Composite functions to model extraterrestrial skydiving

WEBVTT
Kind: captions
Language: en

00:00:00.140 --> 00:00:02.240
- [Instructor] We're told
that Flox is a skydiver

00:00:02.240 --> 00:00:04.450
on the planet Lernon.

00:00:04.450 --> 00:00:09.170
The function. A of w is
equal to 0.2 times W squared,

00:00:09.170 --> 00:00:12.270
gives the area, A, in square meters

00:00:12.270 --> 00:00:17.270
under Flox's parachute when
it has a width of W meters.

00:00:17.570 --> 00:00:18.890
That makes sense.

00:00:18.890 --> 00:00:21.460
The function V of A is
equal to the square root

00:00:21.460 --> 00:00:26.440
of 980 over A gives Flox's maximum speed

00:00:26.440 --> 00:00:29.700
in meters per second when she skydives

00:00:29.700 --> 00:00:34.300
with an area of A square
meters under her parachute.

00:00:34.300 --> 00:00:35.340
All right.

00:00:35.340 --> 00:00:38.940
Write an expression to model
Flox's terminal velocity

00:00:38.940 --> 00:00:42.950
when her parachute is W meters wide,

00:00:42.950 --> 00:00:45.790
and then they want us to
evaluate the terminal velocity

00:00:45.790 --> 00:00:47.670
when her parachute is 14 meters wide.

00:00:47.670 --> 00:00:49.890
Well, let's just focus
on the first part first.

00:00:49.890 --> 00:00:52.624
Pause the video and see if
you can have a go at that.

00:00:52.624 --> 00:00:53.530
All right.

00:00:53.530 --> 00:00:55.490
Now let's just think about
what they're asking us.

00:00:55.490 --> 00:00:59.030
They want us to model terminal velocity

00:00:59.030 --> 00:01:02.700
when her parachute is W meters wide.

00:01:02.700 --> 00:01:04.890
So really what they want us to do

00:01:04.890 --> 00:01:07.810
is come up with a terminal velocity.

00:01:07.810 --> 00:01:12.770
Let's call that V, that
is a function of W,

00:01:12.770 --> 00:01:16.760
that is a function of the
width of her parachute.

00:01:16.760 --> 00:01:20.190
Well, we have a function here
that gives terminal velocity

00:01:20.190 --> 00:01:23.430
as a function of the
area of her parachute,

00:01:23.430 --> 00:01:25.740
but lucky for us, we have another function

00:01:25.740 --> 00:01:30.160
that gives us area as a function of width.

00:01:30.160 --> 00:01:32.450
And so we could say this is
going to be the same thing

00:01:32.450 --> 00:01:36.070
as V of this function, right over here.

00:01:36.070 --> 00:01:40.960
I'll do it in another color, A of w.

00:01:40.960 --> 00:01:44.200
And so that is going to be equal to,

00:01:44.200 --> 00:01:46.510
let me keep the colors consistent.

00:01:46.510 --> 00:01:49.870
Well, everywhere, where I
see an A in this expression,

00:01:49.870 --> 00:01:54.260
I would replace it with A of
W, which is 0.2 W squared.

00:01:54.260 --> 00:01:56.330
So it's going to be
equal to the square root

00:01:56.330 --> 00:02:01.330
of 980 over instead of
A, I am going to write,

00:02:01.990 --> 00:02:05.880
instead of this, I am going
to write 0.2 W squared,

00:02:05.880 --> 00:02:09.140
because that is A as a function of W.

00:02:09.140 --> 00:02:13.310
0.2 W squared.

00:02:13.310 --> 00:02:16.270
So this right over here,
this is an expression

00:02:16.270 --> 00:02:19.920
that models Flox's terminal velocity, V,

00:02:19.920 --> 00:02:24.010
as a function of the
width of her parachute.

00:02:24.010 --> 00:02:27.060
So that's what we have right over there.

00:02:27.060 --> 00:02:28.380
And then the next part, they say,

00:02:28.380 --> 00:02:30.480
what is Flox's terminal
velocity when her parachute

00:02:30.480 --> 00:02:32.670
is 14 meters wide?

00:02:32.670 --> 00:02:35.500
Well, then we just have
to say, okay, W is 14.

00:02:35.500 --> 00:02:37.410
Let's just evaluate this expression.

00:02:37.410 --> 00:02:40.733
So we'll get the square root of 980

00:02:43.480 --> 00:02:48.480
over 0.2 times 14 squared.

00:02:49.350 --> 00:02:52.720
Well, 14 squared is 196

00:02:54.810 --> 00:02:57.800
and this would be equal to the square root

00:02:57.800 --> 00:03:00.799
of, see, 980 divided by 196,

00:03:00.799 --> 00:03:04.340
I believe is exactly five.

00:03:04.340 --> 00:03:06.603
So this would be five divided by 0.2.

00:03:07.510 --> 00:03:10.420
And so, five divided by
essentially one fifth

00:03:10.420 --> 00:03:11.850
is the same thing as five times five.

00:03:11.850 --> 00:03:14.580
So this would be the square root of 25,

00:03:14.580 --> 00:03:16.700
which is equal to five.

00:03:16.700 --> 00:03:19.310
And the terminal velocity, since we gave

00:03:19.310 --> 00:03:22.450
the width in meters,
this is going to give us

00:03:22.450 --> 00:03:24.210
the maximum speed in meters per second.

00:03:24.210 --> 00:03:27.683
So five meters per second, and we're done.


Reading inverse values from a graph

WEBVTT
Kind: captions
Language: en

00:00:00.450 --> 00:00:01.960
- [Instructor] We're told
the following graph shows

00:00:01.960 --> 00:00:04.400
y is equal to f of x, all right.

00:00:04.400 --> 00:00:05.767
And then the first question they say is,

00:00:05.767 --> 00:00:09.450
"What appears to be the
value of f inverse of two?"

00:00:09.450 --> 00:00:12.100
Pause the video and see if
you can have a go at that.

00:00:12.970 --> 00:00:14.760
All right, now let's work
through this together.

00:00:14.760 --> 00:00:15.670
So it's important to realize

00:00:15.670 --> 00:00:18.390
they're not asking us what f of two is.

00:00:18.390 --> 00:00:20.700
They're asking us f inverse of two.

00:00:20.700 --> 00:00:23.880
f of two we would say,
"Okay, when x is equal to two

00:00:23.880 --> 00:00:25.460
that's the input into our function."

00:00:25.460 --> 00:00:28.260
And then the graph tells us that f of two

00:00:28.260 --> 00:00:31.280
it looks like it might be
a little bit more than 2.5.

00:00:31.280 --> 00:00:33.320
Maybe it's approximately 2.6.

00:00:34.250 --> 00:00:35.430
But that's not what they're asking us.

00:00:35.430 --> 00:00:38.970
They're asking us f inverse of two.

00:00:38.970 --> 00:00:41.730
And just as a reminder of
what an inverse function is,

00:00:41.730 --> 00:00:43.940
if we have some input, x,

00:00:43.940 --> 00:00:46.040
and we input into our function, f,

00:00:46.040 --> 00:00:49.420
that is going to output f of x.

00:00:49.420 --> 00:00:51.580
Now, if we were to input f of x

00:00:51.580 --> 00:00:56.270
into the inverse function
for f, then the output here,

00:00:56.270 --> 00:00:59.940
which is going to be f inverse of f of x

00:00:59.940 --> 00:01:02.660
is going to get us back
to this original x,

00:01:02.660 --> 00:01:04.970
is going to be equal to x.

00:01:04.970 --> 00:01:07.690
So we are really, in this scenario,

00:01:07.690 --> 00:01:12.690
dealing with this part of this
chain of inputs and outputs.

00:01:12.930 --> 00:01:16.700
We're saying we want to figure
out what f inverse of two is.

00:01:16.700 --> 00:01:19.570
So this part, right over here
is going to be equal to two.

00:01:19.570 --> 00:01:23.230
So we're saying when f
of x is equal to two,

00:01:23.230 --> 00:01:25.800
what is the corresponding x?

00:01:25.800 --> 00:01:30.350
So when f of x is equal to two,
what is the corresponding X?

00:01:30.350 --> 00:01:31.650
We get four.

00:01:31.650 --> 00:01:32.640
So let's write that down.

00:01:32.640 --> 00:01:36.230
f inverse of two is equal to four.

00:01:36.230 --> 00:01:37.740
So when x is equal to four,

00:01:37.740 --> 00:01:40.240
you input that f of four is equal to two

00:01:40.240 --> 00:01:44.190
or f inverse of two is equal to four.

00:01:44.190 --> 00:01:45.217
Now the next part they say,

00:01:45.217 --> 00:01:49.440
"Sketch the graph of y is
equal to f inverse of x."

00:01:49.440 --> 00:01:50.870
So an important thing to realize

00:01:50.870 --> 00:01:55.180
is if we're saying that
b is equal to f of a

00:01:55.180 --> 00:02:00.180
which implies that the point
a,b is on the graph of f,

00:02:00.510 --> 00:02:02.380
then we're dealing with f inverse.

00:02:02.380 --> 00:02:06.680
We would know that a would
be equal to f inverse of b.

00:02:06.680 --> 00:02:08.660
You can think about swapping these two,

00:02:08.660 --> 00:02:10.090
the a and the b's here,

00:02:10.090 --> 00:02:14.740
which means that b,a
would be on f inverse.

00:02:14.740 --> 00:02:17.646
So any coordinate point that's
on our original graph f,

00:02:17.646 --> 00:02:19.670
if you swap the x and the y,

00:02:19.670 --> 00:02:22.020
that's going to be on our f inverse.

00:02:22.020 --> 00:02:23.930
So let's just pick some points.

00:02:23.930 --> 00:02:25.500
And once again, they're
just saying sketch,

00:02:25.500 --> 00:02:27.300
so it doesn't have to be perfect.

00:02:27.300 --> 00:02:29.610
So if we look at this
point right over here,

00:02:29.610 --> 00:02:34.537
that looks like the point
-10, maybe it looks like 3.4.

00:02:36.330 --> 00:02:38.630
Well, then that means
if we swap that x and y

00:02:38.630 --> 00:02:40.640
it'll be on the graph of f inverse.

00:02:40.640 --> 00:02:45.640
So if we go to 3.4, and then -10,

00:02:45.970 --> 00:02:47.830
so it gets us right about there,

00:02:47.830 --> 00:02:51.090
that would be on the graph of f inverse.

00:02:51.090 --> 00:02:53.927
Now, if we went to, let's say

00:02:53.927 --> 00:02:57.060
let's say this point right over here.

00:02:57.060 --> 00:03:01.470
This is the point -2,3.

00:03:01.470 --> 00:03:04.610
So if -2,3 is on the graph of f,

00:03:04.610 --> 00:03:08.150
then 3,-2 would be on the inverse.

00:03:08.150 --> 00:03:13.150
So 3,-2 would be on the
inverse right over there.

00:03:14.210 --> 00:03:16.160
Let's pick a few more points.

00:03:16.160 --> 00:03:19.290
So we have this point right over here, 4,2

00:03:19.290 --> 00:03:21.100
which would be on the graph of f,

00:03:21.100 --> 00:03:23.770
which means that 2,4 is on the inverse.

00:03:23.770 --> 00:03:28.020
So 2,4, which would be right over there.

00:03:28.020 --> 00:03:29.520
And then if we look at
this point over here

00:03:29.520 --> 00:03:32.700
which looks like roughly nine,

00:03:32.700 --> 00:03:35.620
let's just call it 9,-10

00:03:35.620 --> 00:03:37.530
maybe it's 9.1,-10.

00:03:37.530 --> 00:03:40.080
If that's on the graph of f,

00:03:40.080 --> 00:03:45.080
then if we swap that -10, maybe 9.1

00:03:45.590 --> 00:03:49.970
would be on the inverse
-10,9.1, right over there.

00:03:49.970 --> 00:03:52.170
And so then we could connect the dots

00:03:52.170 --> 00:03:55.330
to try to sketch out
what the inverse function

00:03:55.330 --> 00:03:58.350
is going to look like, the
graph of the inverse function.

00:03:58.350 --> 00:04:03.350
So it's going to look something like that.

00:04:04.580 --> 00:04:06.470
And you might notice, it looks like

00:04:06.470 --> 00:04:10.150
it's a reflection about the line y=x.

00:04:10.150 --> 00:04:12.540
It looks like it's a
reflection about that line,

00:04:12.540 --> 00:04:13.753
which is exactly right.


Using matrices to manipulate data: Game show

WEBVTT
Kind: captions
Language: en

00:00:00.320 --> 00:00:01.410
- [Instructor] We're
told, in the beginning

00:00:01.410 --> 00:00:03.520
of each episode of a certain game show,

00:00:03.520 --> 00:00:07.120
each contestant picks a certain
door out of three doors.

00:00:07.120 --> 00:00:09.070
Then the game show host randomly picks

00:00:09.070 --> 00:00:11.450
one of the two prize bundles.

00:00:11.450 --> 00:00:14.900
After each round, each
contestant receives a prize

00:00:14.900 --> 00:00:16.570
based on the door they picked

00:00:16.570 --> 00:00:18.910
and the bundle the host picked.

00:00:18.910 --> 00:00:21.590
Matrix A represents the possible prizes

00:00:21.590 --> 00:00:22.740
for the first round.

00:00:22.740 --> 00:00:24.460
All right, so for example,

00:00:24.460 --> 00:00:26.690
if the contestant picks door three

00:00:26.690 --> 00:00:30.900
and the host picks bundle
one, the prize is $300,

00:00:30.900 --> 00:00:32.870
but if the contestant picks door three

00:00:32.870 --> 00:00:34.960
and the host picks bundle two,

00:00:34.960 --> 00:00:37.870
the prize would be $0, all right.

00:00:37.870 --> 00:00:40.550
And then they say matrix B
represents the possible prizes

00:00:40.550 --> 00:00:41.700
for the second round.

00:00:41.700 --> 00:00:43.340
All right, that's fair.

00:00:43.340 --> 00:00:46.100
They also tell us the
second round can also

00:00:46.100 --> 00:00:47.970
be a lightning round.

00:00:47.970 --> 00:00:52.420
In this case, the prizes are doubled.

00:00:52.420 --> 00:00:55.330
Matrix C represents the possible prizes

00:00:55.330 --> 00:00:57.540
during a lightning round.

00:00:57.540 --> 00:01:00.110
Complete matrix C.

00:01:00.110 --> 00:01:02.560
So pause this video and see
if you can figure that out

00:01:02.560 --> 00:01:05.190
and then we'll work through this together.

00:01:05.190 --> 00:01:08.860
All right, so matrix C is a
scenario where we're dealing

00:01:08.860 --> 00:01:10.090
with a lightning round,

00:01:10.090 --> 00:01:13.130
and remember in a lightning
round, the prizes are doubled

00:01:13.130 --> 00:01:14.060
but it's the second round.

00:01:14.060 --> 00:01:16.800
It's doubled relative to
what it would have been

00:01:16.800 --> 00:01:18.320
in the second round.

00:01:18.320 --> 00:01:22.330
So what it would have been in
the second round is matrix B.

00:01:22.330 --> 00:01:23.840
So another way to think about it is,

00:01:23.840 --> 00:01:28.840
matrix C is going to be
equal to two times matrix B.

00:01:30.700 --> 00:01:33.404
And we know when we multiply a matrix

00:01:33.404 --> 00:01:36.100
times a scalar like this,
times just a number,

00:01:36.100 --> 00:01:39.180
we just multiply each of
these entries by that number.

00:01:39.180 --> 00:01:40.600
So let's do that.

00:01:40.600 --> 00:01:43.950
If we take $600 and multiply that by two,

00:01:43.950 --> 00:01:46.600
that is going to be $1,200.

00:01:46.600 --> 00:01:47.433
And that makes sense.

00:01:47.433 --> 00:01:50.150
We just said for each
corresponding scenario

00:01:50.150 --> 00:01:52.700
the prizes are doubled
in a lightning round.

00:01:52.700 --> 00:01:54.720
So if the contestant picks door one,

00:01:54.720 --> 00:01:56.970
host picks bundle one, instead of $600

00:01:56.970 --> 00:01:59.210
it's going to be $1,200.

00:01:59.210 --> 00:02:04.210
Keep going, instead of
$200, it's going to be $400.

00:02:04.340 --> 00:02:08.800
All I'm doing here is I'm
multiplying each of these entries

00:02:08.800 --> 00:02:12.950
by two to get the corresponding
entry in matrix C.

00:02:12.950 --> 00:02:15.950
Keep going, instead of $300 here,

00:02:15.950 --> 00:02:19.280
multiply that by two,
you're going to get $600.

00:02:19.280 --> 00:02:24.130
Instead of $300 here,
you're going to get $600.

00:02:25.380 --> 00:02:26.890
We're almost there.

00:02:26.890 --> 00:02:28.110
Instead of $0 here,

00:02:28.110 --> 00:02:31.920
well, zero times two is still $0.

00:02:31.920 --> 00:02:33.860
And then last but not least,

00:02:33.860 --> 00:02:35.810
instead of $400 right over here,

00:02:35.810 --> 00:02:39.600
that times two is going to be $800.

00:02:39.600 --> 00:02:41.050
Now there's one more question

00:02:41.050 --> 00:02:43.160
that they have below the
screen right over here.

00:02:43.160 --> 00:02:45.110
Let me scroll up a little bit.

00:02:45.110 --> 00:02:48.850
So they tell us matrix
D is defined as follows:

00:02:48.850 --> 00:02:51.650
D is equal to A plus B.

00:02:51.650 --> 00:02:54.810
What does matrix D represent?

00:02:54.810 --> 00:02:57.460
So pause the video and think
about that for a second.

00:02:58.330 --> 00:03:00.320
Well, if we add two matrices,

00:03:00.320 --> 00:03:03.050
we're going to add all
the corresponding entries.

00:03:03.050 --> 00:03:07.880
And so what it tells you is
what is the combined prize

00:03:07.880 --> 00:03:10.430
for both rounds one and two

00:03:10.430 --> 00:03:15.240
based on what the contestant
picks and what the host picks.

00:03:15.240 --> 00:03:18.690
So matrix D, that top
left entry will tell you,

00:03:18.690 --> 00:03:21.670
okay, in total, if the
contestant picked door number one

00:03:21.670 --> 00:03:24.000
and the host pick bundle
number one, what would you get?

00:03:24.000 --> 00:03:26.600
Cause it would be $100 dollars plus $600.

00:03:26.600 --> 00:03:29.580
So it would be total
for rounds one and two,

00:03:29.580 --> 00:03:31.410
assuming we don't have the lightning round

00:03:31.410 --> 00:03:33.313
like we had in matrix C.


Using matrices to manipulate data: Pet store

WEBVTT
Kind: captions
Language: en

00:00:00.880 --> 00:00:02.210
- [Instructor] We're told
a certain pet store chain

00:00:02.210 --> 00:00:03.970
has three types of dog food

00:00:03.970 --> 00:00:08.430
and each comes in bags
of two different sizes.

00:00:08.430 --> 00:00:13.170
Matrix A represents the store's
inventory at location A,

00:00:13.170 --> 00:00:18.170
where rows are food types
and columns are bag sizes.

00:00:18.780 --> 00:00:19.990
So let's see, it's store A.

00:00:19.990 --> 00:00:21.810
That's what matrix A is telling us.

00:00:21.810 --> 00:00:26.510
They're telling us we have
three different types of food,

00:00:26.510 --> 00:00:28.110
three different types of dog food,

00:00:28.110 --> 00:00:30.620
and then they each come
in two different sizes.

00:00:30.620 --> 00:00:34.430
So for example, type 1 dog food in size 1,

00:00:34.430 --> 00:00:39.060
they have five bags of
that while type 2 dog food

00:00:39.060 --> 00:00:41.080
in size 2, they have nine bags of that.

00:00:41.080 --> 00:00:42.620
All right. That's fair enough.

00:00:42.620 --> 00:00:46.390
Matrix B represents the store's
inventory at location B.

00:00:46.390 --> 00:00:48.860
All right, same thing for store B.

00:00:48.860 --> 00:00:52.900
Matrix C represents
how many more, or less,

00:00:52.900 --> 00:00:55.570
bags of each type and size there are

00:00:55.570 --> 00:01:00.570
in location A relative to location B.

00:01:00.690 --> 00:01:03.690
Complete matrix C.

00:01:03.690 --> 00:01:06.200
So pause this video and see
if you can have a go at that.

00:01:06.200 --> 00:01:11.083
So we need to fill in the
entries here of matrix C.

00:01:13.490 --> 00:01:15.040
All right, now let's do this together.

00:01:15.040 --> 00:01:17.000
So let me just review
what it just told us.

00:01:17.000 --> 00:01:22.000
Matrix C represents how many
more bags of each type and size

00:01:22.530 --> 00:01:27.530
there are in location A
relative to location B.

00:01:27.750 --> 00:01:30.950
So for example, this first
entry right over here,

00:01:30.950 --> 00:01:35.950
we wanna know how many
more bags of type 1, size 1

00:01:36.100 --> 00:01:40.520
there are in location A than
there are in location B?

00:01:40.520 --> 00:01:43.830
Well, I would take the number
that there are in location A

00:01:43.830 --> 00:01:46.060
and then from that subtract how many

00:01:46.060 --> 00:01:47.180
there are in location B.

00:01:47.180 --> 00:01:50.170
That would tell me how many
more I have in location A.

00:01:50.170 --> 00:01:55.170
So if I take five minus
eight, what am I going to get?

00:01:56.210 --> 00:02:00.760
Well, I'm going to get
negative three right over here,

00:02:00.760 --> 00:02:03.250
and you might already be
recognizing what's happening.

00:02:03.250 --> 00:02:05.200
For every corresponding entry,

00:02:05.200 --> 00:02:07.240
I'm gonna subtract the entry from matrix B

00:02:07.240 --> 00:02:09.530
from the entry in matrix A,

00:02:09.530 --> 00:02:11.290
or another way to think about it is,

00:02:11.290 --> 00:02:16.290
if I take matrix A and
I subtract matrix B,

00:02:16.390 --> 00:02:18.600
I am going to get matrix C.

00:02:18.600 --> 00:02:21.610
I'm just gonna subtract all
of the corresponding entries.

00:02:21.610 --> 00:02:26.230
So if I do seven minus six,
and that is going to be one,

00:02:26.230 --> 00:02:27.380
I'm just gonna color code this.

00:02:27.380 --> 00:02:32.380
If I do three minus 10, that's
going to be negative seven.

00:02:32.950 --> 00:02:36.020
If I do, I'm running out of colors.

00:02:36.020 --> 00:02:39.100
If I do nine minus 12,

00:02:39.100 --> 00:02:44.100
that is also going to be negative three.

00:02:44.320 --> 00:02:48.090
And then if I do this brown color,

00:02:48.090 --> 00:02:52.650
10 minus five, that is
going to be positive five.

00:02:52.650 --> 00:02:57.650
And then if I do 15 minus
nine, that is positive six.

00:02:58.640 --> 00:03:02.470
So we can see that, for
example, type 1, size 2,

00:03:02.470 --> 00:03:06.050
we have one more in store
A than we have in store B.

00:03:06.050 --> 00:03:10.150
But if we think about type 2, size 1,

00:03:10.150 --> 00:03:14.510
it shows us that store A
has actually seven fewer

00:03:14.510 --> 00:03:17.430
of that than store B does.

00:03:17.430 --> 00:03:21.830
Now we have one last question
here that is below the screen,

00:03:21.830 --> 00:03:24.180
but let me scroll down here.

00:03:24.180 --> 00:03:27.730
So they tell us that matrix
D is defined as follows,

00:03:27.730 --> 00:03:29.740
or is defines as follows. (chuckles)

00:03:29.740 --> 00:03:31.150
Make a little grammatical.

00:03:31.150 --> 00:03:36.140
Is defined as follows:
D is equal to A plus B.

00:03:36.140 --> 00:03:38.260
What does matrix D represent?

00:03:38.260 --> 00:03:40.780
So they're not asking us
to calculate A plus B.

00:03:40.780 --> 00:03:42.430
Not asking us to add the matrices,

00:03:42.430 --> 00:03:43.263
but you know how to do it.

00:03:43.263 --> 00:03:44.850
You would add the corresponding entries.

00:03:44.850 --> 00:03:47.403
But what does D represent?

00:03:48.850 --> 00:03:50.940
Well, if you add the
corresponding entries,

00:03:50.940 --> 00:03:53.180
remember, this is the
inventory of store A,

00:03:53.180 --> 00:03:55.350
this is the inventory of store B.

00:03:55.350 --> 00:03:56.510
So if you were to add them,

00:03:56.510 --> 00:03:59.730
the matrix D would tell you
the combined inventories

00:03:59.730 --> 00:04:04.443
of A and B, for each
of the types and sizes.


Magnetism and Magnetic Fields

WEBVTT
Kind: captions
Language: en

00:00:00.450 --> 00:00:02.330
- [Instructor] Let's talk
a little bit about magnets

00:00:02.330 --> 00:00:05.480
and magnetic fields and right
over here we have a picture

00:00:05.480 --> 00:00:08.420
of what today we would call a magnet

00:00:08.420 --> 00:00:11.540
where we have these metal
nails all being attracted

00:00:11.540 --> 00:00:14.100
to the stone and the stone,

00:00:14.100 --> 00:00:17.340
the modern name for it is magnetite,

00:00:17.340 --> 00:00:19.280
and human beings have known

00:00:19.280 --> 00:00:22.530
about magnetite for thousands of years

00:00:22.530 --> 00:00:25.750
in fact the name comes
from Ancient Greece,

00:00:25.750 --> 00:00:27.480
there was the Magnetes people

00:00:27.480 --> 00:00:30.520
who settled actually areas
that are often talked

00:00:30.520 --> 00:00:32.910
about in the origin of the name magnet,

00:00:32.910 --> 00:00:35.940
which are now referred to as Magnesia,

00:00:35.940 --> 00:00:38.270
and depending on the
historical count you see

00:00:38.270 --> 00:00:40.720
it was one of those Magnesia's

00:00:40.720 --> 00:00:43.860
where the Ancient Greeks
were able to find a lot

00:00:43.860 --> 00:00:45.420
of what we would call magnetite,

00:00:45.420 --> 00:00:48.650
and actually the element
magnesium which is not related

00:00:48.650 --> 00:00:51.030
to magnetite was also found in that area,

00:00:51.030 --> 00:00:53.140
which is where it got its name.

00:00:53.140 --> 00:00:55.820
But it wasn't just some interesting thing

00:00:55.820 --> 00:00:57.230
that would attract metal,

00:00:57.230 --> 00:00:59.790
and then they also obviously
observed other properties

00:00:59.790 --> 00:01:04.410
that if you had two pieces
of magnetized magnetite

00:01:04.410 --> 00:01:08.270
it actually turns out that not
all magnetite is magnetized,

00:01:08.270 --> 00:01:10.840
something interesting has
to happen to it we believe

00:01:10.840 --> 00:01:12.120
it's actually lightning strikes

00:01:12.120 --> 00:01:14.330
that magnetizes it naturally,

00:01:14.330 --> 00:01:16.300
that the orientation matters,

00:01:16.300 --> 00:01:19.410
if you're in one orientation
they might attract each other.

00:01:19.410 --> 00:01:21.450
And then if you're in another
orientation if you were

00:01:21.450 --> 00:01:23.930
to spin this one around

00:01:23.930 --> 00:01:26.980
they might actually repel each other.

00:01:26.980 --> 00:01:29.250
And this notion of the orientation

00:01:29.250 --> 00:01:32.410
that there might be some
polarity made it more than just

00:01:32.410 --> 00:01:34.710
an interesting thing to observe,

00:01:34.710 --> 00:01:37.410
an Ancient Han China
roughly 2000 years ago

00:01:37.410 --> 00:01:40.280
they invented the first
compass, where they realized

00:01:40.280 --> 00:01:44.740
that if you took some magnetized magnetite

00:01:44.740 --> 00:01:47.690
which the historical
name for it a lodestone,

00:01:47.690 --> 00:01:49.420
and you allow it to freely move

00:01:49.420 --> 00:01:52.230
and you could do that by
either hanging it from a string

00:01:52.230 --> 00:01:55.290
or have it float on some
still water say in bucket,

00:01:55.290 --> 00:01:58.150
that it will consistently orient itself,

00:01:58.150 --> 00:02:00.370
so that it points in the same direction.

00:02:00.370 --> 00:02:02.950
And so you can use that
for things like navigation

00:02:02.950 --> 00:02:06.460
which the Chinese did
roughly a 1000 years ago,

00:02:06.460 --> 00:02:09.850
and that helped us realize
that the Earth itself is acting

00:02:09.850 --> 00:02:11.410
as a magnet,

00:02:11.410 --> 00:02:15.790
and just like a small magnet
has different sides to it,

00:02:15.790 --> 00:02:18.050
the Earth does too and
that's where the convention

00:02:18.050 --> 00:02:22.220
for a North and South Pole
of a magnet came from.

00:02:22.220 --> 00:02:24.010
But there's probably a
question in your mind

00:02:24.010 --> 00:02:26.420
from the first time that
you noticed a magnet.

00:02:26.420 --> 00:02:29.220
If you have some piece of metal out here,

00:02:29.220 --> 00:02:31.230
that's not touching the magnet

00:02:31.230 --> 00:02:33.020
and in a future video we can even talk

00:02:33.020 --> 00:02:35.950
what touching even means at
a microphone scopic level,

00:02:35.950 --> 00:02:38.840
but if you have a nail out
here that's not touching it,

00:02:38.840 --> 00:02:40.530
but there's some force that's acting

00:02:40.530 --> 00:02:42.840
at a distance on that nail,

00:02:42.840 --> 00:02:46.210
how does that nail know to be
pulled towards that magnet?

00:02:46.210 --> 00:02:48.330
It doesn't have eyes,
it doesn't have ears,

00:02:48.330 --> 00:02:49.420
it doesn't say that there's a magnet

00:02:49.420 --> 00:02:53.070
there I better somehow
move myself towards it.

00:02:53.070 --> 00:02:54.490
There's something about that region

00:02:54.490 --> 00:02:58.220
of space that is
interacting with that nail.

00:02:58.220 --> 00:03:00.000
Or if you think about magnets,

00:03:00.000 --> 00:03:01.670
how does it know the orientation

00:03:01.670 --> 00:03:04.870
of the other magnet to
either be attracted to it

00:03:04.870 --> 00:03:05.703
or repel it?

00:03:05.703 --> 00:03:07.770
How does it even know that
other magnet is there?

00:03:07.770 --> 00:03:11.860
And that's where the concept
of a magnetic field is useful.

00:03:11.860 --> 00:03:14.370
And this was introduced by Michael Faraday

00:03:14.370 --> 00:03:16.520
in the 19th century,

00:03:16.520 --> 00:03:18.180
as a way of at least thinking

00:03:18.180 --> 00:03:21.720
about giving us a framework
for this force at a distance.

00:03:21.720 --> 00:03:24.710
It doesn't exactly tell us what it is,

00:03:24.710 --> 00:03:27.950
but it does gives us a way
of predicting and thinking

00:03:27.950 --> 00:03:30.210
about what is happening.

00:03:30.210 --> 00:03:34.010
And one way to visualize a magnetic field,

00:03:34.010 --> 00:03:38.700
is to take a bar magnet or I
could even say a bar lodestone,

00:03:38.700 --> 00:03:41.300
and put it underneath a piece of paper,

00:03:41.300 --> 00:03:44.750
and then putting metal
filings on top of that piece

00:03:44.750 --> 00:03:47.070
of paper, and you will
see something like this,

00:03:47.070 --> 00:03:50.340
in fact I encourage you to do
this to observe this yourself,

00:03:50.340 --> 00:03:52.150
and you see what look like lines

00:03:52.150 --> 00:03:54.753
that are essentially connecting the North

00:03:54.753 --> 00:03:57.870
and South Poles like that.

00:03:57.870 --> 00:03:59.850
And this notion of field lines we can draw

00:03:59.850 --> 00:04:02.870
it a little bit clearer
in something like this,

00:04:02.870 --> 00:04:05.300
was introduced by Michael
Faraday and he says okay

00:04:05.300 --> 00:04:07.530
there's this thing called a field,

00:04:07.530 --> 00:04:11.560
that tells us for every point
in space around the magnet,

00:04:11.560 --> 00:04:14.270
what it would do to
something that is interactive

00:04:14.270 --> 00:04:17.210
with the magnet another magnet
or maybe a piece of metal,

00:04:17.210 --> 00:04:22.210
so for example, if you were to
put a small bar magnet right

00:04:22.830 --> 00:04:26.920
over here, the North side
is going to be repelled

00:04:26.920 --> 00:04:29.400
from the North side of
this bigger bar magnet,

00:04:29.400 --> 00:04:32.500
and the South side is going
to be attracted to it,

00:04:32.500 --> 00:04:35.470
and so what you could do is
you could put a freely moving,

00:04:35.470 --> 00:04:37.540
magnet or you could put a compass,

00:04:37.540 --> 00:04:40.050
to actually see what the orientation is

00:04:40.050 --> 00:04:42.080
which direction will accomplish point,

00:04:42.080 --> 00:04:44.930
and it will point in the
direction of these field lines,

00:04:44.930 --> 00:04:47.110
and if you were to put it over here,

00:04:47.110 --> 00:04:49.670
it would orient this way if
you allow it to freely move,

00:04:49.670 --> 00:04:51.060
where this is the Northern end,

00:04:51.060 --> 00:04:52.250
and this is the Southern end,

00:04:52.250 --> 00:04:53.660
and if you were to put it here

00:04:53.660 --> 00:04:55.850
and you were allow it to
freely move let's say as part

00:04:55.850 --> 00:04:56.683
of a compass,

00:04:56.683 --> 00:04:58.620
and if this is the Northern end

00:04:58.620 --> 00:04:59.730
and this is the Southern end,

00:04:59.730 --> 00:05:01.220
it would orient like that.

00:05:01.220 --> 00:05:03.320
And so that helps you draw the field lines

00:05:03.320 --> 00:05:04.780
and also know the direction.

00:05:04.780 --> 00:05:07.760
And just by convention the
direction is where coming

00:05:07.760 --> 00:05:10.600
from the North Pole into the South Pole.

00:05:10.600 --> 00:05:13.130
And what Faraday said is not only does

00:05:13.130 --> 00:05:15.610
this tell you the direction
of the magnetic field,

00:05:15.610 --> 00:05:17.300
it tells you the magnitude based

00:05:17.300 --> 00:05:19.380
on the density of these field lines.

00:05:19.380 --> 00:05:23.480
So for example, the magnetic
field is stronger here,

00:05:23.480 --> 00:05:25.040
where in that unit area

00:05:25.040 --> 00:05:27.690
that I just made you have
very dense field lines,

00:05:27.690 --> 00:05:30.910
while it would be weaker right over here,

00:05:30.910 --> 00:05:32.800
I have fewer field lines.

00:05:32.800 --> 00:05:35.030
Now another amazing thing about magnets

00:05:35.030 --> 00:05:36.060
and their polarity

00:05:36.060 --> 00:05:38.650
that people have noticed
since ancient times is,

00:05:38.650 --> 00:05:41.490
you would think at least initially,

00:05:41.490 --> 00:05:44.810
that if this has a North
Pole and a South Pole,

00:05:44.810 --> 00:05:46.780
then maybe you could
separate these two things,

00:05:46.780 --> 00:05:48.680
maybe if you were to break this in half,

00:05:48.680 --> 00:05:51.380
then you could have a
pure North Pole magnet

00:05:51.380 --> 00:05:54.120
and you could have a
pure South Pole magnet,

00:05:54.120 --> 00:05:57.040
but that's not what happens
when you break this in half,

00:05:57.040 --> 00:06:02.040
instead you have two magnets
each with their own North

00:06:02.870 --> 00:06:05.840
and South Poles, and then
you can keep doing this,

00:06:05.840 --> 00:06:08.100
and early scientists just
kept doing it and said

00:06:08.100 --> 00:06:09.380
there might not be any limit

00:06:09.380 --> 00:06:11.220
to how much you can cut this obviously,

00:06:11.220 --> 00:06:13.770
when you keep cutting
and cannot cut any more

00:06:13.770 --> 00:06:15.330
while retaining the properties

00:06:15.330 --> 00:06:18.770
of the magnetite you're getting
down to the molecular level,

00:06:18.770 --> 00:06:20.670
and in future videos
we'll think about how even

00:06:20.670 --> 00:06:25.490
at a molecular level you can
still have a little tiny magnet

00:06:25.490 --> 00:06:28.470
that has a polarity to it.

00:06:28.470 --> 00:06:30.290
Now another really interesting thing

00:06:30.290 --> 00:06:32.220
about magnets is this connection

00:06:32.220 --> 00:06:35.750
between magnetism and electricity.

00:06:35.750 --> 00:06:37.110
People have also observed

00:06:37.110 --> 00:06:40.520
that if you took a current carrying wire

00:06:40.520 --> 00:06:42.130
and the current is going from the positive

00:06:42.130 --> 00:06:47.130
to the negative end so it
is going in this direction,

00:06:47.170 --> 00:06:48.003
and if you make it go

00:06:48.003 --> 00:06:50.680
through a piece of paper
that has metal filings,

00:06:50.680 --> 00:06:54.520
it looks like field
lines are forming here,

00:06:54.520 --> 00:06:57.670
magnetic field lines, it
turns out it does interact

00:06:57.670 --> 00:06:59.770
with the types of things
that tend to interact

00:06:59.770 --> 00:07:01.580
it will interact with magnets

00:07:01.580 --> 00:07:04.810
and so this is a magnetic
field that is being formed

00:07:04.810 --> 00:07:06.640
by an electrical current,

00:07:06.640 --> 00:07:09.970
and so that was the major
clue that the phenomena

00:07:09.970 --> 00:07:13.750
of electricity and magnetism
are in fact related,

00:07:13.750 --> 00:07:15.800
and this relationship is what allows us

00:07:15.800 --> 00:07:18.500
to do things like have electric motors,

00:07:18.500 --> 00:07:23.500
or generate electricity
from wind or water turbines,

00:07:23.600 --> 00:07:25.160
and it's out of the scope of this video

00:07:25.160 --> 00:07:26.830
but I'll give you a little bit of a clue

00:07:26.830 --> 00:07:29.050
of how these things are connected.

00:07:29.050 --> 00:07:31.280
We know that things
they're made up of atoms

00:07:31.280 --> 00:07:34.960
which are made up of particles
like electrons and protons,

00:07:34.960 --> 00:07:37.600
which are charged and
we know that a current,

00:07:37.600 --> 00:07:40.240
an electric current is
based on the movement

00:07:40.240 --> 00:07:42.070
of charged particles,

00:07:42.070 --> 00:07:44.880
that when something is not magnetic,

00:07:44.880 --> 00:07:47.810
say a non-magnetic piece of magnetite,

00:07:47.810 --> 00:07:50.990
all of those charges are
moving more chaotically,

00:07:50.990 --> 00:07:52.670
but if something happens to it,

00:07:52.670 --> 00:07:54.070
maybe a lightning strike,

00:07:54.070 --> 00:07:57.170
it can align how they
move so that they act

00:07:57.170 --> 00:08:01.240
in concert to have a more
coherent magnetic field.

00:08:01.240 --> 00:08:03.760
So leave I'll there I'm
already getting a little bit,

00:08:03.760 --> 00:08:05.600
ahead of our skis so to speak,

00:08:05.600 --> 00:08:08.350
but I do think it's really
interesting to appreciate

00:08:08.350 --> 00:08:10.630
how all of these puzzle pieces that we see

00:08:10.630 --> 00:08:12.393
in nature, fit together.


Introduction to Momentum

WEBVTT
Kind: captions
Language: en

00:00:00.290 --> 00:00:01.123
- [Instructor] In this video,

00:00:01.123 --> 00:00:04.060
we're going to talk a
little bit about momentum.

00:00:04.060 --> 00:00:05.150
And I encourage you to think about

00:00:05.150 --> 00:00:07.780
what does momentum mean
in everyday language.

00:00:07.780 --> 00:00:12.290
If I were to tell you that a
business has a lot of momentum,

00:00:12.290 --> 00:00:14.050
many people would imagine that means

00:00:14.050 --> 00:00:15.520
that it's doing really well,

00:00:15.520 --> 00:00:18.650
it's hard to stop that
business from succeeding.

00:00:18.650 --> 00:00:20.520
If I were to tell you that,

00:00:20.520 --> 00:00:23.330
and let's say a movie star
has a lot of momentum,

00:00:23.330 --> 00:00:25.040
it means that they keep making really,

00:00:25.040 --> 00:00:26.220
really, really good movies,

00:00:26.220 --> 00:00:27.740
it's hard to stop them.

00:00:27.740 --> 00:00:31.150
And in physics, momentum
essentially has that same notion.

00:00:31.150 --> 00:00:34.100
In fact, these everyday
language versions of momentum

00:00:34.100 --> 00:00:36.920
really came from the
physics version of it,

00:00:36.920 --> 00:00:39.860
and an informal definition
you could think of it is

00:00:39.860 --> 00:00:44.860
how hard to stop something.

00:00:45.140 --> 00:00:47.050
Stop something.

00:00:47.050 --> 00:00:48.990
So with that very informal definition,

00:00:48.990 --> 00:00:51.650
we'll get a little bit more
mathy in a few seconds.

00:00:51.650 --> 00:00:53.300
I have two pictures here.

00:00:53.300 --> 00:00:56.920
And let's say both of these vehicles,

00:00:56.920 --> 00:00:59.000
so we have a big 18-wheeler truck here,

00:00:59.000 --> 00:01:01.020
and here we have a Formula One car.

00:01:01.020 --> 00:01:05.170
Let's say that they both have a velocity

00:01:05.170 --> 00:01:08.313
of 20 meters per second.

00:01:09.260 --> 00:01:10.690
Positive 20 meters per second.

00:01:10.690 --> 00:01:12.110
We'll just think in one dimension.

00:01:12.110 --> 00:01:13.410
So let's say it's going to the right

00:01:13.410 --> 00:01:14.660
at 20 meters per second,

00:01:14.660 --> 00:01:17.010
so that's a magnitude and a direction.

00:01:17.010 --> 00:01:18.810
Okay, I could put a positive
here to make it clear

00:01:18.810 --> 00:01:20.680
that I'm giving a direction here.

00:01:20.680 --> 00:01:23.800
So they're both going in
the positive direction

00:01:23.800 --> 00:01:26.500
at 20 meters per second.

00:01:26.500 --> 00:01:29.740
But you can imagine their
masses are very different.

00:01:29.740 --> 00:01:32.120
This large truck would have a mass

00:01:32.120 --> 00:01:37.050
of let's just say 30,000 kilograms.

00:01:37.050 --> 00:01:40.100
That's roughly actually what
an 18-wheeler truck's mass is,

00:01:40.100 --> 00:01:41.490
I looked it up before this video.

00:01:41.490 --> 00:01:42.520
And let's say the mass

00:01:42.520 --> 00:01:46.727
of this Formula One car
is about 1,000 kilograms.

00:01:48.220 --> 00:01:49.730
Now pause this video and think about

00:01:49.730 --> 00:01:53.663
which one do you think
would have more momentum.

00:01:55.660 --> 00:01:57.890
All right, well, you're
probably imagining trying

00:01:57.890 --> 00:01:59.040
to stop either of these,

00:01:59.040 --> 00:02:03.030
and I guess either of these
would be difficult to stop,

00:02:03.030 --> 00:02:07.170
but a truck going at 20 meters
per second seems a lot harder

00:02:07.170 --> 00:02:08.820
than the Formula One car.

00:02:08.820 --> 00:02:10.510
This thing seems like
it would just be able

00:02:10.510 --> 00:02:12.810
to drive through anything.

00:02:12.810 --> 00:02:15.080
So if you picked the truck
having a higher momentum

00:02:15.080 --> 00:02:17.950
when it's going at the same
velocity as a Formula One car,

00:02:17.950 --> 00:02:19.340
you'd be right.

00:02:19.340 --> 00:02:20.490
But an interesting question is well,

00:02:20.490 --> 00:02:22.550
how can I quantify that?

00:02:22.550 --> 00:02:25.330
And that's where we have
the mathematical definition

00:02:25.330 --> 00:02:29.490
of momentum, and momentum is defined

00:02:29.490 --> 00:02:34.463
as mass times velocity.

00:02:35.680 --> 00:02:38.420
And because velocity is a vector,

00:02:38.420 --> 00:02:39.590
it's not just a magnitude,

00:02:39.590 --> 00:02:40.550
we assigned a direction,

00:02:40.550 --> 00:02:42.060
so we said in the positive direction.

00:02:42.060 --> 00:02:42.893
If we were in one dimension,

00:02:42.893 --> 00:02:44.040
if we said in the negative direction,

00:02:44.040 --> 00:02:45.130
we'd be going to the left.

00:02:45.130 --> 00:02:46.940
That's maybe the convention we could use.

00:02:46.940 --> 00:02:50.200
And since it's a scalar, which is mass,

00:02:50.200 --> 00:02:53.340
times a vector, momentum is
also going to be a vector.

00:02:53.340 --> 00:02:57.360
You're going to have a momentum
in a certain direction.

00:02:57.360 --> 00:02:58.670
So pause this video and see

00:02:58.670 --> 00:03:00.180
if you can calculate the momentum

00:03:00.180 --> 00:03:01.800
for each of these vehicles.

00:03:01.800 --> 00:03:03.070
And see also if you can figure out

00:03:03.070 --> 00:03:04.620
what the units are going to be.

00:03:05.770 --> 00:03:07.640
All right, now let's
think about the momentum

00:03:07.640 --> 00:03:09.060
for this 18 wheeler.

00:03:09.060 --> 00:03:11.700
It is going to be its mass,

00:03:11.700 --> 00:03:16.320
which is 30,000 kilograms,

00:03:16.320 --> 00:03:17.990
times its velocity,

00:03:17.990 --> 00:03:22.370
which is positive 20 meters per second.

00:03:22.370 --> 00:03:24.820
And so when I multiply those two things,

00:03:24.820 --> 00:03:25.940
I am going to get,

00:03:25.940 --> 00:03:27.410
let's see, I'm going to get a six

00:03:27.410 --> 00:03:30.590
with one, two, three, four, five zeros.

00:03:30.590 --> 00:03:34.510
One, two, three, four, five zeros.

00:03:34.510 --> 00:03:37.400
600,000, and then the units are

00:03:37.400 --> 00:03:40.080
kilogram times meter per second.

00:03:40.080 --> 00:03:43.890
Kilogram meter per second.

00:03:46.530 --> 00:03:48.470
That's the momentum of the truck.

00:03:48.470 --> 00:03:50.040
Now what about the Formula One car?

00:03:50.040 --> 00:03:53.300
Pause the video and try to
calculate that momentum.

00:03:53.300 --> 00:03:55.100
All right, well, same notion.

00:03:55.100 --> 00:03:56.850
I'll have to squeeze
it in right over here.

00:03:56.850 --> 00:04:01.373
The momentum here is the
mass, 1,000 kilograms.

00:04:02.870 --> 00:04:06.140
1,000 kilograms

00:04:06.140 --> 00:04:07.670
times the velocity,

00:04:07.670 --> 00:04:10.560
positive 20 meters per second.

00:04:10.560 --> 00:04:12.630
I'm stressing the positive
because we're in one dimension,

00:04:12.630 --> 00:04:14.120
and I'm saying positive is one direction,

00:04:14.120 --> 00:04:15.130
negative is the other direction.

00:04:15.130 --> 00:04:16.800
So I am giving it a direction here.

00:04:16.800 --> 00:04:20.940
And so this momentum
is going to be equal to

00:04:20.940 --> 00:04:25.883
20,000 kilogram meters per second.

00:04:28.300 --> 00:04:30.620
Kilogram meters per second.

00:04:30.620 --> 00:04:33.110
And so when you evaluate the momentums,

00:04:33.110 --> 00:04:35.050
it's clear mathematically

00:04:35.050 --> 00:04:38.010
that the truck has much more momentum

00:04:38.010 --> 00:04:40.180
than this Formula One car.

00:04:40.180 --> 00:04:43.350
It has 30 times the momentum.

00:04:43.350 --> 00:04:45.480
Now I know what some
of y'all are thinking.

00:04:45.480 --> 00:04:47.230
Well, this is just from
the reference frame

00:04:47.230 --> 00:04:49.440
if I'm standing still on the ground

00:04:49.440 --> 00:04:52.490
while this truck and car are moving by.

00:04:52.490 --> 00:04:54.540
But what if I were moving along with them?

00:04:54.540 --> 00:04:56.690
Would they have the same momentum?

00:04:56.690 --> 00:04:58.780
And if you were asking that question,

00:04:58.780 --> 00:04:59.613
it's a very good one,

00:04:59.613 --> 00:05:00.490
and the answer is no,

00:05:00.490 --> 00:05:03.570
the momentum depends on
the frame of reference.

00:05:03.570 --> 00:05:04.870
So let's say that you are able

00:05:04.870 --> 00:05:07.100
to travel very quickly in,

00:05:07.100 --> 00:05:09.460
well, let's say you're in a car here.

00:05:09.460 --> 00:05:14.460
And you're also going at
positive 20 meters per second,

00:05:14.560 --> 00:05:17.030
so you're going in the same
direction as the truck.

00:05:17.030 --> 00:05:18.860
Well, relative to you,

00:05:18.860 --> 00:05:21.830
the truck's velocity is now zero.

00:05:21.830 --> 00:05:24.010
So from your frame of reference,

00:05:24.010 --> 00:05:25.610
if you wanted to calculate momentum,

00:05:25.610 --> 00:05:28.470
you'd have the mass of the
truck, 30,000 kilograms.

00:05:28.470 --> 00:05:29.960
But from your frame of reference,

00:05:29.960 --> 00:05:32.350
the velocity would now be zero.

00:05:32.350 --> 00:05:35.720
So if you're traveling in
this car at the same velocity

00:05:35.720 --> 00:05:40.650
as that truck, the momentum
you would calculate is zero.

00:05:40.650 --> 00:05:42.740
Now, what would then have momentum?

00:05:42.740 --> 00:05:43.810
Well, in that world,

00:05:43.810 --> 00:05:47.770
let's say that there's a
big rock over here that,

00:05:47.770 --> 00:05:52.700
let's say this thing's 100,000 kilograms.

00:05:52.700 --> 00:05:54.460
Well, when you are standing on the ground,

00:05:54.460 --> 00:05:56.030
that thing would have no momentum,

00:05:56.030 --> 00:05:58.540
but now that you're moving
20 meters per second

00:05:58.540 --> 00:06:01.470
to the right, positive
20 meters per second,

00:06:01.470 --> 00:06:04.400
the rock in your frame of
reference would look like

00:06:04.400 --> 00:06:06.780
it's going at negative
20 meters per second.

00:06:06.780 --> 00:06:09.550
It would look like it's
going 20 meters per second

00:06:09.550 --> 00:06:10.900
to the left.

00:06:10.900 --> 00:06:13.370
And so this would now, if you're moving

00:06:13.370 --> 00:06:15.870
in the same frame of
reference as that truck,

00:06:15.870 --> 00:06:17.600
this rock would then look like

00:06:17.600 --> 00:06:19.860
it has a very negative momentum,

00:06:19.860 --> 00:06:22.430
and I encourage you to
calculate it if you like.

00:06:22.430 --> 00:06:23.600
But I'll leave you there.

00:06:23.600 --> 00:06:26.060
This is really just an
introduction to momentum.

00:06:26.060 --> 00:06:27.800
But once you get an intuition for it,

00:06:27.800 --> 00:06:30.310
all sorts of interesting
things in physics start

00:06:30.310 --> 00:06:31.483
to emerge from it.


Systems and Objects

WEBVTT
Kind: captions
Language: en

00:00:00.190 --> 00:00:02.650
- [Instructor] Our world is
extraordinarily complicated.

00:00:02.650 --> 00:00:06.060
So in physics we're going to
have to make simplifications.

00:00:06.060 --> 00:00:08.110
Even things in our world that seems simple

00:00:08.110 --> 00:00:09.550
are extraordinarily complicated.

00:00:09.550 --> 00:00:12.720
So consider a basketball
seems simple enough,

00:00:12.720 --> 00:00:15.810
but it's composed of an
extraordinarily large number

00:00:15.810 --> 00:00:19.300
of air molecules bouncing
around inside, colliding

00:00:19.300 --> 00:00:22.080
with the outside leather
and rubber membrane,

00:00:22.080 --> 00:00:26.800
which itself is composed of an
extraordinarily large number

00:00:26.800 --> 00:00:29.470
of atoms and molecules all bonded together

00:00:29.470 --> 00:00:32.020
holding on tight, trying
to prevent themselves

00:00:32.020 --> 00:00:35.990
from being ripped apart and
exploded by the pressure inside.

00:00:35.990 --> 00:00:39.490
So do we have to keep track
of every atom and molecule

00:00:39.490 --> 00:00:42.770
in this ball to include
it in a physics problem?

00:00:42.770 --> 00:00:45.390
Typically not, nor would
we ever really want to.

00:00:45.390 --> 00:00:48.430
I mean, we can't keep track
of all that info, not yet,

00:00:48.430 --> 00:00:50.120
nor would you want to for most scenarios.

00:00:50.120 --> 00:00:52.500
So for instance, if you were an astronaut,

00:00:52.500 --> 00:00:53.670
you went to the moon,

00:00:53.670 --> 00:00:56.120
you took your basketball and
you were going to drop it.

00:00:56.120 --> 00:00:58.650
If all you wanted to know
was how long it's gonna take

00:00:58.650 --> 00:01:01.980
for this ball to strike
the lunar surface below,

00:01:01.980 --> 00:01:04.270
you don't need to know
about the ideal gas law,

00:01:04.270 --> 00:01:06.880
you don't need to know about
the structural integrity

00:01:06.880 --> 00:01:09.310
of the rubber leather membrane.

00:01:09.310 --> 00:01:11.180
You could solve this by
treating the basketball

00:01:11.180 --> 00:01:13.350
as if there was no internal
structure whatsoever.

00:01:13.350 --> 00:01:14.960
Like you were dropping a rock that had

00:01:14.960 --> 00:01:18.600
no interesting internal
structure whatsoever.

00:01:18.600 --> 00:01:21.250
So in physics, the good
news is we can typically get

00:01:21.250 --> 00:01:23.070
away with making a lot of simplifications

00:01:23.070 --> 00:01:24.980
and ignoring the internal structure

00:01:24.980 --> 00:01:27.820
if it isn't relevant to the
problem that we're asking.

00:01:27.820 --> 00:01:30.150
Sometimes it will be relevant though.

00:01:30.150 --> 00:01:32.590
So here was a case where
it wasn't relevant.

00:01:32.590 --> 00:01:34.240
The internal structure wasn't relevant.

00:01:34.240 --> 00:01:36.170
So we could ignore that
internal structure,

00:01:36.170 --> 00:01:38.660
but other questions like
if you were an astronaut,

00:01:38.660 --> 00:01:40.300
I mean if I was an
astronaut, and I was bringing

00:01:40.300 --> 00:01:42.870
my basketball to the moon,
I'd be like, wait a minute,

00:01:42.870 --> 00:01:44.570
there's no atmosphere on the moon.

00:01:44.570 --> 00:01:48.080
That means there is no pressure
pushing in from the outside.

00:01:48.080 --> 00:01:49.980
That means all this air
pressure's still pushing

00:01:49.980 --> 00:01:53.950
out from the inside is my
basketball just gonna explode?

00:01:53.950 --> 00:01:55.500
I'd want to know this before
I brought it out there.

00:01:55.500 --> 00:01:57.170
I don't wanna carry a little bomb out

00:01:57.170 --> 00:01:59.020
that's going to like blow up in my face

00:01:59.020 --> 00:02:00.630
and I don't want to lose a basketball.

00:02:00.630 --> 00:02:02.050
If you wanted to know if your basketball

00:02:02.050 --> 00:02:04.150
was going to explode,
okay now it does depend.

00:02:04.150 --> 00:02:07.140
That question does depend
on the internal structure.

00:02:07.140 --> 00:02:08.570
It depends on the pressure inside

00:02:08.570 --> 00:02:09.800
which is fundamentally related

00:02:09.800 --> 00:02:14.030
to the force of the collisions
between these air molecules

00:02:14.030 --> 00:02:17.690
and the rubber membrane and
then it depends also on,

00:02:17.690 --> 00:02:20.360
well, how strong are the bonds
between these rubber membrane

00:02:20.360 --> 00:02:21.800
and leather molecules?

00:02:21.800 --> 00:02:25.270
How much force can they
withstand before they burst?

00:02:25.270 --> 00:02:27.060
For that question you
would have to consider

00:02:27.060 --> 00:02:28.690
the internal structure.

00:02:28.690 --> 00:02:30.920
So, in some questions you get to ignore

00:02:30.920 --> 00:02:33.370
the internal structure and
other questions you don't.

00:02:33.370 --> 00:02:35.880
It's just context and question-dependent

00:02:35.880 --> 00:02:38.300
and in physics, we have
terminology to sort of

00:02:38.300 --> 00:02:41.270
sort this out and the
terminology we use is the idea

00:02:41.270 --> 00:02:43.240
of a system or the idea of an object.

00:02:43.240 --> 00:02:45.650
So the idea of a system
is just a collection

00:02:45.650 --> 00:02:49.430
of objects, that's the definition
of a system in physics.

00:02:49.430 --> 00:02:50.670
But that begs the question,

00:02:50.670 --> 00:02:52.680
well what do we mean by an object?

00:02:52.680 --> 00:02:55.330
By an object, we mean
anything that you could treat

00:02:55.330 --> 00:02:58.260
as if it had no internal structure.

00:02:58.260 --> 00:03:01.180
We don't mean that objects
have no internal structure.

00:03:01.180 --> 00:03:04.870
They typically do, the only
things that don't truly

00:03:04.870 --> 00:03:06.720
have an internal structures
as far as we know

00:03:06.720 --> 00:03:10.830
are truly fundamental particles
like electrons or neutrinos,

00:03:10.830 --> 00:03:13.510
these fundamental particles
in particle physics

00:03:13.510 --> 00:03:15.950
that as far as we know,
have no internal structure.

00:03:15.950 --> 00:03:18.190
So unless you're doing particle physics,

00:03:18.190 --> 00:03:20.340
you're probably don't have a true object,

00:03:20.340 --> 00:03:22.290
but you can treat things like an object.

00:03:22.290 --> 00:03:24.420
We can treat this
basketball like an object.

00:03:24.420 --> 00:03:26.510
That is to say, we can act as if it has

00:03:26.510 --> 00:03:29.210
no internal structure if
that internal structure

00:03:29.210 --> 00:03:30.690
isn't relevant to the problem.

00:03:30.690 --> 00:03:34.170
So to make this a little more meaningful

00:03:34.170 --> 00:03:35.490
just imagine another example.

00:03:35.490 --> 00:03:40.490
Say you collide two objects
so you collide a putty here.

00:03:40.590 --> 00:03:43.600
Let's say this is three kilogram object

00:03:43.600 --> 00:03:45.300
and it comes in with a certain speed

00:03:45.300 --> 00:03:49.070
and it collides with a
five kilogram object.

00:03:49.070 --> 00:03:52.050
If all you want to know is
when they stick together,

00:03:52.050 --> 00:03:54.160
say these stick together and move off

00:03:54.160 --> 00:03:55.830
with some common speed.

00:03:55.830 --> 00:03:57.950
If all you want to know is
what is that common speed

00:03:57.950 --> 00:04:00.690
that they move off with
after they stick together?

00:04:00.690 --> 00:04:02.050
Notice what you don't need to know.

00:04:02.050 --> 00:04:03.700
I don't need to tell
you that this was made

00:04:03.700 --> 00:04:07.170
out of gold here or that this
one was made out of copper

00:04:07.170 --> 00:04:08.460
as long as you know, the masses and that

00:04:08.460 --> 00:04:10.290
they stick together,
physics will let you solve

00:04:10.290 --> 00:04:13.440
for how fast they'll move off
with a common speed afterward

00:04:13.440 --> 00:04:15.380
if you tell me that they stick together.

00:04:15.380 --> 00:04:17.230
So if that's all you want to know

00:04:17.230 --> 00:04:19.710
doesn't matter what the
internal structure is.

00:04:19.710 --> 00:04:22.070
However, for other questions,
if you wanted to know

00:04:22.070 --> 00:04:25.690
if this was going to set off
some nuclear explosion, okay

00:04:25.690 --> 00:04:28.980
well then it really is going
to matter if these are made

00:04:28.980 --> 00:04:32.320
out of gold, made out of
copper, made out of clay

00:04:32.320 --> 00:04:34.610
or if they're made out
of uranium, so to speak.

00:04:34.610 --> 00:04:37.070
So for that question, you do need to know

00:04:37.070 --> 00:04:38.680
about the internal structure.

00:04:38.680 --> 00:04:40.040
So the idea of a system and the idea

00:04:40.040 --> 00:04:42.580
of an object is an
important one in physics

00:04:42.580 --> 00:04:46.240
and it's not just important
conceptually or abstractly

00:04:46.240 --> 00:04:48.210
it can actually help
you in problem solving.

00:04:48.210 --> 00:04:51.350
So let me show you a more
like, tangible example

00:04:51.350 --> 00:04:52.400
of where this might help you

00:04:52.400 --> 00:04:55.150
in solving a problem you might encounter

00:04:55.150 --> 00:04:56.420
in your physics courses.

00:04:56.420 --> 00:04:58.540
So let's say they're two boxes

00:04:58.540 --> 00:05:00.650
and they're just too big
and unwieldy to handle.

00:05:00.650 --> 00:05:02.350
So you're going to push
them across the floor.

00:05:02.350 --> 00:05:05.190
They're not heavy, they're just
like shaped weird, let's say

00:05:05.190 --> 00:05:07.640
and let's say the floor
has been newly waxed.

00:05:07.640 --> 00:05:09.980
So it's real slick against these boxes

00:05:09.980 --> 00:05:12.650
which are also slick and
there's negligible friction.

00:05:12.650 --> 00:05:14.360
You could ignore the
friction between the boxes

00:05:14.360 --> 00:05:16.220
and the floor, so let's say you come up

00:05:16.220 --> 00:05:17.730
and you're gonna push on these things.

00:05:17.730 --> 00:05:19.570
Push them into the
corner of some warehouse,

00:05:19.570 --> 00:05:22.060
you're working in the
warehouse here, earning

00:05:22.060 --> 00:05:23.610
your pay for the day
and you're going to go

00:05:23.610 --> 00:05:25.890
push these over here and
you're going to exert,

00:05:25.890 --> 00:05:28.310
let's just say nine newtons of force

00:05:28.310 --> 00:05:30.680
on this one kilogram
box and then that pushes

00:05:30.680 --> 00:05:31.930
into the two kilogram box

00:05:31.930 --> 00:05:33.470
and they move off to the right.

00:05:33.470 --> 00:05:35.270
So can we treat this system

00:05:35.270 --> 00:05:37.400
of boxes as if it were a single object?

00:05:37.400 --> 00:05:39.570
Well, like we said,
it's question-dependent.

00:05:39.570 --> 00:05:43.040
If the question we want to ask
is, what's the acceleration

00:05:43.040 --> 00:05:45.630
of these boxes as they slide to the right?

00:05:45.630 --> 00:05:47.560
Well, they're going to
move at the same rate

00:05:47.560 --> 00:05:49.740
because as you push on
this one kilogram box

00:05:49.740 --> 00:05:52.610
that one kilogram box pushes
on the two kilogram box

00:05:52.610 --> 00:05:53.770
and they're going to move together.

00:05:53.770 --> 00:05:57.020
As I keep pushing with nine
newtons, the velocity of both

00:05:57.020 --> 00:05:59.200
of these boxes are going
to be the same to the right

00:05:59.200 --> 00:06:00.790
and the acceleration of the boxes

00:06:00.790 --> 00:06:01.900
are going to be the same to the right.

00:06:01.900 --> 00:06:03.660
They're never going to become separated.

00:06:03.660 --> 00:06:06.060
What that means is the fact
that there were two boxes

00:06:06.060 --> 00:06:09.050
didn't matter, I can treat
this system of two boxes

00:06:09.050 --> 00:06:12.840
as if it were a single three kilogram box.

00:06:12.840 --> 00:06:14.240
I don't even need to know
that they were actually

00:06:14.240 --> 00:06:16.330
a division here, 'cause
they're never going to become

00:06:16.330 --> 00:06:18.960
separated for this question
that I'm asking here.

00:06:18.960 --> 00:06:20.520
So I could treat this whole system

00:06:20.520 --> 00:06:24.330
as if it were just one
big three kilogram object

00:06:24.330 --> 00:06:26.460
and this is an important idea.

00:06:26.460 --> 00:06:28.080
The properties of a system,

00:06:28.080 --> 00:06:30.660
like the mass of the
system, are determined

00:06:30.660 --> 00:06:33.180
by the properties of the
objects in that system.

00:06:33.180 --> 00:06:36.350
So I put a three here and this
is legal, this is allowed.

00:06:36.350 --> 00:06:39.700
The properties of this total
mass of my system is determined

00:06:39.700 --> 00:06:42.080
by the mass of the individual
objects in my system.

00:06:42.080 --> 00:06:44.030
So you really can just add up these masses

00:06:44.030 --> 00:06:47.010
to determine the total mass of the system

00:06:47.010 --> 00:06:49.120
that you're going to be
treating as a single object.

00:06:49.120 --> 00:06:51.210
And now that I get to treat
this as a single object

00:06:51.210 --> 00:06:54.680
I'm in luck, I can use
Newton's second law.

00:06:54.680 --> 00:06:56.470
The acceleration is going
to equal the net force

00:06:56.470 --> 00:06:59.870
over the mass, we'll do this
for the horizontal direction.

00:06:59.870 --> 00:07:01.470
I'm just going to put a mass of three.

00:07:01.470 --> 00:07:04.600
I could ignore the fact
that this was a one and two

00:07:04.600 --> 00:07:06.570
and the total mass of
my system is going to be

00:07:06.570 --> 00:07:09.910
three kilograms and the
only force on my system

00:07:09.910 --> 00:07:11.180
that I'm treating as an object here

00:07:11.180 --> 00:07:12.560
is the nine Newton force.

00:07:12.560 --> 00:07:14.110
I could ignore, in other words,

00:07:14.110 --> 00:07:17.920
I can ignore the internal
forces between these boxes.

00:07:17.920 --> 00:07:19.580
I don't care about the
one pushing on the two

00:07:19.580 --> 00:07:21.820
or the two pushing on the
one I'm treating the system

00:07:21.820 --> 00:07:24.890
like an object and I'm ignoring
that internal structure

00:07:24.890 --> 00:07:26.520
that makes this problem really easy

00:07:26.520 --> 00:07:27.780
when I solve for the acceleration

00:07:27.780 --> 00:07:31.090
I just get three meters
per second squared.

00:07:31.090 --> 00:07:33.330
So for this question I
could treat the system

00:07:33.330 --> 00:07:34.640
as a single object.

00:07:34.640 --> 00:07:37.260
What question would I not
be able to treat the system

00:07:37.260 --> 00:07:38.610
as a single object for?

00:07:38.610 --> 00:07:41.470
Well, if I wanted to know,
let's say the question was

00:07:41.470 --> 00:07:44.270
with how much force does
the one kilogram box exert

00:07:44.270 --> 00:07:46.330
on the two kilogram box?

00:07:46.330 --> 00:07:48.380
And you might think, oh,
it's just nine, but it isn't.

00:07:48.380 --> 00:07:50.420
So stay tuned, hold on.

00:07:50.420 --> 00:07:51.910
It's counterintuitive. I know,

00:07:51.910 --> 00:07:53.590
but the main idea I'm
trying to stress here

00:07:53.590 --> 00:07:58.150
is that this force on two
by one is fundamentally

00:07:58.150 --> 00:08:00.660
a question about an internal force.

00:08:00.660 --> 00:08:02.350
So if the question you're asking is about

00:08:02.350 --> 00:08:04.920
the internal structure,
clearly you're not allowed

00:08:04.920 --> 00:08:07.120
to ignore the internal structure.

00:08:07.120 --> 00:08:10.110
So for this question we
cannot treat the system

00:08:10.110 --> 00:08:13.180
of two boxes as if it were a single mass.

00:08:13.180 --> 00:08:14.980
We'll have to focus on
the internal structure.

00:08:14.980 --> 00:08:18.170
So again, consider this a
one and a two separate boxes

00:08:18.170 --> 00:08:19.530
and we'll do the same formula.

00:08:19.530 --> 00:08:22.910
Acceleration's going to equal
the net force over the mass,

00:08:22.910 --> 00:08:25.700
but this time we do have
to focus on a single mass.

00:08:25.700 --> 00:08:29.870
So we'll focus just on
the two kilogram mass

00:08:29.870 --> 00:08:32.590
the only horizontal force
on this two kilogram mass

00:08:32.590 --> 00:08:34.310
if this really is frictionless

00:08:34.310 --> 00:08:37.860
is this force that we want to
find the force onto by one.

00:08:37.860 --> 00:08:39.470
And that's the only force that's exerted

00:08:39.470 --> 00:08:41.030
on the two kilogram mass.

00:08:41.030 --> 00:08:43.780
This nine newtons is
exerted directly on the one.

00:08:43.780 --> 00:08:45.380
So it's not directly exerted on the two.

00:08:45.380 --> 00:08:46.500
We don't draw that up here.

00:08:46.500 --> 00:08:47.520
We don't include that here.

00:08:47.520 --> 00:08:50.300
These are only forces
directly on the two, and then

00:08:50.300 --> 00:08:52.730
we'd have to put the acceleration
of the two kilogram mass,

00:08:52.730 --> 00:08:55.340
but we already found that this
three was the acceleration

00:08:55.340 --> 00:08:57.590
of the one, the two and the entire system.

00:08:57.590 --> 00:08:59.240
Everything was accelerating
at the same rate.

00:08:59.240 --> 00:09:02.130
So I can put my three meters
per second squared here

00:09:02.130 --> 00:09:04.510
and I find out that the force exerted

00:09:04.510 --> 00:09:08.320
on the two by the one is six newtons.

00:09:08.320 --> 00:09:11.070
So it's not as big and
this isn't surprising.

00:09:11.070 --> 00:09:13.550
It takes more newtons from the left here

00:09:13.550 --> 00:09:16.100
this nine newtons to
accelerate the entire system

00:09:16.100 --> 00:09:18.890
of three kilograms than it does to just

00:09:18.890 --> 00:09:21.080
accelerate the two
kilogram mass over here.

00:09:21.080 --> 00:09:24.300
So the fact that this force
is accelerating less mass

00:09:24.300 --> 00:09:26.030
means it doesn't have to be as big.

00:09:26.030 --> 00:09:28.010
But the key idea is that to find that,

00:09:28.010 --> 00:09:30.410
we could not treat, to
find this force here,

00:09:30.410 --> 00:09:33.640
we could not treat this entire
system as a single mass.

00:09:33.640 --> 00:09:37.550
So recapping, if the question
being asked does not depend

00:09:37.550 --> 00:09:40.190
on the internal structure,
you can simplify your life

00:09:40.190 --> 00:09:42.160
by treating that structure and that system

00:09:42.160 --> 00:09:44.840
as if it were a single
object, in which case,

00:09:44.840 --> 00:09:46.680
the properties of that will be determined

00:09:46.680 --> 00:09:48.920
by the properties of the
objects in that system.

00:09:48.920 --> 00:09:51.590
But if the question being
asked does depend on

00:09:51.590 --> 00:09:53.470
the internal structure,
then you cannot treat

00:09:53.470 --> 00:09:55.420
that system as a single object.

00:09:55.420 --> 00:09:57.953
You will have to focus on
the internal structure.


Contact Forces

WEBVTT
Kind: captions
Language: en

00:00:00.330 --> 00:00:01.370
- [Instructor] There are
a lot of different types

00:00:01.370 --> 00:00:02.550
of forces in physics,

00:00:02.550 --> 00:00:03.540
but for the most part,

00:00:03.540 --> 00:00:07.240
all forces can be
categorized as either being

00:00:07.240 --> 00:00:10.950
a contact force or a long range force.

00:00:10.950 --> 00:00:13.640
So contact forces as the name suggests

00:00:13.640 --> 00:00:14.790
requires the two objects

00:00:14.790 --> 00:00:16.310
that are exerting a force on each other

00:00:16.310 --> 00:00:19.130
to be touching or in contact.

00:00:19.130 --> 00:00:22.010
So tension, the normal
force, frictional forces,

00:00:22.010 --> 00:00:26.000
these are all common everyday
examples of contact forces.

00:00:26.000 --> 00:00:29.260
So, you know, this wire from this crane

00:00:29.260 --> 00:00:31.860
can exert a contact force
i.e. attention force

00:00:31.860 --> 00:00:34.400
on the wrecking ball but that wire

00:00:34.400 --> 00:00:36.810
can only exert that tension
force on the wrecking ball

00:00:36.810 --> 00:00:39.190
if the wire is actually connected to

00:00:39.190 --> 00:00:41.350
i.e. touching the wrecking ball,

00:00:41.350 --> 00:00:43.870
if you forgot to tie the
wire to the wrecking ball,

00:00:43.870 --> 00:00:46.000
that wire is not gonna exert any tension

00:00:46.000 --> 00:00:47.440
on the wrecking ball.

00:00:47.440 --> 00:00:51.310
So these contact forces are
to be distinguished from

00:00:51.310 --> 00:00:52.660
long range forces.

00:00:52.660 --> 00:00:55.950
Sometimes these are called
action at a distance forces

00:00:55.950 --> 00:00:57.520
because they can be exerted

00:00:57.520 --> 00:00:59.690
on objects that are far
away from each other,

00:00:59.690 --> 00:01:01.453
so gravity is a common example,

00:01:02.480 --> 00:01:05.390
the earth can exert a
gravitational force on the moon

00:01:05.390 --> 00:01:07.560
even though the earth and
the moon aren't touching.

00:01:07.560 --> 00:01:09.560
So that's a group long range force.

00:01:09.560 --> 00:01:12.840
Similarly, the electric force
can exert a repulsive force

00:01:12.840 --> 00:01:15.130
on two charges if they're not touching,

00:01:15.130 --> 00:01:17.380
so not a contact force,

00:01:17.380 --> 00:01:19.420
and magnets can attract each other

00:01:19.420 --> 00:01:20.690
even if they're not touching,

00:01:20.690 --> 00:01:24.420
so those are all long range or
action at a distance forces.

00:01:24.420 --> 00:01:26.290
But I'll be honest with you here,

00:01:26.290 --> 00:01:29.410
this distinction is not
nearly as fundamental

00:01:29.410 --> 00:01:31.010
as it might seem at first.

00:01:31.010 --> 00:01:34.620
All of these forces that
we call contact forces

00:01:34.620 --> 00:01:38.880
are really just an enormous
number of long range forces

00:01:38.880 --> 00:01:40.190
in disguise.

00:01:40.190 --> 00:01:42.930
In other words, these contact
forces, tension, normal force

00:01:42.930 --> 00:01:46.740
and friction are all arising
microscopically due to

00:01:46.740 --> 00:01:48.620
a bunch of long range forces

00:01:48.620 --> 00:01:50.910
acting over really short distances.

00:01:50.910 --> 00:01:53.250
So just cause they're
called long range forces

00:01:53.250 --> 00:01:56.640
doesn't mean they can't exert
force over small distances,

00:01:56.640 --> 00:01:59.510
and in fact, all those forces arise,

00:01:59.510 --> 00:02:01.330
you know, cause all these forces to arise.

00:02:01.330 --> 00:02:03.720
So let me go through and explain
how all these come about.

00:02:03.720 --> 00:02:06.000
So we'll start with tension here.

00:02:06.000 --> 00:02:07.450
Where does tension come from?

00:02:07.450 --> 00:02:09.960
Well, tension is the
force exerted by a wire

00:02:09.960 --> 00:02:12.840
or a cable or a string,
something like that.

00:02:12.840 --> 00:02:14.130
And so these strings,

00:02:14.130 --> 00:02:15.780
they're made out of atoms and molecules,

00:02:15.780 --> 00:02:17.400
I'm trying to represent that over here.

00:02:17.400 --> 00:02:20.690
Your string is probably
more than three atoms wide,

00:02:20.690 --> 00:02:22.790
but I didn't wanna have to
draw an enormous number here,

00:02:22.790 --> 00:02:24.550
so I imagine you've got
a certain number of atoms

00:02:24.550 --> 00:02:26.810
and molecules in your string,

00:02:26.810 --> 00:02:29.500
well, these atoms and molecules
are all bonded together,

00:02:29.500 --> 00:02:30.770
chemically bonded together,

00:02:30.770 --> 00:02:33.543
those are all electromagnetic bonds here,

00:02:35.103 --> 00:02:38.200
they don't wanna move away from
their equilibrium position,

00:02:38.200 --> 00:02:40.490
they have a position and if
they get displaced from there,

00:02:40.490 --> 00:02:42.450
they wanna go back to that spot.

00:02:42.450 --> 00:02:44.560
So that's what it means
to be in a solid here.

00:02:44.560 --> 00:02:48.100
So this wire, if you connect
to a heavy load to it

00:02:48.100 --> 00:02:49.200
like a wrecking ball,

00:02:49.200 --> 00:02:51.270
that wrecking ball is gonna
try to rip these atoms

00:02:51.270 --> 00:02:52.540
and molecules apart,

00:02:52.540 --> 00:02:55.230
it's gonna try to pull
them away from each other,

00:02:55.230 --> 00:02:56.500
but they don't wanna move
away from each other,

00:02:56.500 --> 00:02:59.390
in other words, they try
to restore themselves

00:02:59.390 --> 00:03:01.720
as this distance between these atoms

00:03:01.720 --> 00:03:03.180
and molecules gets bigger,

00:03:03.180 --> 00:03:05.720
and it does, you'll stretch
your string or your wire

00:03:05.720 --> 00:03:08.590
sometimes imperceptibly, but a little bit,

00:03:08.590 --> 00:03:10.340
these distances get bigger,

00:03:10.340 --> 00:03:13.090
that force holding them
together gets bigger,

00:03:13.090 --> 00:03:14.900
so more tension force occurs

00:03:14.900 --> 00:03:18.840
and this is the microscopic
origin of that tension force.

00:03:18.840 --> 00:03:21.030
These atoms and molecules
wanna restore themselves

00:03:21.030 --> 00:03:23.890
to their previous length and

00:03:23.890 --> 00:03:25.530
to do that they have to
pull harder and harder.

00:03:25.530 --> 00:03:27.070
Now, this won't last forever,

00:03:27.070 --> 00:03:29.200
you hang a heavy enough load over here,

00:03:29.200 --> 00:03:31.810
you'll overwhelm these
electromagnetic bonds

00:03:31.810 --> 00:03:34.600
and you'll rip these molecules apart

00:03:34.600 --> 00:03:36.260
and that's what happens
when your string breaks.

00:03:36.260 --> 00:03:38.680
So that's the microscopic
origin of tension

00:03:38.680 --> 00:03:42.860
but you don't have to draw an
Avogadro's number of arrows,

00:03:42.860 --> 00:03:45.430
we just represent the
tension with one arrow up,

00:03:45.430 --> 00:03:47.290
it turns out you can pretty much summarize

00:03:47.290 --> 00:03:51.250
all of those microscopic
electromagnetic chemical bonds

00:03:51.250 --> 00:03:54.420
with one arrow that we call tension.

00:03:54.420 --> 00:03:56.580
So how about the normal force,
where does that come from?

00:03:56.580 --> 00:03:58.060
Well, this is kind of the opposite.

00:03:58.060 --> 00:03:59.600
Tension is a pulling force,

00:03:59.600 --> 00:04:02.930
the normal force is the force
that tries to prevent two

00:04:02.930 --> 00:04:05.390
objects from getting
smashed into each other.

00:04:05.390 --> 00:04:07.100
So now instead of the atoms and molecules

00:04:07.100 --> 00:04:08.470
trying to get ripped apart,

00:04:08.470 --> 00:04:11.300
the atoms and molecules
in this green box here

00:04:11.300 --> 00:04:13.960
due to its weight are
trying to get shoved into

00:04:13.960 --> 00:04:15.590
the atoms and molecules this table,

00:04:15.590 --> 00:04:17.330
so I've tried to represent that here,

00:04:17.330 --> 00:04:19.340
again, the box and the
table are made out of more

00:04:19.340 --> 00:04:21.500
than these number of atoms and molecules,

00:04:21.500 --> 00:04:24.290
but you've got your atoms
molecules of the box,

00:04:24.290 --> 00:04:26.820
atoms and molecules of the table,

00:04:26.820 --> 00:04:28.900
they won't get moved into each other,

00:04:28.900 --> 00:04:31.110
there's gonna be an electron cloud

00:04:31.110 --> 00:04:33.120
around these atoms and
molecules of the box

00:04:33.120 --> 00:04:35.010
and similarly for the table,

00:04:35.010 --> 00:04:37.420
there's gonna be an
electromagnetic repulsion

00:04:37.420 --> 00:04:38.890
when they try to overlap

00:04:38.890 --> 00:04:40.840
and other quantum mechanical effects,

00:04:40.840 --> 00:04:43.820
it turns out it's surprisingly
complicated to explain

00:04:43.820 --> 00:04:47.020
why matter is solid and it
can't penetrate each other,

00:04:47.020 --> 00:04:50.240
but the enormous number of
electromagnetic interactions

00:04:50.240 --> 00:04:52.970
and other quantum mechanical
effects between these

00:04:52.970 --> 00:04:56.190
atoms and molecules are
the microscopic origin

00:04:56.190 --> 00:04:57.200
of the normal force.

00:04:57.200 --> 00:04:59.880
So again, it's, you know,
action at a distance

00:04:59.880 --> 00:05:01.710
over a small scale,

00:05:01.710 --> 00:05:04.583
which really bugs people out,
they're like, wait a minute,

00:05:05.659 --> 00:05:07.550
are two things ever actually touching?

00:05:07.550 --> 00:05:09.300
You know, as you sit in a chair,

00:05:09.300 --> 00:05:12.440
do the atoms and molecules of your pants

00:05:12.440 --> 00:05:14.690
actually physically touch?

00:05:14.690 --> 00:05:17.800
Hard to actually define
what it means touching here,

00:05:17.800 --> 00:05:20.440
you know, you've got these
amorphous electron clouds,

00:05:20.440 --> 00:05:22.430
how do you define
whether they're touching?

00:05:22.430 --> 00:05:24.930
Hard to do, but good news,
we don't have to do it,

00:05:24.930 --> 00:05:27.400
we can actually just
summarize microscopically

00:05:27.400 --> 00:05:29.620
all of these microscopic interactions

00:05:29.620 --> 00:05:33.960
as one big normal force and that helps us

00:05:33.960 --> 00:05:36.330
both calculationally and conceptually

00:05:36.330 --> 00:05:38.470
knock it to at last here.

00:05:38.470 --> 00:05:39.760
Now, you might be disturbed here,

00:05:39.760 --> 00:05:40.650
you might be like, wait a minute,

00:05:40.650 --> 00:05:43.160
this whole video is about contact forces,

00:05:43.160 --> 00:05:43.993
you're telling me,

00:05:43.993 --> 00:05:46.310
we don't even know if two
surfaces are in contact,

00:05:46.310 --> 00:05:48.170
well, I'm saying it's hard to define

00:05:48.170 --> 00:05:50.370
but here's a good way to define it,

00:05:50.370 --> 00:05:52.370
your pants, atoms and molecules

00:05:52.370 --> 00:05:56.050
are contacting the seat's
atom molecules as soon as

00:05:56.050 --> 00:05:57.650
you notice that force

00:05:57.650 --> 00:05:59.760
preventing them from
moving into each other.

00:05:59.760 --> 00:06:01.830
So as soon as you could
detect this normal force,

00:06:01.830 --> 00:06:03.040
that's as good a way as any

00:06:03.040 --> 00:06:06.460
to define two surfaces
as being in contact.

00:06:06.460 --> 00:06:08.680
So let's look at some other forces.

00:06:08.680 --> 00:06:10.640
So how about the frictional force?

00:06:10.640 --> 00:06:13.180
What are the microscopic
origins of the frictional force?

00:06:13.180 --> 00:06:15.800
Well, you know, the frictional
force is the force that

00:06:15.800 --> 00:06:19.660
resist two surfaces from being
dragged across each other.

00:06:19.660 --> 00:06:21.220
Why is there a resistive force?

00:06:21.220 --> 00:06:23.920
Well, if you zoomed in on these surfaces,

00:06:23.920 --> 00:06:25.800
a table, no matter how smooth it looks,

00:06:25.800 --> 00:06:27.400
even if you just wiped it down,

00:06:27.400 --> 00:06:28.570
if you zoomed in close enough,

00:06:28.570 --> 00:06:30.570
you'd be shocked at all the little

00:06:30.570 --> 00:06:32.890
crevices and cracks and valleys involved,

00:06:32.890 --> 00:06:34.600
the whole world you don't know about

00:06:34.600 --> 00:06:36.280
unless you look at it microscopically,

00:06:36.280 --> 00:06:39.380
and similarly for this purple
box, maybe it's cardboard,

00:06:39.380 --> 00:06:41.960
if you zoomed in microscopically,

00:06:41.960 --> 00:06:45.730
again, it's astonishing how
not smooth those surfaces are.

00:06:45.730 --> 00:06:48.467
So obviously, if you tried
to drag one across the other

00:06:48.467 --> 00:06:49.970
and these are bumping into each other,

00:06:49.970 --> 00:06:52.570
these hills and valleys are
running into each other,

00:06:52.570 --> 00:06:53.940
that's gonna be a problem

00:06:53.940 --> 00:06:55.650
that's gonna cause a resistive force.

00:06:55.650 --> 00:06:59.150
You might break this, you
know, yellow hill off,

00:06:59.150 --> 00:07:00.500
sometimes they just bust off,

00:07:00.500 --> 00:07:04.270
yep, that's gonna be a
resistive cause of friction.

00:07:04.270 --> 00:07:05.380
Sometimes they don't bust off,

00:07:05.380 --> 00:07:07.410
maybe they just like bend and bounce back

00:07:07.410 --> 00:07:08.980
but even if they do,

00:07:08.980 --> 00:07:10.500
still gonna cause a frictional force

00:07:10.500 --> 00:07:12.300
and add to this friction,

00:07:12.300 --> 00:07:15.330
and it's not just that
but sometimes even like

00:07:15.330 --> 00:07:17.800
the atoms and molecules
in the surface over here,

00:07:17.800 --> 00:07:19.380
look, this pot doesn't look too bad,

00:07:19.380 --> 00:07:20.960
it looks like they could
slide across each other

00:07:20.960 --> 00:07:22.300
pretty well,

00:07:22.300 --> 00:07:25.760
but there can be adhesion
like molecular bonds that form

00:07:25.760 --> 00:07:28.110
between those atoms and molecules
that are near each other,

00:07:28.110 --> 00:07:30.410
that can also contribute to friction.

00:07:30.410 --> 00:07:32.350
So again, astonishingly complicated,

00:07:32.350 --> 00:07:35.080
there's actually lots of
questions to still be answered

00:07:35.080 --> 00:07:36.220
in studying friction,

00:07:36.220 --> 00:07:39.150
and the study of friction
is called tribology.

00:07:39.150 --> 00:07:42.210
Shockingly, a lot of
questions to this day but,

00:07:42.210 --> 00:07:44.110
the good news is you can summarize

00:07:44.110 --> 00:07:47.840
all of those microscopic
interactions as one force

00:07:47.840 --> 00:07:50.090
we call friction that resists

00:07:50.090 --> 00:07:52.570
the two surfaces from
sliding over each other.

00:07:52.570 --> 00:07:54.970
So you don't have to do
a lot of calculations

00:07:54.970 --> 00:07:57.750
and microscopically
zoom in on the surface,

00:07:57.750 --> 00:07:59.510
we can pretty much account for all of it

00:07:59.510 --> 00:08:01.930
by simply drawing it as
one big resistive force

00:08:01.930 --> 00:08:03.770
of friction backwards.

00:08:03.770 --> 00:08:06.670
So recapping, contact
forces are those forces

00:08:06.670 --> 00:08:09.510
that require the two objects
interacting to be touching

00:08:09.510 --> 00:08:11.190
for that force to occur,

00:08:11.190 --> 00:08:13.860
but we've seen that these
contact forces are actually due

00:08:13.860 --> 00:08:17.580
to a mind bogglingly large
number of long range forces

00:08:17.580 --> 00:08:19.790
all acting over a very short distance,

00:08:19.790 --> 00:08:22.210
but you can summarize all
those long range forces

00:08:22.210 --> 00:08:24.560
as a single contact force

00:08:24.560 --> 00:08:28.083
when doing most introductory
physics problems.


Using matrices to represent data: Payoffs

WEBVTT
Kind: captions
Language: en

00:00:00.400 --> 00:00:01.280
- [Instructor] We're told Violet

00:00:01.280 --> 00:00:05.210
and Lennox play an elaborated
version of rock paper scissors

00:00:05.210 --> 00:00:08.280
where each combination of shape choices

00:00:08.280 --> 00:00:10.780
earns a different number
of points for the winner.

00:00:10.780 --> 00:00:12.840
So rock paper scissors, the game,

00:00:12.840 --> 00:00:16.410
of course, where rock beats scissors,

00:00:16.410 --> 00:00:19.470
scissors beats paper,
and paper beats rock,

00:00:19.470 --> 00:00:21.560
and then they give us
this elaborate version

00:00:21.560 --> 00:00:22.393
right over here.

00:00:22.393 --> 00:00:24.350
When Violet wins, she gets two points.

00:00:24.350 --> 00:00:27.000
When Lennox wins with
rock, she gets three.

00:00:27.000 --> 00:00:29.700
When Lennox wins with
paper, she gets two points.

00:00:29.700 --> 00:00:32.290
When Lennox wins with
scissors, she gets one point.

00:00:32.290 --> 00:00:35.230
And if they choose the same
shape, nobody gets any points

00:00:35.230 --> 00:00:37.290
'cause no one wins that round.

00:00:37.290 --> 00:00:40.980
Complete the matrix so it
represents their scoring system.

00:00:40.980 --> 00:00:45.140
It shows the number of points Violet gets,

00:00:45.140 --> 00:00:48.590
a negative number means
Lennox gets those points,

00:00:48.590 --> 00:00:51.390
where rows are Violet's chosen shape

00:00:51.390 --> 00:00:54.650
and columns are Lennox's chosen shape.

00:00:54.650 --> 00:00:56.670
So here we have the
matrix right over here.

00:00:56.670 --> 00:00:58.940
I encourage you to pause this video

00:00:58.940 --> 00:01:00.670
and see if you can have
a go at this on your own

00:01:00.670 --> 00:01:02.090
if you have a piece of paper
in front of you, alright?

00:01:02.090 --> 00:01:04.210
I encourage you to get a piece of paper.

00:01:04.210 --> 00:01:06.450
All right, now let's do this together.

00:01:06.450 --> 00:01:08.040
So, how many points,

00:01:08.040 --> 00:01:10.570
remember, this matrix is
how many points Violet gets.

00:01:10.570 --> 00:01:14.290
And if Lennox gets points, then
it's a negative for Violet.

00:01:14.290 --> 00:01:18.180
So, if Violet chooses a rock
and Lennox chooses a rock,

00:01:18.180 --> 00:01:20.130
so that is this entry right over here,

00:01:20.130 --> 00:01:20.963
what's going to happen?

00:01:20.963 --> 00:01:22.890
How many points is Violet going to get?

00:01:23.830 --> 00:01:25.860
Well, if both players
choose the same shape,

00:01:25.860 --> 00:01:28.320
nobody gets any points.

00:01:28.320 --> 00:01:31.700
So if they both get rock, rock
will get a zero right there.

00:01:31.700 --> 00:01:33.590
And we also know that's going to be true

00:01:33.590 --> 00:01:35.900
if Violet picks paper
and Lennox picks paper,

00:01:35.900 --> 00:01:38.060
you're gonna get a zero
points for Violet there.

00:01:38.060 --> 00:01:41.170
And if they both pick
scissors, that entry there,

00:01:41.170 --> 00:01:43.550
you're also going to get a zero.

00:01:43.550 --> 00:01:46.280
All right, now, what if Violet picks rock

00:01:46.280 --> 00:01:48.460
and Lennox picks paper?

00:01:48.460 --> 00:01:49.293
What should I put there?

00:01:49.293 --> 00:01:50.320
Pause the video and think about it.

00:01:50.320 --> 00:01:52.763
Violet picks rock and Lennox picks paper.

00:01:53.860 --> 00:01:57.460
Well, we know that paper beats rock.

00:01:57.460 --> 00:02:01.530
So this is a situation where
Lennox wins with paper.

00:02:01.530 --> 00:02:03.780
And so that's this
scenario right over here,

00:02:03.780 --> 00:02:06.140
so Lennox will get two points.

00:02:06.140 --> 00:02:07.550
So if Lennox gets two points,

00:02:07.550 --> 00:02:09.840
remember this matrix is
what does Violet get,

00:02:09.840 --> 00:02:12.300
Violet gets negative two
points right over here

00:02:12.300 --> 00:02:14.110
'cause Lennox got them.

00:02:14.110 --> 00:02:15.730
All right, now what about this entry?

00:02:15.730 --> 00:02:17.640
What does that represent?

00:02:17.640 --> 00:02:19.750
Well, that represents Violet picking rock

00:02:19.750 --> 00:02:22.530
and Lennox picking scissors.

00:02:22.530 --> 00:02:25.680
And we know that rock beats scissors

00:02:25.680 --> 00:02:27.610
so this is a situation where Violet wins,

00:02:27.610 --> 00:02:31.610
and we know whenever Violet
wins, she gets two points.

00:02:31.610 --> 00:02:34.690
So this will be a positive
two points right over here.

00:02:34.690 --> 00:02:36.180
Now, what about this entry?

00:02:36.180 --> 00:02:38.200
What does that represent?

00:02:38.200 --> 00:02:40.550
Well, that represents Violet picking paper

00:02:40.550 --> 00:02:42.400
and Lennox picking rock.

00:02:42.400 --> 00:02:44.510
And we know that paper beats rock,

00:02:44.510 --> 00:02:47.410
so this is another
situation where Violet wins

00:02:47.410 --> 00:02:50.350
and she gets two points in
any scenario where she wins.

00:02:50.350 --> 00:02:51.780
So that's two points.

00:02:51.780 --> 00:02:54.390
And now, what about this one over here?

00:02:54.390 --> 00:02:57.163
Pause this video and think
about what number goes there.

00:02:58.340 --> 00:03:01.040
Well, this is a situation
where Violet picks paper

00:03:01.040 --> 00:03:03.130
and Lennox picks scissors.

00:03:03.130 --> 00:03:05.270
We know scissors beats paper

00:03:05.270 --> 00:03:06.950
'cause it can cut it up, I guess,

00:03:06.950 --> 00:03:09.360
and so Lennox has won with scissors.

00:03:09.360 --> 00:03:12.070
And we see here, I'll do
it in a different color,

00:03:12.070 --> 00:03:15.180
Lennox wins with scissors,
she gets one point.

00:03:15.180 --> 00:03:17.320
So you might be tempted
to write a one here

00:03:17.320 --> 00:03:19.410
but remember, that's
Lennox getting a point.

00:03:19.410 --> 00:03:21.610
So this is all about how
many points does Violet get

00:03:21.610 --> 00:03:23.910
and we said that would
be negative one point

00:03:23.910 --> 00:03:26.290
if it's going to Lennox.

00:03:26.290 --> 00:03:30.200
And then, let's think
about this last row here.

00:03:30.200 --> 00:03:31.650
What does this entry represent?

00:03:31.650 --> 00:03:32.740
What number should go there?

00:03:32.740 --> 00:03:34.490
Pause the video and think about it.

00:03:35.520 --> 00:03:37.460
Well, this is Violet picking scissors

00:03:37.460 --> 00:03:39.070
and Lennox picking rock.

00:03:39.070 --> 00:03:41.150
Now, we know that rock beats scissors

00:03:41.150 --> 00:03:42.890
'cause I guess it can bash it up,

00:03:42.890 --> 00:03:46.700
and so Lennox in this
scenario has won with rock,

00:03:46.700 --> 00:03:49.130
and we know, I'll pick another color here,

00:03:49.130 --> 00:03:52.500
when Lennox wins with rock,
she gets three points.

00:03:52.500 --> 00:03:54.300
So Lennox is getting three points here.

00:03:54.300 --> 00:03:56.030
This matrix is all about
what does Violet get

00:03:56.030 --> 00:03:58.000
so we wanna put a negative three here

00:03:58.000 --> 00:03:59.690
'cause that's three points for Lennox.

00:03:59.690 --> 00:04:03.590
Remember, a negative number
means Lennox gets those points.

00:04:03.590 --> 00:04:07.873
And one last entry, what do
you think should go there?

00:04:09.550 --> 00:04:11.860
Well, this is Violet picking scissors

00:04:11.860 --> 00:04:14.550
and Lennox picking paper.

00:04:14.550 --> 00:04:17.230
So we know that scissors beats paper

00:04:17.230 --> 00:04:18.330
'cause it can cut it up,

00:04:18.330 --> 00:04:21.040
and we know that in any
situation where Violet wins,

00:04:21.040 --> 00:04:23.740
'cause she won with scissors
here, she gets two points.

00:04:23.740 --> 00:04:27.320
So that is two points, just like that.

00:04:27.320 --> 00:04:28.620
So we filled in the matrix

00:04:28.620 --> 00:04:30.230
and now we have to answer this question.

00:04:30.230 --> 00:04:33.980
Assuming Lennox picks her
shape entirely at random,

00:04:33.980 --> 00:04:38.090
what shape should Violet
choose to maximize her chances

00:04:38.090 --> 00:04:40.130
of getting the most points?

00:04:40.130 --> 00:04:43.000
So pause the video and see
if this matrix is helpful

00:04:43.000 --> 00:04:44.800
for figuring out the answer to that.

00:04:46.190 --> 00:04:47.150
All right.

00:04:47.150 --> 00:04:50.030
So, this obviously isn't
an exercise on probability,

00:04:50.030 --> 00:04:52.210
but just as a little bit of a review,

00:04:52.210 --> 00:04:55.330
one way to think about it
is when Violet picks rock,

00:04:55.330 --> 00:04:57.010
here are the scenarios,

00:04:57.010 --> 00:04:59.400
here are the outcomes that might happen.

00:04:59.400 --> 00:05:02.430
Now, they're telling us
that Lennox picks at random

00:05:02.430 --> 00:05:04.840
so there would be a 1/3
chance that Lennox picks rock,

00:05:04.840 --> 00:05:07.270
1/3 paper, 1/3 scissors.

00:05:07.270 --> 00:05:09.490
And since these are equally likely

00:05:09.490 --> 00:05:12.330
'cause they're saying that
Lennox is picking at random,

00:05:12.330 --> 00:05:13.310
you can get the,

00:05:13.310 --> 00:05:16.070
what is sometimes known as
the expected value here,

00:05:16.070 --> 00:05:19.000
by taking the average
of these three numbers.

00:05:19.000 --> 00:05:19.940
Another way to think about it would be

00:05:19.940 --> 00:05:24.940
1/3 X 0 + 1/3 X -2 + 1/3 X 2.

00:05:25.700 --> 00:05:27.900
If you wanna go dig deeper
into expected value,

00:05:27.900 --> 00:05:29.880
there's a lot of that on Khan Academy,

00:05:29.880 --> 00:05:31.910
but we can really just take
the average of these numbers.

00:05:31.910 --> 00:05:33.300
Add them up and divide by three

00:05:33.300 --> 00:05:34.660
is another way to think about it.

00:05:34.660 --> 00:05:37.000
And so, here the expected value

00:05:37.000 --> 00:05:39.110
is going to be if we take the sum,

00:05:39.110 --> 00:05:41.260
we get 0 + -2 + 2.

00:05:41.260 --> 00:05:42.780
Well, that all sums up to 0,

00:05:42.780 --> 00:05:45.370
divided by 3, you get to 0.

00:05:45.370 --> 00:05:48.090
So I'll just write this 0
here as the expected value

00:05:48.090 --> 00:05:51.820
when Violet picks rock and
Lennox picks at random.

00:05:51.820 --> 00:05:56.130
Now, in this second scenario,
let's take the average.

00:05:56.130 --> 00:05:58.370
If we add all three of these up,

00:05:58.370 --> 00:06:02.070
you get 2 + 0 + -1,
which is 1, divide by 3,

00:06:02.070 --> 00:06:05.640
you're gonna get 1/3 as the expected value

00:06:05.640 --> 00:06:07.210
of the points for Violet.

00:06:07.210 --> 00:06:09.800
And then in that last scenario,

00:06:09.800 --> 00:06:13.270
if you add all of these up, you get -1,

00:06:13.270 --> 00:06:14.270
divide by 3 is -1/3.

00:06:16.300 --> 00:06:19.690
So it looks like the best
expected value for Violet,

00:06:19.690 --> 00:06:22.110
assuming that Lennox is
going to pick at random,

00:06:22.110 --> 00:06:23.510
is to go with paper.

00:06:23.510 --> 00:06:26.640
You have a positive 1/3 expected value.

00:06:26.640 --> 00:06:29.220
So what shape should Violet
choose to maximize her chances

00:06:29.220 --> 00:06:30.940
of getting the most points?

00:06:30.940 --> 00:06:32.150
Paper.

00:06:32.150 --> 00:06:33.630
Now, of course that's assuming Lennox

00:06:33.630 --> 00:06:35.050
always picks at random.

00:06:35.050 --> 00:06:36.730
Obviously, if Lennox catches on

00:06:36.730 --> 00:06:38.030
that Violet keeps picking paper,

00:06:38.030 --> 00:06:39.970
Lennox would adjust their strategy.

00:06:39.970 --> 00:06:41.670
But that gets a little bit deeper.