that's out of

00:06:47.455 --> 00:06:50.180
the scope of this lecture, but
let's say that the object's

00:06:50.180 --> 00:06:52.580
velocity is like that.

00:06:52.580 --> 00:06:54.420
And I don't know, let's say this
is a planet, or it's some

00:06:54.420 --> 00:06:57.830
kind of weird forcefield,
or something.

00:06:57.830 --> 00:07:02.520
If this forcefield is a weak
forcefield, if this object is

00:07:02.520 --> 00:07:06.100
moving really fast, and this
is a weak forcefield that's

00:07:06.100 --> 00:07:08.540
always pulling in towards it,
the object's path is going to

00:07:08.540 --> 00:07:09.790
look something like this.

00:07:14.250 --> 00:07:15.490
It's not going to be a circle.

00:07:15.490 --> 00:07:16.750
It's going to kind of
spiral outwards.

00:07:20.530 --> 00:07:22.620
So that's kind of a
weak inward force.

00:07:22.620 --> 00:07:25.693
If it's a really strong inward
force, the object's path is

00:07:25.693 --> 00:07:26.650
going to look like this.

00:07:26.650 --> 00:07:29.250
And I'll do that in yellow.

00:07:29.250 --> 00:07:31.290
It's actually going
to fall in.

00:07:31.290 --> 00:07:34.310
It's going to spiral
into the object.

00:07:34.310 --> 00:07:37.540
And if it's just right, given
how far the object is, and

00:07:37.540 --> 00:07:40.100
I'll give you a formula for
what just right implies.

00:07:40.100 --> 00:07:43.680
But if the inward force is just
right, the object's going

00:07:43.680 --> 00:07:46.960
to essentially orbit
around the subject.

00:07:46.960 --> 00:07:48.380
It's going to go in
a perfect circle.

00:07:48.380 --> 00:07:53.620
Its path will be like that.

00:07:53.620 --> 00:07:56.460
So let's think about what that
inward force has to be.

00:07:56.460 --> 00:07:58.730
And we're not going to-- well,
this could have been gravity,

00:07:58.730 --> 00:08:01.870
this inward force, but I won't
work with gravity yet, because

00:08:01.870 --> 00:08:04.300
gravity, actually the force
changes depending on how close

00:08:04.300 --> 00:08:05.190
you get to it.

00:08:05.190 --> 00:08:07.020
So I won't deal with
gravity yet.

00:08:07.020 --> 00:08:10.970
But it could've been anything,
it's just a tractor beam.

00:08:10.970 --> 00:08:15.360
So let's try to get an intuition
of how strong that

00:08:15.360 --> 00:08:20.280
inward force is, and what
variables it's dependent on.

00:08:20.280 --> 00:08:26.100
So let's say I have a let's
say this is my hand, and I

00:08:26.100 --> 00:08:27.350
have a string.

00:08:30.920 --> 00:08:32.950
And on that string,
I have a rock.

00:08:39.049 --> 00:08:41.980
And my question is, if I want
this rock just spinning around

00:08:41.980 --> 00:08:44.400
in a circle, and it'll spin
around in a perfect circle.

00:08:44.400 --> 00:08:46.720
We've all done this before.

00:08:46.720 --> 00:08:51.180
If I want the rock to spin
around in that circle, what

00:08:51.180 --> 00:08:54.330
has to be the force that
I pull on this string?

00:08:54.330 --> 00:08:58.320
Or essentially, how much does
the inward acceleration have

00:08:58.320 --> 00:09:00.280
to be on that rock?

00:09:00.280 --> 00:09:04.870
Let's get an intuition for
maybe what has to happen.

00:09:04.870 --> 00:09:08.430
So think about it, if the rock
is moving, let's say that

00:09:08.430 --> 00:09:10.750
right here, at this point,
its velocity looks

00:09:10.750 --> 00:09:12.700
something like that.

00:09:12.700 --> 00:09:15.310
At this point, it's velocity
looks something like this.

00:09:15.310 --> 00:09:18.480
And then when the rock is
here, its velocity looks

00:09:18.480 --> 00:09:19.730
something like this.

00:09:21.950 --> 00:09:24.650
So to move from this velocity
to this velocity, if the

00:09:24.650 --> 00:09:27.990
magnitude is really big, I'm
going to have to apply more

00:09:27.990 --> 00:09:29.990
acceleration to change
direction.

00:09:29.990 --> 00:09:32.170
Hopefully that makes
a little sense.

00:09:32.170 --> 00:09:34.130
To go from this velocity
vector to

00:09:34.130 --> 00:09:35.730
this velocity vector.

00:09:35.730 --> 00:09:38.390
If this vector was even bigger,
if to say the vector

00:09:38.390 --> 00:09:41.330
looked like this, it
had more magnitude.

00:09:41.330 --> 00:09:43.910
I would have to pull inward
with even more force and

00:09:43.910 --> 00:09:47.000
accelerate inward at an
even higher rate.

00:09:47.000 --> 00:09:50.290
So definitely, the higher the
velocity the more I'm going to

00:09:50.290 --> 00:09:51.410
have to pull in.

00:09:51.410 --> 00:09:54.400
And also the higher the
velocity, the shorter the time

00:09:54.400 --> 00:09:55.580
between this and this.

00:09:55.580 --> 00:09:57.620
This object is moving
around in a circle.

00:09:57.620 --> 00:09:59.540
So there's kind of two
components that are

00:09:59.540 --> 00:10:02.180
affecting-- velocity affects
the acceleration

00:10:02.180 --> 00:10:03.840
I need in two ways.

00:10:03.840 --> 00:10:05.620
Well, actually I'm running out
of time, so I'll continue this

00:10:05.620 --> 00:10:07.220
in the next video.


Projectile motion with ordered set notation

WEBVTT
Kind: captions
Language: en

00:00:00.790 --> 00:00:01.530
Welcome back.

00:00:01.530 --> 00:00:03.920
I now want to introduce you
really just to a different

00:00:03.920 --> 00:00:06.330
notation for writing vectors,
and then we'll do that same

00:00:06.330 --> 00:00:08.540
problem or a slight variation
on that problem

00:00:08.540 --> 00:00:09.680
using the new notation.

00:00:09.680 --> 00:00:11.760
This is just to expose you to
things, so that you don't get

00:00:11.760 --> 00:00:14.830
confused if your teacher uses a
different notation than what

00:00:14.830 --> 00:00:15.770
I've been doing.

00:00:15.770 --> 00:00:18.090
So when we did the unit vectors,
we learned that we

00:00:18.090 --> 00:00:21.550
can express a vector as a
component of its x- and

00:00:21.550 --> 00:00:22.130
y-components.

00:00:22.130 --> 00:00:30.270
So let's say I had a vector--
let me just pick a random

00:00:30.270 --> 00:00:31.090
vector just to show you.

00:00:31.090 --> 00:00:37.780
So say I had vector a and that
equals 2 times the unit vector

00:00:37.780 --> 00:00:41.930
i plus 3 times the
unit vector j.

00:00:41.930 --> 00:00:43.430
That's the unit vector notation,
and I actually

00:00:43.430 --> 00:00:45.090
looked it up on Wikipedia, and
they actually called it the

00:00:45.090 --> 00:00:46.060
engineering notation.

00:00:46.060 --> 00:00:49.170
That's probably why I used it
because I am an engineer, or I

00:00:49.170 --> 00:00:52.830
was an engineer before
managing money.

00:00:52.830 --> 00:00:55.350
But another way to write this,
and I call this the bracket

00:00:55.350 --> 00:00:59.770
notation, or the ordered pair
notation, is you could also

00:00:59.770 --> 00:01:01.020
write it like this.

00:01:03.140 --> 00:01:05.050
We have this one bracket.

00:01:05.050 --> 00:01:09.180
That's the x-component, that's
the y-component.

00:01:09.180 --> 00:01:11.130
It almost looks like a
coordinate pair, but since

00:01:11.130 --> 00:01:12.990
they have the brackets, you
know it's a vector.

00:01:12.990 --> 00:01:15.290
But you would draw it
the exact same way.

00:01:15.290 --> 00:01:17.570
So given that, let's do
that same problem

00:01:17.570 --> 00:01:19.150
that we had just done.

00:01:19.150 --> 00:01:20.660
Hopefully, this make
sense to you.

00:01:20.660 --> 00:01:22.230
It's just a different
way of writing it.

00:01:22.230 --> 00:01:24.170
Instead of an i and a j, you
just write these brackets.

00:01:24.170 --> 00:01:26.611
Instead of a plus, you
write a comma.

00:01:26.611 --> 00:01:29.120
Let me clear this.

00:01:29.120 --> 00:01:30.230
I'm going to do a slight
variation.

00:01:30.230 --> 00:01:32.140
This was actually the second
part of that problem.

00:01:32.140 --> 00:01:33.560
My cousin gave these
problems to me.

00:01:33.560 --> 00:01:36.890
They're pretty good, so I figure
I'd stick with them.

00:01:36.890 --> 00:01:42.000
So in the old problem, let me
draw my coordinate axes again.

00:01:44.885 --> 00:01:47.370
That's the y-axis.

00:01:47.370 --> 00:01:48.915
That's the x-axis.

00:01:53.670 --> 00:01:56.740
So in the old problem, I started
off with a ball that

00:01:56.740 --> 00:01:57.970
was 4 feet off the ground.

00:01:57.970 --> 00:02:00.310
So let's say that's 4.

00:02:00.310 --> 00:02:07.760
And I hit it at 120 feet per
second at a 30-degree angle.

00:02:07.760 --> 00:02:10.710
So that's a 30-degree
angle like that.

00:02:13.332 --> 00:02:17.170
Its' a 30-degree angle
to the horizontal.

00:02:17.170 --> 00:02:23.580
And there's a fence 350 feet
away that's 30 feet high.

00:02:23.580 --> 00:02:24.660
It's roughly around there.

00:02:24.660 --> 00:02:26.020
That's 30.

00:02:26.020 --> 00:02:27.845
And what we need to do is figure
out whether the ball

00:02:27.845 --> 00:02:28.770
can clear the fence.

00:02:28.770 --> 00:02:31.190
We figured out the last time
when we used the unit vector

00:02:31.190 --> 00:02:33.430
notation that it doesn't
clear the fence.

00:02:33.430 --> 00:02:36.180
But in this problem, or the
second part of this problem,

00:02:36.180 --> 00:02:38.300
they said that there's
a 5 meter per second

00:02:38.300 --> 00:02:40.340
wind gust to the right.

00:02:40.340 --> 00:02:45.260
So there's a wind gust of 5
meters per second right when I

00:02:45.260 --> 00:02:46.190
hit the ball.

00:02:46.190 --> 00:02:48.305
And you could go into the
complications of how much does

00:02:48.305 --> 00:02:49.620
that accelerate the ball?

00:02:49.620 --> 00:02:51.410
Or what's the air resistance
of the ball?

00:02:51.410 --> 00:02:53.410
I think for the simplicity of
the problem, they're just

00:02:53.410 --> 00:02:56.670
saying that the x-component of
the ball's velocity right

00:02:56.670 --> 00:03:00.370
after you hit it increases
by 5 meters per second.

00:03:00.370 --> 00:03:01.540
I think that's their point.

00:03:01.540 --> 00:03:03.690
So let's go back and do the
problem the exact same way

00:03:03.690 --> 00:03:05.240
that we did it the last
time, but we'll

00:03:05.240 --> 00:03:07.090
use a different notation.

00:03:07.090 --> 00:03:09.750
So we can write that equation
that I had written before,

00:03:09.750 --> 00:03:16.110
that the position at any given
time as a function of t is

00:03:16.110 --> 00:03:19.870
equal to the initial position--
that's an i right

00:03:19.870 --> 00:03:24.480
there-- plus the initial
velocity.

00:03:24.480 --> 00:03:26.300
These are all vectors.

00:03:26.300 --> 00:03:36.040
Initial velocity times t plus
the acceleration vector over

00:03:36.040 --> 00:03:38.820
2t squared.

00:03:38.820 --> 00:03:40.330
So what's the initial
position?

00:03:40.330 --> 00:03:43.140
And now we're going to use
some of our new notation.

00:03:43.140 --> 00:03:50.690
The initial position when I hit
the ball, its x-component

00:03:50.690 --> 00:03:52.630
is 0, right?

00:03:52.630 --> 00:03:55.130
It's almost like its coordinate,
and they're not

00:03:55.130 --> 00:03:57.010
that different of a notation.

00:03:57.010 --> 00:04:00.920
And then the y-position is 4.

00:04:00.920 --> 00:04:02.510
Easy enough.

00:04:02.510 --> 00:04:05.710
What's its initial velocity?

00:04:05.710 --> 00:04:07.470
Let me do it.

00:04:07.470 --> 00:04:10.810
So we can split it up into the
x- and the y-components.

00:04:10.810 --> 00:04:18.420
The y-component is 120 sine of
30 degrees and then the x

00:04:18.420 --> 00:04:24.240
component is 120 cosine
of 30 degrees.

00:04:24.240 --> 00:04:26.410
That's just the x-component
after I hit it.

00:04:26.410 --> 00:04:27.860
But then they say there's
this wind gust so it's

00:04:27.860 --> 00:04:29.220
going to be plus 5.

00:04:29.220 --> 00:04:31.020
I think that's their point when
they say that there's

00:04:31.020 --> 00:04:33.130
this wind gust. They say that
right when you hit it, for

00:04:33.130 --> 00:04:35.150
some reason in the x-direction,
it accelerates a

00:04:35.150 --> 00:04:38.750
little bit by 5 meters
per second.

00:04:38.750 --> 00:04:42.140
So the velocity vector.

00:04:42.140 --> 00:04:44.140
This notation actually is
better, because it takes less

00:04:44.140 --> 00:04:46.170
space up, and you don't have
all these i's and j's and

00:04:46.170 --> 00:04:47.880
pluses confusing everything.

00:04:47.880 --> 00:04:49.690
So the initial velocity
vector, what's its

00:04:49.690 --> 00:04:50.690
x-component?

00:04:50.690 --> 00:04:52.590
It's 120 cosine of 30.

00:04:52.590 --> 00:04:58.310
Cosine of 30 is square root of
3/2 times 120 is 60 square

00:04:58.310 --> 00:05:00.370
roots of 3, and then
you add 5 to it.

00:05:00.370 --> 00:05:00.990
So what is that?

00:05:00.990 --> 00:05:03.650
Let me just solve
it right now.

00:05:03.650 --> 00:05:15.130
So 3 times the square root
of 3 times 60 plus 5.

00:05:15.130 --> 00:05:16.730
So let's just round up
and make it easier.

00:05:16.730 --> 00:05:18.315
It's 109 meters per second.

00:05:18.315 --> 00:05:21.940
108.9, so let's just say 109.

00:05:21.940 --> 00:05:26.010
So the x-component of
the velocity is 109.

00:05:26.010 --> 00:05:29.030
And the y-component was just
120 times the sine of 30.

00:05:29.030 --> 00:05:32.660
Well, sine of 30 is 1/2,
so this is 60.

00:05:32.660 --> 00:05:36.050
Oh, sorry, this should be
brackets, although some people

00:05:36.050 --> 00:05:38.060
actually write the parentheses
there so it looks just like

00:05:38.060 --> 00:05:39.930
coordinates, but I like to keep
it with these brackets so

00:05:39.930 --> 00:05:41.410
that you don't think that these
are coordinates since

00:05:41.410 --> 00:05:43.750
you know these are vectors.

00:05:43.750 --> 00:05:45.790
And a position vector is really
the same thing as a

00:05:45.790 --> 00:05:46.600
position coordinate.

00:05:46.600 --> 00:05:49.270
But a velocity vector is
obviously not a coordinate.

00:05:49.270 --> 00:05:51.640
What's the acceleration
vector?

00:05:51.640 --> 00:05:53.505
Well, the acceleration vector,
as we said, goes straight--

00:05:53.505 --> 00:05:54.540
that's not straight down.

00:05:54.540 --> 00:05:59.520
This is straight down at minus
32 feet per second squared.

00:05:59.520 --> 00:06:01.880
That's the acceleration
of gravity on Earth.

00:06:01.880 --> 00:06:06.785
So the acceleration vector
is equal to -- it has no

00:06:06.785 --> 00:06:12.390
x-component and its y-component
is minus 32.

00:06:12.390 --> 00:06:14.560
So now let's put these back
in that original equation.

00:06:14.560 --> 00:06:17.650
So our position vector, and
I'll switch colors to keep

00:06:17.650 --> 00:06:19.680
things from getting
monotonous.

00:06:19.680 --> 00:06:23.320
Our position vector-- these are
little arrows or one-sided

00:06:23.320 --> 00:06:33.040
arrows-- equals my initial
position, and that's 0, 4 plus

00:06:33.040 --> 00:06:43.660
my initial velocity vector,
109, 60 times t, and I'm

00:06:43.660 --> 00:06:49.640
running out of space, plus at
squared over two, so t squared

00:06:49.640 --> 00:06:59.740
over 2 times my acceleration
vector, 0 minus 32.

00:06:59.740 --> 00:07:02.350
This is actually a little
cleaner way of writing it, but

00:07:02.350 --> 00:07:03.800
this is exactly what
we did when we did

00:07:03.800 --> 00:07:04.600
it with unit vectors.

00:07:04.600 --> 00:07:07.030
Instead of writing i's and j's,
we're just writing the

00:07:07.030 --> 00:07:08.670
numbers in brackets here.

00:07:08.670 --> 00:07:10.510
So let's see if we can
simplify this.

00:07:10.510 --> 00:07:14.090
So let me write it in a
different color, so that you

00:07:14.090 --> 00:07:15.440
know I'm doing.

00:07:15.440 --> 00:07:26.060
OK, so our position vector t
is equal to 0, 4 plus-- and

00:07:26.060 --> 00:07:28.680
now we can distribute this t,
multiply it times both of

00:07:28.680 --> 00:07:40.990
these-- plus 109t, 60t plus--
and we can distribute this t

00:07:40.990 --> 00:07:42.030
squared over 2.

00:07:42.030 --> 00:07:43.970
Well, that times 0 is 0.

00:07:43.970 --> 00:07:51.210
And then that times minus
32 is minus 16t squared.

00:07:51.210 --> 00:07:53.540
Now we can add the vectors.

00:07:53.540 --> 00:07:57.330
So the position at any t.

00:07:57.330 --> 00:07:59.950
So let's add all the
x-components of the vectors.

00:07:59.950 --> 00:08:06.720
0, 109t, 0, so we
just get 109t.

00:08:06.720 --> 00:08:09.110
And then what's the
y-components?

00:08:09.110 --> 00:08:23.320
4 plus 60t minus 16t squared.

00:08:23.320 --> 00:08:24.710
And there we go.

00:08:24.710 --> 00:08:27.730
We've defined the position
vector at a

00:08:27.730 --> 00:08:28.880
function of any time.

00:08:28.880 --> 00:08:29.770
So let's solve the problem.

00:08:29.770 --> 00:08:32.260
Now that they have this wind
gust and our x velocity's

00:08:32.260 --> 00:08:34.380
going a little faster, let's see
if we can clear the fence.

00:08:34.380 --> 00:08:37.610
So how long does it take to
get to 350 feet in the

00:08:37.610 --> 00:08:38.450
x-direction?

00:08:38.450 --> 00:08:42.159
Well, this number right
here has to equal 350.

00:08:42.159 --> 00:08:47.480
So we have 109t has to
be equal to 350.

00:08:47.480 --> 00:08:49.820
And so what's 350
divided by 109?

00:08:49.820 --> 00:08:59.040
350 divided by 109 is equal
to 3.2 seconds.

00:08:59.040 --> 00:09:03.070
t is equal to 3.2 seconds.

00:09:03.070 --> 00:09:06.120
And so what's the height
at 3.2 seconds?

00:09:06.120 --> 00:09:07.370
So let's square that.

00:09:09.740 --> 00:09:21.235
3.2 times 3.2 equals times
16 equals 164.

00:09:21.235 --> 00:09:25.000
So this equals 164.

00:09:25.000 --> 00:09:27.546
And then what's 60 times 3.2?

00:09:27.546 --> 00:09:33.680
60 times 3.2 is equal to 192.

00:09:33.680 --> 00:09:35.090
So what do we get?

00:09:35.090 --> 00:09:50.990
We get 192 plus 4 minus
164 is equal to 32.

00:09:50.990 --> 00:10:00.350
So our position vector at time
3.2 seconds is equal to 350

00:10:00.350 --> 00:10:08.150
feet in the x-direction and 32
feet in the y-direction, and

00:10:08.150 --> 00:10:10.970
that will clear that
30-foot fence.

00:10:10.970 --> 00:10:13.860
Our ball's going to be two
feet above the fence.

00:10:13.860 --> 00:10:15.390
Hope I didn't confuse
you too much.

00:10:15.390 --> 00:10:16.980
See you soon.


Unit vector notation (part 2)

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:01.790
Welcome back.

00:00:01.790 --> 00:00:04.700
In the last video, I at the
end of the video, like I

00:00:04.700 --> 00:00:06.700
always do in the attempt to
confuse you, I told you that

00:00:06.700 --> 00:00:09.060
if I had two vectors-- And let
me just make up some new ones,

00:00:09.060 --> 00:00:12.520
so I can draw them visually
in a second or two.

00:00:12.520 --> 00:00:14.920
Let's call the first vector a.

00:00:14.920 --> 00:00:15.910
Let me do a different color.

00:00:15.910 --> 00:00:19.490
This toothpaste color is
getting monotonous.

00:00:19.490 --> 00:00:21.880
Let me do something that
looks relaxing.

00:00:21.880 --> 00:00:25.910
Let's call a first vector a and,
I don't know, let's make

00:00:25.910 --> 00:00:31.500
it interesting, let me say it's
minus 3 times the unit

00:00:31.500 --> 00:00:37.520
vector i plus 2 times
the unit vector j.

00:00:37.520 --> 00:00:40.570
And then I have vector b.

00:00:40.570 --> 00:00:48.120
And that is equal to, 2i, so two
times the unit vector i.

00:00:48.120 --> 00:00:54.100
Plus, 4 times the
unit vector j.

00:00:54.100 --> 00:00:56.550
In the last video I said, well,
the whole reason why

00:00:56.550 --> 00:00:59.680
this unit vector notation is
even -- Well, one of the

00:00:59.680 --> 00:01:01.860
reasons, we'll see that there
many reasons why it's useful.

00:01:01.860 --> 00:01:04.390
One of the really cool things
about it is, before when we

00:01:04.390 --> 00:01:07.170
added vectors, we would put them
head to tails, and then

00:01:07.170 --> 00:01:08.960
draw it visually, and then
we had this new vector.

00:01:08.960 --> 00:01:10.980
And we really had no way
of expressing it

00:01:10.980 --> 00:01:12.130
without drawing it.

00:01:12.130 --> 00:01:15.850
But when we write things as
multiples of the unit vectors.

00:01:15.850 --> 00:01:17.020
We don't have to draw it.

00:01:17.020 --> 00:01:19.290
And it's actually very
easy to add vectors.

00:01:19.290 --> 00:01:19.890
And how do we do it?

00:01:19.890 --> 00:01:23.050
We just add the x components,
and we add the y components.

00:01:23.050 --> 00:01:28.100
So we said that these two
vectors, a plus b, these

00:01:28.100 --> 00:01:30.320
little weird arrows on top,
that's just saying that those

00:01:30.320 --> 00:01:31.790
are vectors.

00:01:31.790 --> 00:01:33.040
That's equals.

00:01:38.290 --> 00:01:43.640
So it's minus 3, plus 2i, and
I'm going to arbitrarily

00:01:43.640 --> 00:01:46.150
switch colors, because it's
getting monotonous.

00:01:46.150 --> 00:01:49.450
Plus 2 plus 4j.

00:01:49.450 --> 00:01:51.680
We just added the x
components, or the

00:01:51.680 --> 00:01:52.800
multiples of i.

00:01:52.800 --> 00:01:55.440
And we added the y components,
or just the multiples of j.

00:01:55.440 --> 00:01:58.180
Because i was the unit vector in
the x direction, and j was

00:01:58.180 --> 00:02:00.050
the unit vector in
the y direction.

00:02:00.050 --> 00:02:02.560
And we get, what's
minus 3 plus 2?

00:02:02.560 --> 00:02:03.600
That's minus 1.

00:02:03.600 --> 00:02:05.190
We get minus 1i.

00:02:05.190 --> 00:02:06.510
That could just be minus i.

00:02:06.510 --> 00:02:09.720
But I'll write the 1 because
we're just getting warmed up

00:02:09.720 --> 00:02:10.419
with unit vectors.

00:02:10.419 --> 00:02:15.410
So minus 1i plus 6j.

00:02:15.410 --> 00:02:18.990
And when I did that, you might
say, well, Sal, I can't just

00:02:18.990 --> 00:02:19.790
take your word for it.

00:02:19.790 --> 00:02:23.910
Because you seem
not someone who

00:02:23.910 --> 00:02:27.090
should be believed blindly.

00:02:27.090 --> 00:02:31.200
So I think that's a valid
opinion to have. So I will

00:02:31.200 --> 00:02:34.330
show you that this works, by
adding the vectors visually.

00:02:34.330 --> 00:02:35.410
So let's draw it.

00:02:35.410 --> 00:02:37.640
And I think this will give you
a little better sense of unit

00:02:37.640 --> 00:02:38.910
vectors generally.

00:02:38.910 --> 00:02:40.635
Let me draw the axes.

00:02:44.140 --> 00:02:47.820
So that's my y-axis.

00:02:47.820 --> 00:02:49.070
Let me draw my x-axis.

00:02:52.700 --> 00:02:55.400
I have to make sure have enough
space to draw the unit

00:02:55.400 --> 00:02:57.890
vectors that we've drawn,
or to draw the

00:02:57.890 --> 00:02:59.140
vectors that we've drawn.

00:03:01.690 --> 00:03:03.990
Just to show that the axes go
on forever, I have to draw

00:03:03.990 --> 00:03:05.270
that arrow.

00:03:05.270 --> 00:03:13.433
All right, so let's say
this is 1, 2, 3.

00:03:13.433 --> 00:03:21.140
This is 1, 2, 3, 4.

00:03:21.140 --> 00:03:30.360
And I draw 1, 2, 3, 4, 5, 6.

00:03:30.360 --> 00:03:33.110
I think we should be able
to now add them.

00:03:33.110 --> 00:03:35.020
I didn't have to waste all
this space down here.

00:03:35.020 --> 00:03:39.200
So let's just first draw the
vectors, minus 3i plus 2j.

00:03:39.200 --> 00:03:46.070
So minus 3i, just this right
here, is going to be a vector

00:03:46.070 --> 00:03:47.700
that looks something
like this.

00:03:47.700 --> 00:03:50.600
So it's just minus 3 times
the x vector, so

00:03:50.600 --> 00:03:52.320
it'll go to the left.

00:03:52.320 --> 00:03:55.780
Because i is 1 in the
positive direction.

00:03:55.780 --> 00:03:58.180
If we put a negative there,
it flips it over.

00:03:58.180 --> 00:03:59.690
Let me use a different color.

00:03:59.690 --> 00:04:05.940
So this is minus 3i,
and then plus 2j.

00:04:05.940 --> 00:04:07.560
So plus 2j looks like this.

00:04:11.590 --> 00:04:14.300
If we were to add those two
vectors visually, we can put

00:04:14.300 --> 00:04:15.170
them head to tails.

00:04:15.170 --> 00:04:17.750
And the way we can do that, we
can either shift this vector

00:04:17.750 --> 00:04:19.730
up like this, and
draw it up here.

00:04:19.730 --> 00:04:21.910
Or we could shift this vector,
and put its tail

00:04:21.910 --> 00:04:22.810
its vector's head.

00:04:22.810 --> 00:04:25.010
But either way, let's
shift this one up.

00:04:25.010 --> 00:04:27.950
So if we shifted up like that.

00:04:27.950 --> 00:04:30.160
Remember, we're just doing the
head to tails, visual addition

00:04:30.160 --> 00:04:31.480
method of vectors.

00:04:31.480 --> 00:04:34.340
So I just put this tail
to this head.

00:04:34.340 --> 00:04:35.260
And what do we get?

00:04:35.260 --> 00:04:37.400
So vector a will look like this,
and I'm going to do it

00:04:37.400 --> 00:04:40.360
in the same color as vector a
because I have a feeling that

00:04:40.360 --> 00:04:41.710
this diagram might
get complicated.

00:04:44.900 --> 00:04:46.150
Well, I wanted to use
the line tool.

00:04:50.180 --> 00:04:53.700
OK, so this is vector a.

00:04:53.700 --> 00:04:55.740
That's what vector
a looks like.

00:04:55.740 --> 00:04:56.880
And so we worked backwards.

00:04:56.880 --> 00:04:59.260
I gave you the x component
and the y component.

00:04:59.260 --> 00:05:01.310
And then I added them together
by doing the head to tails

00:05:01.310 --> 00:05:04.990
method, and so this is what
vector a would look like.

00:05:04.990 --> 00:05:07.440
And, instead of drawing it, a
very easy representation is

00:05:07.440 --> 00:05:09.990
exactly what we did up here,
a unit vector notation.

00:05:09.990 --> 00:05:11.800
And what's vector b look like?

00:05:11.800 --> 00:05:15.320
So it's 2i-- I'm going to do a
completely different color.

00:05:15.320 --> 00:05:17.680
It's 2i, so it's this vector.

00:05:17.680 --> 00:05:19.500
2 times unit vector i.

00:05:19.500 --> 00:05:20.990
That's this.

00:05:20.990 --> 00:05:24.170
Plus 4j, 1, 2, 3, 4.

00:05:24.170 --> 00:05:26.870
So it looks like this.

00:05:26.870 --> 00:05:29.240
And let's take this one and
shift it over to the left, so

00:05:29.240 --> 00:05:31.170
we can put its tail to the
vector's head, so it would

00:05:31.170 --> 00:05:34.140
look like this.

00:05:34.140 --> 00:05:38.370
So vector b will look --
I'll do it in red.

00:05:38.370 --> 00:05:39.760
And I'll use a line tool.

00:05:39.760 --> 00:05:43.540
Vector b looks like this.

00:05:46.450 --> 00:05:48.500
I just put its components head
to tails, and that's how

00:05:48.500 --> 00:05:50.170
I got vector b.

00:05:50.170 --> 00:05:53.480
And if I were to add
them visually.

00:05:53.480 --> 00:05:55.640
I would do it the same way that
I added its components.

00:05:55.640 --> 00:05:58.120
I would put the tail of one
vector to the head of the

00:05:58.120 --> 00:06:00.100
other, and see if you get
the resulting vector.

00:06:00.100 --> 00:06:00.930
So you could do it either way.

00:06:00.930 --> 00:06:02.270
Let's shift this a vector.

00:06:02.270 --> 00:06:04.760
Let's shift it in
this direction.

00:06:04.760 --> 00:06:07.250
Remember, vectors, we're
just giving the

00:06:07.250 --> 00:06:08.350
magnitude of direction.

00:06:08.350 --> 00:06:10.880
We're not necessarily giving
a starting point.

00:06:10.880 --> 00:06:12.300
So you can shift them.

00:06:12.300 --> 00:06:15.600
You just can't change their
orientation or their

00:06:15.600 --> 00:06:16.850
magnitudes.

00:06:16.850 --> 00:06:18.930
And that's actually how you add
them, you shift them, and

00:06:18.930 --> 00:06:19.970
put them head to tails.

00:06:19.970 --> 00:06:21.690
That's when you add
them visually.

00:06:21.690 --> 00:06:25.780
Let's put that a
vector up here.

00:06:25.780 --> 00:06:30.015
So if we have the a vector, it
looks something like this.

00:06:40.270 --> 00:06:41.700
And I want it to
work out right.

00:06:41.700 --> 00:06:45.640
So the a vector looks
something like that.

00:06:45.640 --> 00:06:49.280
And remember, all I did was I
took the same vector, and I

00:06:49.280 --> 00:06:49.940
just shifted it.

00:06:49.940 --> 00:06:52.310
So that it can start
at the head.

00:06:52.310 --> 00:06:55.160
So its tail can start at the
head of the b vector.

00:06:55.160 --> 00:06:58.060
I just shifted the a vector, so
this is still the a vector.

00:06:58.060 --> 00:06:59.030
By moving the vector
around, you

00:06:59.030 --> 00:07:00.340
haven't changed the vector.

00:07:00.340 --> 00:07:02.980
I would only change the vector,
if I scaled it, if I

00:07:02.980 --> 00:07:06.180
made it bigger or smaller, if
I changed its orientation.

00:07:06.180 --> 00:07:09.600
And so visually, this is b, this
is a, so if I add a to b,

00:07:09.600 --> 00:07:14.880
the resulting vector, going head
to tails-- i'll do it in

00:07:14.880 --> 00:07:17.700
this green color --would
look like this.

00:07:20.280 --> 00:07:23.710
It would look like that.

00:07:23.710 --> 00:07:25.940
So here we took all this
trouble, and I had to draw

00:07:25.940 --> 00:07:27.630
these straight lines
to visually

00:07:27.630 --> 00:07:28.710
add these two vectors.

00:07:28.710 --> 00:07:31.140
This green vector is a plus b.

00:07:31.140 --> 00:07:33.170
Let's see if this green
vector is the same

00:07:33.170 --> 00:07:35.540
thing that we got here.

00:07:35.540 --> 00:07:39.070
Let's see if it's the
same thing as this.

00:07:39.070 --> 00:07:44.660
So we got negative 1 times
i, so negative 1 is here.

00:07:44.660 --> 00:07:47.440
And then we have 6j.

00:07:47.440 --> 00:07:48.530
Let me do it in another color.

00:07:48.530 --> 00:07:51.330
6j would look like this.

00:07:51.330 --> 00:07:52.520
6j looks like that.

00:07:52.520 --> 00:07:54.160
You put them heads to tails.

00:07:54.160 --> 00:07:58.020
And it would be something
like this.

00:07:58.020 --> 00:07:59.320
And that is the green vector.

00:07:59.320 --> 00:08:01.840
And actually, just so you know,
I know it didn't line up

00:08:01.840 --> 00:08:04.660
perfectly, and that's because
I'm not drawing neatly, but

00:08:04.660 --> 00:08:08.420
these two points should actually
be here if I were to

00:08:08.420 --> 00:08:09.980
have drawn this better.

00:08:09.980 --> 00:08:11.930
But I know this is
very confusing, I

00:08:11.930 --> 00:08:12.840
had all these colors.

00:08:12.840 --> 00:08:15.620
But the whole point of it is,
I wanted to show that you

00:08:15.620 --> 00:08:19.440
could visually draws vectors,
and then shift them around,

00:08:19.440 --> 00:08:20.840
and then put them
heads to tails.

00:08:20.840 --> 00:08:22.440
And then get the resulting
vector.

00:08:22.440 --> 00:08:25.170
That's one way to add vectors,
there's still no way to

00:08:25.170 --> 00:08:27.280
analytically represent it.

00:08:27.280 --> 00:08:30.510
Or you could just write any
vector as its x and y

00:08:30.510 --> 00:08:33.340
components, and then the sum of
the vectors is just going

00:08:33.340 --> 00:08:36.409
to be the sum of the x's
and the sum of the y's.

00:08:36.409 --> 00:08:39.049
And that's a much cleaner, and
a much easier, and much less

00:08:39.049 --> 00:08:44.840
prone to error, way of adding
or subtracting two vectors.

00:08:44.840 --> 00:08:46.600
So hopefully that
was convincing.

00:08:46.600 --> 00:08:51.660
That a plus b really
is this vector.

00:08:51.660 --> 00:08:53.470
If it wasn't, I'm sorry.

00:08:53.470 --> 00:08:55.620
And I hope I didn't
confuse you more.

00:08:55.620 --> 00:08:58.240
But now that we have this out
of the way, and hopefully

00:08:58.240 --> 00:09:00.860
you're convinced that unit
vector notation is useful.

00:09:00.860 --> 00:09:04.050
We can move on and maybe try
to do some of our old

00:09:04.050 --> 00:09:06.370
projectile motion problems
using this notation.

00:09:06.370 --> 00:09:09.140
And maybe it'll let us
to do a little bit of

00:09:09.140 --> 00:09:10.590
extra stuff with it.

00:09:10.590 --> 00:09:12.100
See you soon.


Unit vector notation

WEBVTT
Kind: captions
Language: en

00:00:01.230 --> 00:00:03.140
Good afternoon.

00:00:03.140 --> 00:00:06.410
We've done a lot of
work with vectors.

00:00:06.410 --> 00:00:10.050
In a lot of the problems, when
we launch something into--- In

00:00:10.050 --> 00:00:11.920
the projectile motion problems,
or when you were

00:00:11.920 --> 00:00:16.620
doing the incline plane
problems. I always gave you a

00:00:16.620 --> 00:00:18.370
vector, like I would draw
a vector like this.

00:00:18.370 --> 00:00:20.440
I would say something
has a velocity of

00:00:20.440 --> 00:00:21.420
10 meters per second.

00:00:21.420 --> 00:00:22.900
It's at a 30 degree angle.

00:00:22.900 --> 00:00:25.060
And then I would break it up
into the x and y components.

00:00:25.060 --> 00:00:31.550
So if I called this vector v, I
would use a notation, v sub

00:00:31.550 --> 00:00:36.910
x, and the v sub x would have
been this vector right here.

00:00:36.910 --> 00:00:39.200
v sub x would've been this
vector down here.

00:00:39.200 --> 00:00:40.740
The x component of the vector.

00:00:43.280 --> 00:00:45.575
And then v sub y would have been
the y component of the

00:00:45.575 --> 00:00:49.800
vector, and it would have
been this vector.

00:00:49.800 --> 00:00:56.390
So this was v sub x,
this was v sub y.

00:00:56.390 --> 00:00:58.610
And hopefully by now, it's
second nature of how we would

00:00:58.610 --> 00:01:02.260
figure these things out. v sub
x would be 10 times cosine of

00:01:02.260 --> 00:01:03.860
this angle.

00:01:03.860 --> 00:01:08.200
10 cosine of 30 degrees, which I
think is square root of 3/2,

00:01:08.200 --> 00:01:09.890
but we're not worried about
that right now.

00:01:09.890 --> 00:01:14.810
And v sub y would be 10 times
the sine of that angle.

00:01:14.810 --> 00:01:17.740
This hopefully should be
second nature to you.

00:01:17.740 --> 00:01:20.560
If it's not, you can just go
through SOH-CAH-TOA and say,

00:01:20.560 --> 00:01:24.180
well, the sine of 30 degrees
is the opposite of the

00:01:24.180 --> 00:01:24.890
hypotenuse.

00:01:24.890 --> 00:01:25.820
And you would get
back to this.

00:01:25.820 --> 00:01:28.030
But we've reviewed all of that,
and you should review

00:01:28.030 --> 00:01:31.160
the initial vector videos.

00:01:31.160 --> 00:01:34.590
But what I want you to do now,
because this is useful for

00:01:34.590 --> 00:01:37.430
simple projectile motion
problems-- But once we start

00:01:37.430 --> 00:01:40.700
dealing with more complicated
vectors-- and maybe we're

00:01:40.700 --> 00:01:43.190
dealing with multi-dimensional
of vectors, three-dimensional

00:01:43.190 --> 00:01:46.170
vectors, or we start doing
linear algebra, where we do

00:01:46.170 --> 00:01:50.790
end dimensional factors --we
need a coherent way, an

00:01:50.790 --> 00:01:52.450
analytical way, instead of
having to always draw a

00:01:52.450 --> 00:01:55.660
picture of representing
vectors.

00:01:55.660 --> 00:01:58.700
So what we do is, we use
something I call, and I think

00:01:58.700 --> 00:02:00.630
everyone calls it, unit
vector notation.

00:02:00.630 --> 00:02:01.860
So what does that mean?

00:02:01.860 --> 00:02:03.530
So we define these
unit vectors.

00:02:03.530 --> 00:02:06.070
Let me draw some axes.

00:02:06.070 --> 00:02:07.720
And it's important to keep in
mind, this might seem a little

00:02:07.720 --> 00:02:09.699
confusing at first, but this
is no different than what

00:02:09.699 --> 00:02:12.350
we've been doing in our physics
problem so far.

00:02:12.350 --> 00:02:20.980
Let me draw the axes
right there.

00:02:20.980 --> 00:02:29.600
Let's say that this is 1,
this is 0, this is 2.

00:02:29.600 --> 00:02:30.450
0, 1, 2.

00:02:30.450 --> 00:02:32.410
I don't know if must been
writing an Arabic or

00:02:32.410 --> 00:02:33.820
something, going backwards.

00:02:33.820 --> 00:02:36.640
This is 0, 1, 2,
that's not 20.

00:02:36.640 --> 00:02:42.240
And then let's say this is 1,
this is 2, in the y direction.

00:02:42.240 --> 00:02:45.200
I'm going to define what I call
the unit vectors in two

00:02:45.200 --> 00:02:46.020
dimensions.

00:02:46.020 --> 00:02:48.720
So I'm going to first
define a vector.

00:02:48.720 --> 00:02:51.970
I'll call this vector i.

00:02:51.970 --> 00:02:53.220
And this is the vector.

00:02:58.175 --> 00:03:02.000
It just goes straight in the x
direction, has no y component,

00:03:02.000 --> 00:03:04.140
and it has the magnitude of 1.

00:03:04.140 --> 00:03:06.450
And so this is i.

00:03:06.450 --> 00:03:10.470
We denote the unit vector
by putting this little

00:03:10.470 --> 00:03:11.620
cap on top of it.

00:03:11.620 --> 00:03:12.540
There's multiple notations.

00:03:12.540 --> 00:03:15.480
Sometimes in the book, you'll
see this i without the cap,

00:03:15.480 --> 00:03:16.410
and it's just boldface.

00:03:16.410 --> 00:03:17.370
There's some other notations.

00:03:17.370 --> 00:03:22.640
But if you see i, and not in
the imaginary number sense,

00:03:22.640 --> 00:03:25.255
you should realize that that's
the unit vector.

00:03:25.255 --> 00:03:28.960
It has magnitude 1 and it's
completely in the x direction.

00:03:28.960 --> 00:03:32.200
And I'm going to define another
vector, and that one

00:03:32.200 --> 00:03:33.710
is called j.

00:03:33.710 --> 00:03:37.750
And that is the same thing
but in the y direction.

00:03:37.750 --> 00:03:40.200
That is the vector j.

00:03:40.200 --> 00:03:42.600
You put a little cap over it.

00:03:42.600 --> 00:03:44.290
So why did I do this?

00:03:44.290 --> 00:03:46.000
Well, if I'm dealing with
two dimensions.

00:03:46.000 --> 00:03:48.325
And as later we'll see in three
dimensions, so there

00:03:48.325 --> 00:03:50.090
will actually be a third
dimension and we'll call that

00:03:50.090 --> 00:03:52.070
k, but don't worry about
that right now.

00:03:52.070 --> 00:03:56.480
But if we're dealing in two
dimensions, we can define any

00:03:56.480 --> 00:04:01.800
vector in terms of some sum
of these two vectors.

00:04:01.800 --> 00:04:03.270
So how does that work?

00:04:03.270 --> 00:04:07.780
Well, this vector here,
let's call it v.

00:04:07.780 --> 00:04:10.860
This vector, v, is
the sum of its x

00:04:10.860 --> 00:04:12.200
component plus its y component.

00:04:12.200 --> 00:04:13.450
When you add vectors, you
can put them head

00:04:13.450 --> 00:04:14.600
to tail like this.

00:04:14.600 --> 00:04:15.320
And that's the sum.

00:04:15.320 --> 00:04:18.589
So hopefully knowing what we
already know, we knew that the

00:04:18.589 --> 00:04:21.079
vector, v, is equal to its x

00:04:21.079 --> 00:04:26.910
component plus its y component.

00:04:26.910 --> 00:04:28.270
When you add vectors, you
essentially just put

00:04:28.270 --> 00:04:29.080
them head to tails.

00:04:29.080 --> 00:04:33.840
And then the resulting sum
is where you end up.

00:04:33.840 --> 00:04:36.340
It would be if you added this
vector, and then you put this

00:04:36.340 --> 00:04:37.170
tail to this head.

00:04:37.170 --> 00:04:37.910
And you end up there.

00:04:37.910 --> 00:04:38.480
So you end up there.

00:04:38.480 --> 00:04:40.300
So that's the vector.

00:04:40.300 --> 00:04:46.820
So can we define v sub x as some
multiple of i, of this

00:04:46.820 --> 00:04:48.160
unit vector?

00:04:48.160 --> 00:04:49.070
Well, sure.

00:04:49.070 --> 00:04:53.610
v sub x completely goes
in the x direction.

00:04:53.610 --> 00:04:56.590
But it doesn't have
a magnitude of 1.

00:04:56.590 --> 00:05:01.150
It has a magnitude of 10
cosine 30 degrees.

00:05:01.150 --> 00:05:02.870
So its magnitude is ten.

00:05:02.870 --> 00:05:05.340
Let me draw the unit
vector up here.

00:05:05.340 --> 00:05:07.640
This is the unit vector i.

00:05:07.640 --> 00:05:10.550
It's going to look something
like this and this.

00:05:10.550 --> 00:05:13.120
So v sub x is in the exact same
direction, and it's just

00:05:13.120 --> 00:05:14.820
a scaled version of
this unit vector.

00:05:14.820 --> 00:05:19.120
And what multiple is it
of that unit vector?

00:05:19.120 --> 00:05:21.320
Well, the unit vector has
a magnitude of 1.

00:05:21.320 --> 00:05:23.720
This has a magnitude of 10
cosine of 30 degrees.

00:05:23.720 --> 00:05:27.130
I think that's like, 5
square roots of 3, or

00:05:27.130 --> 00:05:28.070
something like that.

00:05:28.070 --> 00:05:34.200
So we can write v sub x-- I keep
switching colors to keep

00:05:34.200 --> 00:05:35.850
things interesting.

00:05:35.850 --> 00:05:42.680
We can write v sub x is equal
to 10 cosine of 30 degrees

00:05:42.680 --> 00:05:45.610
times-- that's the degrees
--times the unit vector i--

00:05:45.610 --> 00:05:49.070
let me stay in that color, so
you don't confused --times the

00:05:49.070 --> 00:05:50.420
unit vector i.

00:05:50.420 --> 00:05:52.010
Does that make sense?

00:05:52.010 --> 00:05:54.970
Well, the unit vector i goes in
the exact same direction.

00:05:54.970 --> 00:05:57.980
But the x component of this
vector is just a lot longer.

00:05:57.980 --> 00:06:01.630
It's 10 cosine 30
degrees long.

00:06:01.630 --> 00:06:05.080
And that's equal to-- cosine of
30 degrees is square root

00:06:05.080 --> 00:06:11.580
of 3/2 --so that's 5 square
roots of 3 i.

00:06:11.580 --> 00:06:18.150
Similary, we can write the y
component of this vector as

00:06:18.150 --> 00:06:19.400
some multiple of j.

00:06:23.110 --> 00:06:28.490
So we could say v sub y, the y
component-- Well, what is sine

00:06:28.490 --> 00:06:29.250
of 30 degrees?

00:06:29.250 --> 00:06:31.400
Sine of 30 degrees is 1/2.

00:06:31.400 --> 00:06:35.470
1/2 times 10, so this is 5.

00:06:35.470 --> 00:06:39.740
So the y component goes
completely in the y direction.

00:06:39.740 --> 00:06:42.880
So it's just going to be a
multiple of this vector j, of

00:06:42.880 --> 00:06:44.410
the unit vector j.

00:06:44.410 --> 00:06:45.350
And what multiple is it?

00:06:45.350 --> 00:06:48.240
Well, it has length 5,
while the unit vector

00:06:48.240 --> 00:06:49.550
has just length 1.

00:06:49.550 --> 00:06:54.590
So it's just 5 times
the unit vector j.

00:06:54.590 --> 00:06:56.240
So how can we write vector v?

00:06:56.240 --> 00:06:59.030
Well, we know the vector v is
the sum of its x component and

00:06:59.030 --> 00:07:00.930
its y component.

00:07:00.930 --> 00:07:03.840
And we also know, so this
is a whole vector v.

00:07:03.840 --> 00:07:04.880
What's its x component?

00:07:04.880 --> 00:07:07.130
Its x component can be
written as a multiple

00:07:07.130 --> 00:07:08.710
of the x unit vector.

00:07:08.710 --> 00:07:10.490
That's that right there.

00:07:10.490 --> 00:07:15.080
So you can write it as
5 square roots of 3

00:07:15.080 --> 00:07:19.330
i plus its y component.

00:07:19.330 --> 00:07:21.530
So what's its y component?

00:07:21.530 --> 00:07:24.880
Well, its y component is just
a multiple of the y unit

00:07:24.880 --> 00:07:27.160
vector, which is called
j, with the little

00:07:27.160 --> 00:07:28.680
funny hat on top.

00:07:28.680 --> 00:07:29.730
And that's just this.

00:07:29.730 --> 00:07:31.050
It's 5 times j.

00:07:34.970 --> 00:07:37.770
So what we've done now, by
defining these unit vectors--

00:07:37.770 --> 00:07:39.390
And I can switch this
color just so you

00:07:39.390 --> 00:07:41.640
remember this is i.

00:07:41.640 --> 00:07:43.450
This unit vector is this.

00:07:43.450 --> 00:07:46.860
Using unit vectors in two
dimensions, and we can

00:07:46.860 --> 00:07:50.140
eventually do them in multiple
dimensions, we can

00:07:50.140 --> 00:07:55.130
analytically express any
two dimensional vector.

00:07:55.130 --> 00:07:57.680
Instead of having to always draw
it like we did before,

00:07:57.680 --> 00:08:00.710
and having to break out
its components and

00:08:00.710 --> 00:08:01.810
always do it visually.

00:08:01.810 --> 00:08:05.660
We can stay in analytical mode
and non graphical mode.

00:08:05.660 --> 00:08:09.930
And what makes this very useful
is that if I can write

00:08:09.930 --> 00:08:13.480
a vector in this format, I can
add them and subtract them

00:08:13.480 --> 00:08:18.830
without having to resort
to visual means.

00:08:18.830 --> 00:08:20.080
And what do I mean by that?

00:08:23.180 --> 00:08:27.710
So if I had to find some vector
a, is equal to, I don't

00:08:27.710 --> 00:08:33.870
know, 2i plus 3j.

00:08:33.870 --> 00:08:37.980
And I have some other
vector b.

00:08:37.980 --> 00:08:39.980
This little arrow just
means it's a vector.

00:08:39.980 --> 00:08:42.409
Sometimes you'll see it
as a whole arrow.

00:08:42.409 --> 00:08:52.800
As, I don't know, 10i plus 2j.

00:08:52.800 --> 00:08:55.080
If I were to say what's
the sum of these two

00:08:55.080 --> 00:08:57.650
vectors a plus b?

00:08:57.650 --> 00:09:00.460
Before we had this unit vector
notation, we would have to

00:09:00.460 --> 00:09:02.320
draw them, and put them
heads to tails.

00:09:02.320 --> 00:09:04.300
And you had to do it visually,
and it would take

00:09:04.300 --> 00:09:04.870
you a lot of time.

00:09:04.870 --> 00:09:07.580
But once you have it broken up
into the x and y components,

00:09:07.580 --> 00:09:10.170
you can just separately add
the x and y components.

00:09:10.170 --> 00:09:18.330
So vector a plus vector b,
that's just 2 plus 10 times i

00:09:18.330 --> 00:09:22.660
plus 3 plus 2 times j.

00:09:22.660 --> 00:09:27.820
And that's equal
to 12i plus 5j.

00:09:27.820 --> 00:09:29.790
And something you might want to
do, maybe I'll do it in the

00:09:29.790 --> 00:09:32.820
future video, is actually draw
out these two vectors and add

00:09:32.820 --> 00:09:33.420
them visually.

00:09:33.420 --> 00:09:37.210
And you'll see that you
get this exact answer.

00:09:37.210 --> 00:09:40.050
And as we go into further
videos, or future videos,

00:09:40.050 --> 00:09:42.590
you'll see how this is super
useful once we start doing

00:09:42.590 --> 00:09:45.090
more complicated physics
problems, or once we start

00:09:45.090 --> 00:09:47.140
doing physics with calculus.

00:09:47.140 --> 00:09:50.840
Anyway, I'm about to run out
of time on the ten minutes.

00:09:50.840 --> 00:09:52.090
So I'll see you in
the next video.


Introduction to the yield curve

WEBVTT
Kind: captions
Language: en

00:00:00.770 --> 00:00:01.750
Welcome back.

00:00:01.750 --> 00:00:04.780
Before we proceed further and
get a little bit better

00:00:04.780 --> 00:00:08.000
understanding of why maybe some
of these investors were

00:00:08.000 --> 00:00:10.330
so keen on investing in mortgage
backed securities,

00:00:10.330 --> 00:00:13.330
essentially loaning this money
to all these people who are

00:00:13.330 --> 00:00:16.090
buying these ever appreciating
houses, I think we need to a

00:00:16.090 --> 00:00:18.110
few more tools in
our tool belt.

00:00:18.110 --> 00:00:19.910
So I'm going to introduce
you to the concept

00:00:19.910 --> 00:00:21.710
of the yield curve.

00:00:21.710 --> 00:00:22.890
You might have heard
this before.

00:00:22.890 --> 00:00:25.970
You might have heard people
on CNBC talk about it.

00:00:25.970 --> 00:00:29.460
And hopefully, after about the
next five or ten minutes, you

00:00:29.460 --> 00:00:32.600
will know a lot about
the yield curve.

00:00:32.600 --> 00:00:34.290
So when most people talk about
the yield curve, they're

00:00:34.290 --> 00:00:35.950
talking about the treasury
yield curve.

00:00:35.950 --> 00:00:37.930
And what does that mean?

00:00:37.930 --> 00:00:39.580
What is even a treasury?

00:00:39.580 --> 00:00:42.230
So these treasury securities,
whether they're T-Bills,

00:00:42.230 --> 00:00:45.445
treasury bills, treasury notes,
or treasury bonds.

00:00:57.010 --> 00:01:01.130
These are loans to the
federal government.

00:01:01.130 --> 00:01:04.720
And these are considered
risk-free.

00:01:04.720 --> 00:01:08.090
Because if you lend to the
federal government and they're

00:01:08.090 --> 00:01:10.840
running short of cash, all they
have to do is increase

00:01:10.840 --> 00:01:15.590
taxes on us the people and they
can pay back your debt.

00:01:15.590 --> 00:01:19.400
So in dollar denominated terms,
the treasury bills,

00:01:19.400 --> 00:01:22.060
notes, and bonds are about as
safe as you can get in terms

00:01:22.060 --> 00:01:24.420
of lending your money
to anyone.

00:01:24.420 --> 00:01:26.540
So when most people talk about
the yield curve, they're

00:01:26.540 --> 00:01:28.630
talking about the risk-free
yield curve.

00:01:28.630 --> 00:01:34.260
And they're talking about the
curve for treasuries.

00:01:34.260 --> 00:01:36.770
So first, a little bit
of definitions.

00:01:36.770 --> 00:01:40.470
What is the difference between
treasury bills, treasury

00:01:40.470 --> 00:01:43.480
notes, and treasury bonds?

00:01:43.480 --> 00:01:45.790
They're pretty much all loans
to the government.

00:01:45.790 --> 00:01:48.210
But they're loans for different
amounts of time.

00:01:48.210 --> 00:01:55.580
So if I give a loan to the
government for $1,000 for six

00:01:55.580 --> 00:01:57.500
months, that will be
a treasury bill.

00:01:57.500 --> 00:02:00.710
So I will give the government
$1,000, the government would

00:02:00.710 --> 00:02:02.180
give me a treasury bill.

00:02:02.180 --> 00:02:04.340
And that treasury bill from the
government is essentially

00:02:04.340 --> 00:02:08.229
just an IOU saying that I'm
going to give you your money

00:02:08.229 --> 00:02:12.740
back in six months with
interest. Similarly, if it's

00:02:12.740 --> 00:02:15.010
three months, it's a three
month treasury bill.

00:02:15.010 --> 00:02:19.590
Treasury notes are loans that
are from one year to 10 years.

00:02:19.590 --> 00:02:24.110
So on this graph that we're
going to make using the actual

00:02:24.110 --> 00:02:29.060
yield curve rates, from zero
to one year-- and actually

00:02:29.060 --> 00:02:31.490
there's no zero year
treasury bill.

00:02:31.490 --> 00:02:34.250
Actually, the shortest
one is one month.

00:02:34.250 --> 00:02:36.830
This would be something like
here on our graph.

00:02:36.830 --> 00:02:42.810
So from one month to one year,
these are T-bills.

00:02:42.810 --> 00:02:46.090
And this is just definitional.

00:02:46.090 --> 00:02:49.560
Then from one year to 10
year, these are notes.

00:02:49.560 --> 00:02:55.670
Actually, I believe the one
year itself is a note.

00:02:55.670 --> 00:02:57.440
Up to one year is a bill.

00:02:57.440 --> 00:02:58.520
Although, I might be
wrong with that.

00:02:58.520 --> 00:02:59.490
Correct me if I'm wrong.

00:02:59.490 --> 00:03:00.700
That's just a definitional
thing.

00:03:00.700 --> 00:03:03.950
From one to 10 year, these
are called notes.

00:03:03.950 --> 00:03:06.850
And then when you go beyond
10 years, these are called

00:03:06.850 --> 00:03:09.470
treasury bonds.

00:03:09.470 --> 00:03:12.930
These are just definitional
things to worry about.

00:03:12.930 --> 00:03:15.060
So with that out of the way,
let's talk about what the

00:03:15.060 --> 00:03:17.840
yield curve is.

00:03:17.840 --> 00:03:19.570
I'll just give you a simple
thought experiment.

00:03:19.570 --> 00:03:24.260
If I'm lending money to someone
for a month versus

00:03:24.260 --> 00:03:28.020
lending money to that person for
a year, in which situation

00:03:28.020 --> 00:03:31.080
am I probably taking
on more risk?

00:03:31.080 --> 00:03:36.040
Well, sure, if I'm lending
someone for a month, I know

00:03:36.040 --> 00:03:37.870
only so much can happen
in that month.

00:03:37.870 --> 00:03:41.920
So I would expect to be paid
less interest. Not just

00:03:41.920 --> 00:03:46.530
obviously in dollar terms, but
even adjusted for time, I

00:03:46.530 --> 00:03:50.370
would expect less interest
for that month.

00:03:50.370 --> 00:03:52.410
And this is actually an
important point to make.

00:03:52.410 --> 00:03:55.880
When I say that I'm charging
6% interest for that month,

00:03:55.880 --> 00:03:57.730
that doesn't mean that after a
month the person is going to

00:03:57.730 --> 00:03:59.650
pay me 6% on my money.

00:03:59.650 --> 00:04:03.040
It means that if I were to give
that money to somebody

00:04:03.040 --> 00:04:04.820
for a month, and they
were to pay it back.

00:04:04.820 --> 00:04:06.970
And then I were to give that
money to, say, that same

00:04:06.970 --> 00:04:09.240
person, or another person, for
a month, and I were to keep

00:04:09.240 --> 00:04:12.770
doing that for a year, then in
aggregate I would get 6%.

00:04:12.770 --> 00:04:16.310
So that 6%, no matter what
duration we talk about,

00:04:16.310 --> 00:04:19.550
whether one month, one year,
five years, 15 years, when we

00:04:19.550 --> 00:04:22.190
talk about the interest rate,
that's the rate that on

00:04:22.190 --> 00:04:24.160
average we would
get for a year.

00:04:24.160 --> 00:04:26.440
It's the annualized
interest rate.

00:04:26.440 --> 00:04:27.740
So going back to my question.

00:04:27.740 --> 00:04:30.970
If lend someone money, even the
government, for a month,

00:04:30.970 --> 00:04:32.870
there's usually less
risk in that.

00:04:32.870 --> 00:04:34.510
Because only so much
could happen in a

00:04:34.510 --> 00:04:35.570
month versus in a year.

00:04:35.570 --> 00:04:38.190
In a year there might be more
inflation, the dollar might

00:04:38.190 --> 00:04:42.120
collapse, I might be passing on
better investments, I might

00:04:42.120 --> 00:04:44.880
need the cash in a year's time,
while I have a lot of

00:04:44.880 --> 00:04:47.630
confidence that I don't need
the cash in a month's time.

00:04:47.630 --> 00:04:51.690
So in general, you expect less
interest when you loan money

00:04:51.690 --> 00:04:55.910
for a shorter period time than
a longer period of time.

00:04:55.910 --> 00:04:57.645
And so let's draw the
yield curve and see

00:04:57.645 --> 00:04:58.490
if this holds true.

00:04:58.490 --> 00:05:00.830
So I actually went
to the treasury

00:05:00.830 --> 00:05:04.400
website, so that's treas.gov.

00:05:04.400 --> 00:05:05.490
And this is the yield curve.

00:05:05.490 --> 00:05:08.000
So they say on March
14, so this is

00:05:08.000 --> 00:05:09.860
the most recent number.

00:05:09.860 --> 00:05:10.830
And I'm going to plot this.

00:05:10.830 --> 00:05:15.520
They say, if you lend money to
the government for one month,

00:05:15.520 --> 00:05:17.830
you'll get 1.2% on that money.

00:05:17.830 --> 00:05:20.740
And remember, if it's $1,000
it's not like I'm going to get

00:05:20.740 --> 00:05:23.830
1.2% on that $1,000 just
after a month.

00:05:23.830 --> 00:05:26.320
If I kept doing it for a year,
this is an annualized number,

00:05:26.320 --> 00:05:27.950
I'll get 1.2%.

00:05:27.950 --> 00:05:31.400
And so for three months, I
get a little bit less.

00:05:31.400 --> 00:05:33.080
And then for six months
I get more.

00:05:33.080 --> 00:05:35.710
And then it does seem that the
overall trend is that I expect

00:05:35.710 --> 00:05:39.700
more and more money as I lend
money to the government for

00:05:39.700 --> 00:05:41.530
larger and larger
periods of time.

00:05:41.530 --> 00:05:44.490
And this is a little interesting
anomaly that you

00:05:44.490 --> 00:05:46.610
get a little bit more
interest for one

00:05:46.610 --> 00:05:47.500
month than three months.

00:05:47.500 --> 00:05:51.810
And we'll do a more advanced
presentation later as to why

00:05:51.810 --> 00:05:56.170
you might get lower yields for
longer duration investments.

00:05:56.170 --> 00:05:58.195
That's called an inverted
yield curve.

00:05:58.195 --> 00:06:02.160
So let's just plot this and
see what it looks like.

00:06:02.160 --> 00:06:03.530
So you saw where
I got my data.

00:06:03.530 --> 00:06:06.000
So they say for one month
I'd get 1.2%.

00:06:06.000 --> 00:06:07.030
So this is one month.

00:06:07.030 --> 00:06:09.800
It'd be right about here.

00:06:09.800 --> 00:06:12.370
Three months I get about
the same thing.

00:06:12.370 --> 00:06:14.720
For six months I get 1.32%.

00:06:14.720 --> 00:06:16.910
Maybe that's like here.

00:06:16.910 --> 00:06:20.520
One year, I get one 1.37%.

00:06:20.520 --> 00:06:22.460
Maybe it's here.

00:06:22.460 --> 00:06:25.880
Five years, I get 2.37%.

00:06:25.880 --> 00:06:27.130
So that's maybe like here.

00:06:30.280 --> 00:06:31.760
And these aren't all
of the durations.

00:06:31.760 --> 00:06:34.700
I'm just for simplicity not
going to do all of them.

00:06:34.700 --> 00:06:37.030
For 10 years, 3.44%.

00:06:37.030 --> 00:06:40.850
So maybe that's here.

00:06:40.850 --> 00:06:44.410
For 20 years, I get 4.3%.

00:06:44.410 --> 00:06:45.870
Like that.

00:06:45.870 --> 00:06:50.540
And then for 30 years,
I get 4.35%.

00:06:50.540 --> 00:06:53.400
So the current yield curve looks
something like this.

00:07:03.480 --> 00:07:06.140
And so you now hopefully at
least understand what the

00:07:06.140 --> 00:07:06.830
yield curve is.

00:07:06.830 --> 00:07:09.960
All it is, is using
a simple graph.

00:07:09.960 --> 00:07:12.950
Someone can look at that graph
and say, well, in general what

00:07:12.950 --> 00:07:17.340
type of rates am I getting for
lending to the government?

00:07:17.340 --> 00:07:19.710
On a risk-free free basis, or
as risk-free as anything we

00:07:19.710 --> 00:07:23.280
can expect, what type of rates
am I getting when I lend to

00:07:23.280 --> 00:07:25.100
the government for different
periods of time?

00:07:25.100 --> 00:07:26.730
And that's what the yield
curve tells us.

00:07:26.730 --> 00:07:29.120
And in general, it's
upwardly sloping.

00:07:29.120 --> 00:07:31.380
Because, as I said, when you
lend money for a longer period

00:07:31.380 --> 00:07:34.000
of time, you're kind of
taking on more risk.

00:07:34.000 --> 00:07:36.770
There's a lot more that you
feel that could happen.

00:07:36.770 --> 00:07:38.020
You might need that cash.

00:07:40.630 --> 00:07:41.480
There might be inflation.

00:07:41.480 --> 00:07:42.560
The dollar might devalue.

00:07:42.560 --> 00:07:44.690
There's a lot of things
that could happen.

00:07:44.690 --> 00:07:47.120
So the next question
is, well, what

00:07:47.120 --> 00:07:49.500
determines this yield curve?

00:07:49.500 --> 00:07:55.890
Well, when the treasury, the
government, needs to borrow

00:07:55.890 --> 00:07:59.370
money, what it does is say, hey
everyone we need to borrow

00:07:59.370 --> 00:08:00.990
a billion dollars from
you, because we

00:08:00.990 --> 00:08:02.510
can't control are spending.

00:08:02.510 --> 00:08:04.920
And they say we're going to
borrow a billion dollars in

00:08:04.920 --> 00:08:05.960
one month notes.

00:08:05.960 --> 00:08:07.490
So this is one month notes.

00:08:07.490 --> 00:08:08.790
They're going to borrow
a billion dollars.

00:08:08.790 --> 00:08:10.370
And they have an auction.

00:08:10.370 --> 00:08:14.340
And the world, investors from
everywhere, they go in, they

00:08:14.340 --> 00:08:16.460
say, well, this is a
safe place to put

00:08:16.460 --> 00:08:17.850
my cash for a month.

00:08:17.850 --> 00:08:21.490
And depending on the demand,
that determines the rate.

00:08:21.490 --> 00:08:24.800
So if there are a lot of people
who want to buy those

00:08:24.800 --> 00:08:28.940
one month treasuries, the rate
might be a little bit lower.

00:08:28.940 --> 00:08:30.290
Does that make sense to you?

00:08:30.290 --> 00:08:31.170
Think about it.

00:08:31.170 --> 00:08:33.900
If a lot of people want to buy
it, there's a lot of demand

00:08:33.900 --> 00:08:35.390
relative to the supply.

00:08:35.390 --> 00:08:38.840
So the government has to pay a
lower interest rate on it.

00:08:38.840 --> 00:08:42.159
Similarly, if for whatever
reason people don't want to

00:08:42.159 --> 00:08:44.630
keep their money in the dollar,
they think the U.S.

00:08:44.630 --> 00:08:47.860
might default on their debt
one day, and not that many

00:08:47.860 --> 00:08:51.480
people want to invest in the
treasury, then that auction,

00:08:51.480 --> 00:08:53.510
the government is going to have
to pay a higher interest

00:08:53.510 --> 00:08:56.340
rate to people for them
to loan money to it.

00:08:56.340 --> 00:08:59.310
So maybe then the auction
ends up up here.

00:08:59.310 --> 00:09:02.230
And similarly, the government
does auctions for all of the

00:09:02.230 --> 00:09:03.380
different durations.

00:09:03.380 --> 00:09:05.150
And duration, I just mean
the time period you're

00:09:05.150 --> 00:09:06.320
getting the loan for.

00:09:06.320 --> 00:09:08.370
So they do it for one month,
three months, six months, one

00:09:08.370 --> 00:09:10.940
year, two year, three
year, et cetera.

00:09:10.940 --> 00:09:17.930
Once the government has done
that auction-- You give the

00:09:17.930 --> 00:09:19.010
money to the government,
they give you an

00:09:19.010 --> 00:09:20.740
IOU called a T-bill.

00:09:20.740 --> 00:09:22.100
Then you could trade it
with other people.

00:09:22.100 --> 00:09:24.930
And that's going to determine
the rate in the short term.

00:09:24.930 --> 00:09:26.510
So the government does
the auction.

00:09:26.510 --> 00:09:29.310
But then after the auction,
and a lot of people had

00:09:29.310 --> 00:09:32.990
demand, but then a lot of
people get freaked out.

00:09:32.990 --> 00:09:36.180
And the public markets, when you
try to sell that treasury,

00:09:36.180 --> 00:09:37.480
will then expect.

00:09:37.480 --> 00:09:38.200
a higher yield.

00:09:38.200 --> 00:09:39.690
I know that might be a little
complicated now.

00:09:39.690 --> 00:09:42.510
And I always start to jumble
things when I run out of time.

00:09:42.510 --> 00:09:45.780
But hopefully at this point you
have a sense of what the

00:09:45.780 --> 00:09:46.590
yield curve is.

00:09:46.590 --> 00:09:48.210
You have a sense of what
treasury bills, treasury

00:09:48.210 --> 00:09:49.820
notes, and treasury bonds are.

00:09:49.820 --> 00:09:52.440
And you have some intuition
on why the yield

00:09:52.440 --> 00:09:54.180
curve has this shape.

00:09:54.180 --> 00:09:56.110
See you in the next video.