Example 4: Factoring quadratics as a perfect square of a difference: (a-b)^2

WEBVTT
Kind: captions
Language: en

00:00:00.800 --> 00:00:05.580
Factor 25x squared
minus 30x plus 9.

00:00:05.580 --> 00:00:08.590
And we have a leading
coefficient that's not a 1,

00:00:08.590 --> 00:00:11.130
and it doesn't look like
there are any common factors.

00:00:11.130 --> 00:00:17.050
Both 25 and 30 are divisible by
5, but 9 isn't divisible by 5.

00:00:17.050 --> 00:00:19.510
We could factor
this by grouping.

00:00:19.510 --> 00:00:22.080
But if we look a little
bit more carefully here,

00:00:22.080 --> 00:00:23.530
see something interesting.

00:00:23.530 --> 00:00:28.410
25 is a perfect square, and 25x
squared is a perfect square.

00:00:28.410 --> 00:00:30.390
It's the square of 5x.

00:00:30.390 --> 00:00:32.720
And then nine is also
a perfect square.

00:00:32.720 --> 00:00:34.374
It's the square of
3, or actually, it

00:00:34.374 --> 00:00:35.790
could be the square
of negative 3.

00:00:35.790 --> 00:00:38.730
This could also be the
square of negative 5x.

00:00:38.730 --> 00:00:42.210
Maybe, just maybe this
could be a perfect square.

00:00:42.210 --> 00:00:43.930
Let's just think
about what happens

00:00:43.930 --> 00:00:46.910
when we take the perfect square
of a binomial, especially

00:00:46.910 --> 00:00:50.000
when the coefficient on
the x term is not a 1.

00:00:50.000 --> 00:00:54.130
If we have ax plus b
squared, what will this look

00:00:54.130 --> 00:00:56.810
like when we expand
this into a trinomial?

00:00:56.810 --> 00:01:02.440
Well, this is the same
thing as ax plus b times ax

00:01:02.440 --> 00:01:07.370
plus b, which is the same
thing as ax times ax.

00:01:07.370 --> 00:01:15.310
Ax times ax is a squared x
squared plus ax times b, which

00:01:15.310 --> 00:01:22.430
is abx plus b times
ax, which is another.

00:01:22.430 --> 00:01:27.450
You You could call
it bax or abx, plus b

00:01:27.450 --> 00:01:31.140
times b, so plus b squared.

00:01:31.140 --> 00:01:36.110
This is equal to a squared
x squared plus-- these two

00:01:36.110 --> 00:01:42.060
are the same term--
2abx plus c squared.

00:01:42.060 --> 00:01:46.450
This is what happens when
you square a binomial.

00:01:46.450 --> 00:01:48.600
Now, this pattern seems
to work out pretty good.

00:01:48.600 --> 00:01:50.520
Let me rewrite our
problem right below it.

00:01:50.520 --> 00:01:57.260
We have 25x squared
minus 30x plus 9.

00:01:57.260 --> 00:01:59.240
If this is a
perfect square, then

00:01:59.240 --> 00:02:04.480
that means that the a squared
part right over here is 25.

00:02:04.480 --> 00:02:06.190
And then that means
that the b squared

00:02:06.190 --> 00:02:13.630
part-- let me do this in
a different color-- is 9.

00:02:13.630 --> 00:02:19.900
That tells us that a
could be plus or minus 5

00:02:19.900 --> 00:02:25.460
and that b could
be plus or minus 3.

00:02:25.460 --> 00:02:28.990
Now let's see if this gels
with this middle term.

00:02:28.990 --> 00:02:33.030
For this middle
term to work out--

00:02:33.030 --> 00:02:36.420
I'm trying to look for good
colors-- 2ab, this part

00:02:36.420 --> 00:02:41.400
right over here, needs to
be equal to negative 30.

00:02:41.400 --> 00:02:44.780
Or another way-- let me write
it over here-- 2ab needs

00:02:44.780 --> 00:02:46.450
to be equal to negative 30.

00:02:46.450 --> 00:02:50.570
Or if we divide
both sides by 2, ab

00:02:50.570 --> 00:02:55.950
needs to be equal
to negative 15.

00:02:55.950 --> 00:02:58.610
That tells us that the
product is negative.

00:02:58.610 --> 00:03:02.120
One has to be positive,
and one has to be negative.

00:03:02.120 --> 00:03:05.175
Now, lucky for us the
product of 5 and 3 is 15.

00:03:05.175 --> 00:03:07.550
If we make one of them positive
and one of them negative,

00:03:07.550 --> 00:03:09.170
we'll get up to negative 15.

00:03:09.170 --> 00:03:11.460
It looks like things
are going to work out.

00:03:11.460 --> 00:03:16.660
We could select a is
equal to positive 5,

00:03:16.660 --> 00:03:19.830
and b is equal to negative 3.

00:03:19.830 --> 00:03:24.450
Those would work out to ab
being equal to negative 15.

00:03:24.450 --> 00:03:27.910
Or we could make a is
equal to negative 5,

00:03:27.910 --> 00:03:31.020
and b is equal to positive 3.

00:03:31.020 --> 00:03:32.770
Either of these will work.

00:03:32.770 --> 00:03:34.480
If we factor this
out, this could

00:03:34.480 --> 00:03:38.670
be either a is negative--
let's do this first one.

00:03:38.670 --> 00:03:41.670
It could either be a
is 5, b is negative 3.

00:03:41.670 --> 00:03:46.290
This could either be
5x minus 3 squared.

00:03:46.290 --> 00:03:51.030
a is 5, b is negative 3.

00:03:51.030 --> 00:03:52.470
It could be that.

00:03:52.470 --> 00:03:55.631
Or you could have-- we could
switch the signs on the two

00:03:55.631 --> 00:03:56.130
terms.

00:03:56.130 --> 00:04:01.480
Or a could be negative 5,
and b could be positive 3.

00:04:01.480 --> 00:04:07.330
Or it could be negative
5x plus 3 squared.

00:04:07.330 --> 00:04:09.870
Either of these
are possible ways

00:04:09.870 --> 00:04:11.302
to factor this term out here.

00:04:11.302 --> 00:04:13.010
And you say wait, how
does this work out?

00:04:13.010 --> 00:04:15.650
How can both of these
multiply to the same thing?

00:04:15.650 --> 00:04:18.300
Well, this term, remember,
this negative 5x plus 3,

00:04:18.300 --> 00:04:19.720
we could factor
out a negative 1.

00:04:19.720 --> 00:04:24.400
So this right here is the same
thing as negative 1 times 5x

00:04:24.400 --> 00:04:27.800
minus 3, the whole
thing squared.

00:04:27.800 --> 00:04:31.900
And that's the same thing
as negative 1 squared times

00:04:31.900 --> 00:04:34.630
5x minus 3 squared.

00:04:34.630 --> 00:04:37.650
And negative 1 squared
is clearly equal to 1.

00:04:37.650 --> 00:04:41.170
That's why this and
this are the same thing.

00:04:41.170 --> 00:04:44.460
This comes out to the same
thing as 5x minus 3 squared,

00:04:44.460 --> 00:04:47.020
which is the same thing
as that over there.

00:04:47.020 --> 00:04:51.650
Either of these are
possible answers.


Example 1: Factoring trinomials with a common factor

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:07.582
Factor 8k squared
minus 24k minus 144.

00:00:07.582 --> 00:00:09.040
Now the first thing
we can do here,

00:00:09.040 --> 00:00:11.020
just eyeballing
each of these terms,

00:00:11.020 --> 00:00:14.820
if we want to simplify it a
good bit is all of these terms

00:00:14.820 --> 00:00:15.930
are divisible by 8.

00:00:15.930 --> 00:00:19.770
Clearly, 8k squared is divisible
by 8, 24 is divisible by 8,

00:00:19.770 --> 00:00:24.130
and 144-- it might not be as
obvious is divisible by 8--

00:00:24.130 --> 00:00:25.400
but it looks like it is.

00:00:25.400 --> 00:00:29.600
8 goes into one and 144,
8 goes into 14 one time.

00:00:29.600 --> 00:00:31.730
1 times 8 is 8.

00:00:31.730 --> 00:00:33.800
Subtract, you get a 6.

00:00:33.800 --> 00:00:35.450
14 minus 8 is 6.

00:00:35.450 --> 00:00:36.880
Bring down the 4.

00:00:36.880 --> 00:00:40.020
8 goes into 64 eight times.

00:00:40.020 --> 00:00:42.430
So it goes into 144 18 times.

00:00:42.430 --> 00:00:44.560
So let's just factor
out an 8 of this.

00:00:44.560 --> 00:00:46.530
And then that will
simplify our expression.

00:00:46.530 --> 00:00:48.890
It will actually give us
a leading 1 coefficient.

00:00:48.890 --> 00:00:55.900
So this will become 8 times k
squared minus 24 divided by 8

00:00:55.900 --> 00:01:04.590
is 3k minus 18.

00:01:04.590 --> 00:01:08.150
Now we have to factor
this business in here.

00:01:08.150 --> 00:01:13.800
And remember if anything has the
form x squared plus bx plus c,

00:01:13.800 --> 00:01:15.910
where you have a leading
1 coefficient-- this

00:01:15.910 --> 00:01:18.190
is implicitly a one--
we have that here

00:01:18.190 --> 00:01:20.220
in this expression
in parentheses.

00:01:20.220 --> 00:01:22.160
Then we literally
just need to-- and we

00:01:22.160 --> 00:01:23.670
can do this multiple
ways-- but we

00:01:23.670 --> 00:01:26.610
need to find two
numbers whose sum is

00:01:26.610 --> 00:01:28.450
equal to the coefficient on x.

00:01:28.450 --> 00:01:31.840
So two numbers whose sum
is equal to negative 3

00:01:31.840 --> 00:01:36.240
and whose product is equal
to the constant term.

00:01:36.240 --> 00:01:39.370
And whose product is
equal to negative 18.

00:01:39.370 --> 00:01:43.410
So let's just think about the
factors of negative 18 here.

00:01:43.410 --> 00:01:46.140
Let's see if we can do
something interesting.

00:01:46.140 --> 00:01:47.855
So it could be 1.

00:01:47.855 --> 00:01:50.480
And since it's negative, one of
the numbers has to be positive,

00:01:50.480 --> 00:01:53.739
one has to be negative 1 and
18 is if it was positive.

00:01:53.739 --> 00:01:56.280
And then one of these could be
positive and then one of these

00:01:56.280 --> 00:01:57.630
could be negative.

00:01:57.630 --> 00:02:01.740
But no matter what
if this is negative

00:02:01.740 --> 00:02:04.950
and this is positive
then they add up to 17.

00:02:04.950 --> 00:02:07.290
If you switch them, then
they add up to negative 17.

00:02:07.290 --> 00:02:08.169
So those won't work.

00:02:08.169 --> 00:02:10.720
So either we could
write it this way,

00:02:10.720 --> 00:02:13.765
positive or negative 1, and
then negative or positive 18

00:02:13.765 --> 00:02:15.640
to show that they have
to be different signs.

00:02:15.640 --> 00:02:16.990
So those don't work.

00:02:16.990 --> 00:02:19.720
Then you have positive
or negative 3.

00:02:19.720 --> 00:02:21.820
And then negative
or positive 6, just

00:02:21.820 --> 00:02:23.870
to know that they
are different signs.

00:02:23.870 --> 00:02:26.330
So if you have positive
3 and negative 6,

00:02:26.330 --> 00:02:28.360
they add up to
negative 3 which is

00:02:28.360 --> 00:02:30.020
what we need them to add up to.

00:02:30.020 --> 00:02:33.280
And clearly, positive 3 and
negative 6, their product

00:02:33.280 --> 00:02:34.250
is negative 18.

00:02:34.250 --> 00:02:35.690
So it works.

00:02:35.690 --> 00:02:37.810
So we're going to
go with positive 3

00:02:37.810 --> 00:02:41.030
and negative 6 as
our two numbers.

00:02:41.030 --> 00:02:45.610
Now, for this example-- just
for the sake of this example--

00:02:45.610 --> 00:02:46.720
We'll do this by grouping.

00:02:46.720 --> 00:02:49.770
So what we can do is we can
separate this middle term right

00:02:49.770 --> 00:02:53.750
here as the sum
of 3k negative 6k.

00:02:53.750 --> 00:02:59.619
So I could write the negative
3k as plus 3k minus 6k.

00:02:59.619 --> 00:03:01.160
And then let me
write the rest of it.

00:03:01.160 --> 00:03:05.460
So we have k squared up
here, plus 3k minus 6k,

00:03:05.460 --> 00:03:07.640
which is the same thing
as this over here.

00:03:07.640 --> 00:03:09.840
And then we have minus 18.

00:03:09.840 --> 00:03:13.240
And then all of that's
being multiplied by 8.

00:03:13.240 --> 00:03:15.790
Now we're ready to
group this thing.

00:03:15.790 --> 00:03:18.620
We can group these first two
terms, they're both divisible

00:03:18.620 --> 00:03:19.610
by k.

00:03:19.610 --> 00:03:22.270
And then we can group-- let
me put a positive sign--

00:03:22.270 --> 00:03:24.530
let's group these
second two terms.

00:03:24.530 --> 00:03:28.790
So then we have 8 times--
I'll write brackets here

00:03:28.790 --> 00:03:30.670
instead of drawing
double parentheses.

00:03:30.670 --> 00:03:32.320
Brackets are really
just parentheses

00:03:32.320 --> 00:03:35.110
that look a little
bit more serious.

00:03:35.110 --> 00:03:38.990
Now let's factor out a k
from this term right here.

00:03:38.990 --> 00:03:40.740
I'm going to do this
in a different color.

00:03:40.740 --> 00:03:42.010
Let's factor out the k here.

00:03:42.010 --> 00:03:46.540
So this is k times k plus 3.

00:03:46.540 --> 00:03:47.480
And then we have plus.

00:03:47.480 --> 00:03:48.938
And then over here
it looks like we

00:03:48.938 --> 00:03:51.339
could factor out a negative 6.

00:03:51.339 --> 00:03:52.880
So let's factor
out-- I'm going to do

00:03:52.880 --> 00:03:55.090
this in a different color--
let's factor out a negative 6

00:03:55.090 --> 00:03:55.590
over here.

00:03:55.590 --> 00:04:00.910
So plus negative
6 times k plus 3.

00:04:04.300 --> 00:04:07.360
So now it looks like we
can factor out a k plus 3.

00:04:07.360 --> 00:04:11.180
There's a k plus 3 times k, and
then we have a k plus 3 times

00:04:11.180 --> 00:04:12.140
the negative 6.

00:04:12.140 --> 00:04:15.880
So let's factor that out.

00:04:15.880 --> 00:04:18.579
So we have this 8 out
front, that's not changing.

00:04:18.579 --> 00:04:20.120
So let me write that
in the brackets.

00:04:20.120 --> 00:04:22.390
We're factoring out a k plus 3.

00:04:22.390 --> 00:04:26.400
So then we have the k plus
3 that we factored out.

00:04:26.400 --> 00:04:30.980
And then inside of that
we just have left this k.

00:04:30.980 --> 00:04:32.900
Instead of writing
plus negative 6,

00:04:32.900 --> 00:04:40.010
I could just write k minus 6.

00:04:40.010 --> 00:04:42.430
We factor out the k
plus 3 and we're done.

00:04:42.430 --> 00:04:43.780
And then we can rewrite this.

00:04:43.780 --> 00:04:46.360
The way we wrote it here
it's 8 times the product of k

00:04:46.360 --> 00:04:48.260
plus 3 times k minus 6.

00:04:48.260 --> 00:04:50.450
But we know from the
properties of multiplication,

00:04:50.450 --> 00:04:54.470
this is the exact same
thing as 8 times k

00:04:54.470 --> 00:04:58.760
plus 3 times k minus 6.

00:04:58.760 --> 00:05:00.522
And we are done.


Example 1: Factoring a difference of squares with two variables

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:03.890
Factor x squared
minus 49y squared.

00:00:03.890 --> 00:00:05.920
So what's interesting
here is that well x

00:00:05.920 --> 00:00:07.620
squared is clearly
a perfect square.

00:00:07.620 --> 00:00:08.820
It's the square of x.

00:00:08.820 --> 00:00:12.150
And 49y squared is
also a perfect square.

00:00:12.150 --> 00:00:14.880
It's the square of 7y.

00:00:14.880 --> 00:00:17.090
So it looks like we might
have a special form here.

00:00:17.090 --> 00:00:18.590
And to remind
ourselves, let's think

00:00:18.590 --> 00:00:24.200
about what happens if we take
a plus b times a minus b.

00:00:24.200 --> 00:00:25.780
I'm just doing it
in the general case

00:00:25.780 --> 00:00:27.580
so we can see a pattern here.

00:00:27.580 --> 00:00:29.640
So over here, this
would be a times a,

00:00:29.640 --> 00:00:33.980
which would be a squared
plus a times negative b,

00:00:33.980 --> 00:00:39.210
which would be negative ab plus
b times a or a times b again,

00:00:39.210 --> 00:00:40.940
which would be ab.

00:00:40.940 --> 00:00:43.010
And then you have
b times negative b,

00:00:43.010 --> 00:00:45.410
so it would b minus b squared.

00:00:45.410 --> 00:00:47.010
Now these middle two
terms cancel out.

00:00:47.010 --> 00:00:49.530
Negative ab plus
ab, they cancel out

00:00:49.530 --> 00:00:53.030
and you're left with just
a squared minus b squared.

00:00:53.030 --> 00:00:54.780
And that's the exact
pattern we have here.

00:00:54.780 --> 00:00:58.300
We have an a squared
minus a b squared.

00:00:58.300 --> 00:01:06.890
So in this case, a is equal
to x and b is equal to 7y.

00:01:06.890 --> 00:01:11.570
So we have x squared minus
7y, the whole thing squared.

00:01:11.570 --> 00:01:16.607
So we can expand this as
the difference of squares,

00:01:16.607 --> 00:01:18.190
or actually this
thing right over here

00:01:18.190 --> 00:01:19.398
is the difference of squares.

00:01:19.398 --> 00:01:21.090
So we expand this like this.

00:01:21.090 --> 00:01:29.514
So this will be equal to x
plus 7y times x minus 7y.

00:01:29.514 --> 00:01:30.930
And once again,
we're just pattern

00:01:30.930 --> 00:01:33.280
matching based on this
realization right here.

00:01:33.280 --> 00:01:35.345
If I take a plus
b times a minus b,

00:01:35.345 --> 00:01:37.270
I get a difference of squares.

00:01:37.270 --> 00:01:38.990
This is a difference of squares.

00:01:38.990 --> 00:01:40.570
So when I factor
it, it must come out

00:01:40.570 --> 00:01:42.111
to the result of
something that looks

00:01:42.111 --> 00:01:47.866
like a plus b times a minus b
or x plus 7y times x minus 7y.


Introduction to grouping

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:03.847
Factor t squared
plus 8t plus 15.

00:00:03.847 --> 00:00:05.430
So let's just think
about what happens

00:00:05.430 --> 00:00:10.270
if we multiply two binomials
t plus a times t plus b.

00:00:10.270 --> 00:00:11.850
And I'm using t
here because t is

00:00:11.850 --> 00:00:14.890
the variable in the polynomial
that we need to factor.

00:00:14.890 --> 00:00:16.810
So if you multiply
this out, applying

00:00:16.810 --> 00:00:20.220
the distributive property
twice or using FOIL,

00:00:20.220 --> 00:00:23.240
you get t times t,
which is t squared.

00:00:23.240 --> 00:00:27.020
Plus t times b, which is bt.

00:00:27.020 --> 00:00:30.030
Plus a times t, which is at.

00:00:30.030 --> 00:00:32.432
Plus a times b, which is ab.

00:00:32.432 --> 00:00:34.140
We essentially multiplied
every term here

00:00:34.140 --> 00:00:36.410
by every term over there.

00:00:36.410 --> 00:00:39.950
And then we have two t terms,
I guess you could call them,

00:00:39.950 --> 00:00:41.810
this bt plus at.

00:00:41.810 --> 00:00:43.210
So we could combine those.

00:00:43.210 --> 00:00:46.890
And we get t squared
plus a plus bt--

00:00:46.890 --> 00:00:51.870
I could have written this as
b plus at as well-- plus ab.

00:00:51.870 --> 00:00:54.750
Now, if we compare this
to this right over here,

00:00:54.750 --> 00:00:56.750
we see that we have
a similar pattern.

00:00:56.750 --> 00:01:00.670
Our coefficient on the
second degree term is 1.

00:01:00.670 --> 00:01:03.750
Our coefficient on the
second degree term here is 1.

00:01:03.750 --> 00:01:05.260
We didn't have to write it.

00:01:05.260 --> 00:01:08.320
Then a plus b is the
coefficient on t.

00:01:08.320 --> 00:01:12.550
So 8 right over here,
this 8 could be a plus b.

00:01:12.550 --> 00:01:18.680
And then finally, our constant
term, ab, that could be 15.

00:01:18.680 --> 00:01:20.880
So if we wanted to
factor this out,

00:01:20.880 --> 00:01:24.670
we just have to find and a and
a b where their product is 15

00:01:24.670 --> 00:01:27.382
and their sum is 8 in general.

00:01:27.382 --> 00:01:29.340
In general, if you ever
see-- and I'll write it

00:01:29.340 --> 00:01:32.690
in the more traditional
with an x variable--

00:01:32.690 --> 00:01:38.120
if you see anything of the
form x squared plus bx plus c,

00:01:38.120 --> 00:01:40.620
the coefficient here is 1.

00:01:40.620 --> 00:01:43.930
Then you just have
to find two numbers

00:01:43.930 --> 00:01:46.370
whose sum is equal to
this thing right here

00:01:46.370 --> 00:01:48.840
and whose product is equal
to that thing right there.

00:01:48.840 --> 00:01:54.680
Whose sum is equal to 8 and
whose product is equal to 15.

00:01:54.680 --> 00:01:58.050
So what are two numbers that add
up to 8 whose products are 15?

00:01:58.050 --> 00:02:01.390
So if you just factor
15, it could be 1 and 15.

00:02:01.390 --> 00:02:03.340
Those don't add up
to 8 in any way.

00:02:03.340 --> 00:02:06.270
3 and 5, those do add up to 8.

00:02:06.270 --> 00:02:09.270
So a and b could be 3 and 5.

00:02:09.270 --> 00:02:11.800
So a and b, this
could be 3 times 5,

00:02:11.800 --> 00:02:15.250
and then 8 is 3 plus 5.

00:02:15.250 --> 00:02:17.510
Now we could just go straight
and factor this and say,

00:02:17.510 --> 00:02:20.400
hey, this is t plus
3 times t plus 5,

00:02:20.400 --> 00:02:22.320
since we already
figured out a and b are.

00:02:22.320 --> 00:02:26.600
But what I want to do is kind
of factor this by grouping.

00:02:26.600 --> 00:02:29.430
So I'm essentially going
to go in reverse steps

00:02:29.430 --> 00:02:30.610
from what I just showed you.

00:02:30.610 --> 00:02:33.290
So this polynomial
right here, I'm

00:02:33.290 --> 00:02:37.020
going to write it as t squared
plus-- instead of writing

00:02:37.020 --> 00:02:41.090
8t I'm going to write it
as the sum of at plus bt,

00:02:41.090 --> 00:02:43.870
or as the sum of 3t plus 5t.

00:02:43.870 --> 00:02:47.570
So plus 3t plus 5t.

00:02:47.570 --> 00:02:49.760
So essentially
I've started here.

00:02:49.760 --> 00:02:51.760
And then I'm going to
this step where I break up

00:02:51.760 --> 00:02:55.380
that middle term into
the coefficients that

00:02:55.380 --> 00:02:56.640
add up to the 8.

00:02:56.640 --> 00:02:58.980
And then finally plus 15.

00:02:58.980 --> 00:03:01.440
And now I can factor
it by grouping.

00:03:01.440 --> 00:03:04.610
These two guys right over
here have a common factor t.

00:03:04.610 --> 00:03:07.480
And then these two guys over
here have a common factor 5.

00:03:07.480 --> 00:03:10.431
So let's factor out the t
in this first expression

00:03:10.431 --> 00:03:12.430
right over here, or this
part of the expression.

00:03:12.430 --> 00:03:15.680
So it's t times t plus 3.

00:03:15.680 --> 00:03:18.400
Plus-- and then over here,
if you factor out a 5

00:03:18.400 --> 00:03:22.660
you get a 5-- times t plus 3.

00:03:22.660 --> 00:03:25.170
5t divided by 5 is t.

00:03:25.170 --> 00:03:27.110
15 divided by 5 is 3.

00:03:27.110 --> 00:03:30.040
And now you can
factor out a t plus 3.

00:03:30.040 --> 00:03:32.230
You have a t plus
3 being multiplied

00:03:32.230 --> 00:03:33.720
times both of these terms.

00:03:33.720 --> 00:03:36.340
So let's factor that out.

00:03:36.340 --> 00:03:43.925
So it becomes t plus
3 times t plus 5.

00:03:43.925 --> 00:03:45.830
Let me write that plus
a little bit neater.

00:03:45.830 --> 00:03:51.440
Times t plus 5.

00:03:51.440 --> 00:03:53.120
And we are done.

00:03:53.120 --> 00:03:55.590
But we actually didn't even
have to do this grouping step,

00:03:55.590 --> 00:03:57.551
although hopefully you
see that it does work.

00:03:57.551 --> 00:03:59.550
We could have just said,
look, from this pattern

00:03:59.550 --> 00:04:02.640
over here I have two
numbers that add up to 8

00:04:02.640 --> 00:04:03.950
and their product is 15.

00:04:03.950 --> 00:04:06.550
So this must be t
plus 3 times t plus 5.

00:04:06.550 --> 00:04:11.050
Or t plus 5 times t
plus 3, either way.


Example 3: Factor a quartic binomial with two variables by taking a common factor

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:02.750
Factor out the
greatest common factor.

00:00:02.750 --> 00:00:05.890
And the expression they give us
is 4x to the fourth y plus 8x

00:00:05.890 --> 00:00:06.927
to the third y.

00:00:06.927 --> 00:00:09.510
When they say to factor out the
greatest common factor they're

00:00:09.510 --> 00:00:12.380
essentially telling us, find
the greatest common factor of 4x

00:00:12.380 --> 00:00:14.150
to the fourth and
8x to the third y

00:00:14.150 --> 00:00:16.500
and factor it out
of this expression.

00:00:16.500 --> 00:00:18.390
Or kind of undistribute it.

00:00:18.390 --> 00:00:20.360
And to find that
greatest common factor--

00:00:20.360 --> 00:00:21.990
and I always put
it in quotes when

00:00:21.990 --> 00:00:23.655
we speak in algebraic terms.

00:00:23.655 --> 00:00:25.030
Because we don't
really know what

00:00:25.030 --> 00:00:27.212
x and y are, whether
they're positive or negative

00:00:27.212 --> 00:00:29.170
or whether they're greater
than or less than 1.

00:00:29.170 --> 00:00:32.420
So it's not always going to be
the greatest absolute number.

00:00:32.420 --> 00:00:34.610
But it's the greatest
in it contains the most

00:00:34.610 --> 00:00:38.630
terms of these two expressions,
these two monomials.

00:00:38.630 --> 00:00:44.830
So if we were to essentially
factor out 4x to the fourth y

00:00:44.830 --> 00:00:45.940
it would look like this.

00:00:45.940 --> 00:00:47.700
We would do the
prime factorization

00:00:47.700 --> 00:00:53.830
of 4, which is just 2 times 2,
times x to the fourth, which

00:00:53.830 --> 00:01:02.260
is x times x times
x times x, times y.

00:01:02.260 --> 00:01:04.300
We just expanded
it out as a product

00:01:04.300 --> 00:01:06.360
of its basic constituents.

00:01:06.360 --> 00:01:09.284
Now let's do the
same thing for 8--

00:01:09.284 --> 00:01:13.520
I'll color code it--
8x to the third.

00:01:13.520 --> 00:01:15.990
Let me do it in similar colors.

00:01:15.990 --> 00:01:24.140
So in this situation we
have 8x to the 1/3 y.

00:01:24.140 --> 00:01:28.230
So the prime factorization
of 8 is 2 times 2 times 2.

00:01:28.230 --> 00:01:31.890
It's 2 times 2 times 2.

00:01:31.890 --> 00:01:34.860
Prime, or I should say
the factorization of x

00:01:34.860 --> 00:01:36.470
to the third, or
the expansion of it,

00:01:36.470 --> 00:01:39.950
is just times x
times x times x. x

00:01:39.950 --> 00:01:42.120
multiplied by
itself three times.

00:01:42.120 --> 00:01:44.950
And then we are multiplying
everything by a y here,

00:01:44.950 --> 00:01:46.640
times y.

00:01:46.640 --> 00:01:49.140
So what factors are
common to both of these?

00:01:49.140 --> 00:01:51.350
And we want to include as
many of them as possible

00:01:51.350 --> 00:01:54.320
to find this greatest
common factor.

00:01:54.320 --> 00:01:57.030
So we have two 2's
here, three 2's here.

00:01:57.030 --> 00:02:01.850
So we only have two 2's
in common in both of them.

00:02:01.850 --> 00:02:04.100
We have four x's here,
only three x's here.

00:02:04.100 --> 00:02:06.630
So we only have
three x's in common.

00:02:06.630 --> 00:02:09.710
Three x's and three x's.

00:02:09.710 --> 00:02:11.300
And we have a y
here and a y here.

00:02:11.300 --> 00:02:14.840
So y is common to
both expressions.

00:02:14.840 --> 00:02:19.340
So the greatest common factor
here is going to be 2 times 2.

00:02:19.340 --> 00:02:24.140
So it's going to be 2 times 2
times x times x times x times

00:02:24.140 --> 00:02:24.790
y.

00:02:24.790 --> 00:02:29.310
Or 4x to the third y.

00:02:29.310 --> 00:02:31.890
So this is what we
want to factor out.

00:02:31.890 --> 00:02:38.360
So that means we can write this
thing as-- if we factor out

00:02:38.360 --> 00:02:43.255
a 4x to the third y,
where essentially we

00:02:43.255 --> 00:02:45.740
have to divide each of
these by 4x to the third.

00:02:45.740 --> 00:02:46.970
We're factoring it out.

00:02:46.970 --> 00:02:48.020
So let me rewrite this.

00:02:48.020 --> 00:02:54.850
So this is 4x to the fourth
y plus 8x to the third y.

00:02:54.850 --> 00:03:01.667
And we're going to divide each
of these by 4x to the third y.

00:03:01.667 --> 00:03:03.250
And hopefully this
makes sense to you.

00:03:03.250 --> 00:03:04.740
If we were to
multiply this out, we

00:03:04.740 --> 00:03:07.574
would distribute this 4x to
the third y on each terms.

00:03:07.574 --> 00:03:09.490
And then it would cancel
with the denominator.

00:03:09.490 --> 00:03:13.059
You would have the same thing in
the numerator and denominator.

00:03:13.059 --> 00:03:15.100
And then you would get
this expression over here.

00:03:15.100 --> 00:03:16.870
So hopefully this
makes sense that these

00:03:16.870 --> 00:03:18.426
are the exact same expression.

00:03:18.426 --> 00:03:20.050
But when you write
it this way, then it

00:03:20.050 --> 00:03:23.530
becomes pretty clear that
this is 4x to the third y.

00:03:23.530 --> 00:03:26.040
And then you just simplify
each of these expressions.

00:03:26.040 --> 00:03:28.100
4 cancels with 4.
x to the fourth

00:03:28.100 --> 00:03:31.290
divided by x to the third is x.

00:03:31.290 --> 00:03:33.180
y divided by y is just 1.

00:03:33.180 --> 00:03:36.900
So you have x plus
8 divided by 4 is 2.

00:03:36.900 --> 00:03:39.250
x to the third divided
by x to the third is 1.

00:03:39.250 --> 00:03:41.030
Y divided by y is 1.

00:03:41.030 --> 00:03:43.110
So x plus 2.

00:03:43.110 --> 00:03:46.060
Another way to see what's left
over when you factor it out

00:03:46.060 --> 00:03:48.610
is if you were to take
out the common factor.

00:03:48.610 --> 00:03:51.330
So we took out this and this.

00:03:51.330 --> 00:03:53.890
What was left over
in 4x to the fourth y

00:03:53.890 --> 00:03:55.230
when we took this stuff out?

00:03:55.230 --> 00:03:57.234
When we undistributed it?

00:03:57.234 --> 00:03:58.650
Well, the only
thing that was left

00:03:58.650 --> 00:04:00.610
was this x right over here.

00:04:00.610 --> 00:04:02.110
Let me do that in another color.

00:04:02.110 --> 00:04:04.490
The only thing that
was left was this x.

00:04:04.490 --> 00:04:07.090
So that's why we just
have that x over there.

00:04:07.090 --> 00:04:12.020
When we factored everything
out of the 8x to the third y

00:04:12.020 --> 00:04:13.840
we factored all this
other stuff out.

00:04:13.840 --> 00:04:16.200
We factored out
4x to the third y.

00:04:16.200 --> 00:04:22.760
We factored it out, so
all we had left was the 2.

00:04:22.760 --> 00:04:24.410
Now in general, you
don't always have

00:04:24.410 --> 00:04:26.020
to go through this process.

00:04:26.020 --> 00:04:28.890
You could have done it this way,
but this really hopefully makes

00:04:28.890 --> 00:04:30.790
it clear exactly
what we're doing.

00:04:30.790 --> 00:04:33.240
You could have said,
look, 4x to the fourth y

00:04:33.240 --> 00:04:35.830
plus 8x to the third y.

00:04:35.830 --> 00:04:37.700
You could have said,
well let's see,

00:04:37.700 --> 00:04:39.870
the largest number that's
divisible into both 4

00:04:39.870 --> 00:04:41.240
an 8 is 4.

00:04:41.240 --> 00:04:42.926
So let's factor
out a 4 out here.

00:04:42.926 --> 00:04:45.550
The largest multiple of x that's
divisible into x to the fourth

00:04:45.550 --> 00:04:48.008
and x to the third, well that's
going to be x to the third.

00:04:48.008 --> 00:04:49.800
And you put an x to
the third out here.

00:04:49.800 --> 00:04:51.300
And you say, well,
the largest thing

00:04:51.300 --> 00:04:53.312
that's divisible both
into y and y is just y.

00:04:53.312 --> 00:04:55.770
So you could have done it a
little bit faster in your head.

00:04:55.770 --> 00:04:57.228
So you factor out
a 4x to the third

00:04:57.228 --> 00:04:59.880
and you say, OK, if I
take out a 4 out of here,

00:04:59.880 --> 00:05:00.807
then this becomes a 1.

00:05:00.807 --> 00:05:02.890
If I take an x to the third
out of x to the fourth

00:05:02.890 --> 00:05:04.780
I'll just have an x left over.

00:05:04.780 --> 00:05:06.510
And then if I take a
y out of the y then

00:05:06.510 --> 00:05:07.490
I just have a 1 there.

00:05:07.490 --> 00:05:09.480
So this term becomes x.

00:05:09.480 --> 00:05:11.250
And then if I take
a 4x to the third y

00:05:11.250 --> 00:05:13.560
out of here, if I
take a 4 out of an 8

00:05:13.560 --> 00:05:14.807
I just have a 2 left over.

00:05:14.807 --> 00:05:16.890
If I take an x to the third
out of x to the third,

00:05:16.890 --> 00:05:17.580
that's just 1.

00:05:17.580 --> 00:05:20.360
If I take y out of
y, that's just one.

00:05:20.360 --> 00:05:21.997
So I'm just left with x plus 2.

00:05:21.997 --> 00:05:24.330
Eventually you'll just do
this in your head a little bit

00:05:24.330 --> 00:05:25.040
faster.

00:05:25.040 --> 00:05:27.820
But hopefully this
makes everything clear.


Finding the greatest common factor of two monomials

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:04.320
Find the greatest common
factor of these monomials.

00:00:04.320 --> 00:00:06.970
Now, the greatest common
factor of anything

00:00:06.970 --> 00:00:11.571
is the largest factor
that's divisible into both.

00:00:11.571 --> 00:00:14.070
If we're talking about just
pure numbers, into both numbers,

00:00:14.070 --> 00:00:16.390
or in this case,
into both monomials.

00:00:16.390 --> 00:00:18.070
Now, we have to be
a little bit careful

00:00:18.070 --> 00:00:20.570
when we talk about
greatest in the context

00:00:20.570 --> 00:00:22.897
of algebraic
expressions like this.

00:00:22.897 --> 00:00:24.730
Because it's greatest
from the point of view

00:00:24.730 --> 00:00:29.190
that it includes the most
factors of each of these

00:00:29.190 --> 00:00:29.850
monomials.

00:00:29.850 --> 00:00:32.560
It's not necessarily the
greatest possible number

00:00:32.560 --> 00:00:35.150
because maybe some
of these variables

00:00:35.150 --> 00:00:36.820
could take on
negative values, maybe

00:00:36.820 --> 00:00:38.490
they're taking on
values less than 1.

00:00:38.490 --> 00:00:39.906
So if you square
it, it's actually

00:00:39.906 --> 00:00:41.300
going to become
a smaller number.

00:00:41.300 --> 00:00:43.970
But I think without getting
too much into the weeds there,

00:00:43.970 --> 00:00:46.570
I think if we just kind of
run through the process of it,

00:00:46.570 --> 00:00:48.450
you'll understand it
a little bit better.

00:00:48.450 --> 00:00:50.107
So to find the
greatest common factor,

00:00:50.107 --> 00:00:51.690
let's just essentially
break down each

00:00:51.690 --> 00:00:53.900
of these numbers
into what we could

00:00:53.900 --> 00:00:55.536
call their prime factorization.

00:00:55.536 --> 00:00:57.910
But it's kind of a combination
of the prime factorization

00:00:57.910 --> 00:00:59.650
of the numeric
parts of the number,

00:00:59.650 --> 00:01:01.300
plus essentially
the factorization

00:01:01.300 --> 00:01:02.720
of the variable parts.

00:01:05.260 --> 00:01:08.130
If we were to
write 10cd squared,

00:01:08.130 --> 00:01:09.850
we can rewrite
that as the product

00:01:09.850 --> 00:01:11.910
of the prime factors of 10.

00:01:11.910 --> 00:01:14.440
The prime factorization
of 10 is just 2 times 5.

00:01:14.440 --> 00:01:16.470
Those are both prime numbers.

00:01:16.470 --> 00:01:19.930
So 10 can be broken
down as 2 times 5.

00:01:19.930 --> 00:01:22.740
c can only be broken down by c.

00:01:22.740 --> 00:01:26.730
We don't know anything else
that c can be broken into.

00:01:26.730 --> 00:01:28.650
So 2 times 5 times c.

00:01:28.650 --> 00:01:34.800
But then the d squared can
be rewritten as d times d.

00:01:34.800 --> 00:01:38.920
This is what I mean by writing
this monomial essentially

00:01:38.920 --> 00:01:40.970
as the product of
its constituents.

00:01:40.970 --> 00:01:43.470
For the numeric part of
it, it's the constituents

00:01:43.470 --> 00:01:44.950
of the prime factors.

00:01:44.950 --> 00:01:46.630
And for the rest
of it, we're just

00:01:46.630 --> 00:01:48.770
kind of expanding
out the exponents.

00:01:48.770 --> 00:01:52.560
Now, let's do that for 25c
to the third d squared.

00:01:52.560 --> 00:01:55.250
So 25 right here,
that's 5 times 5.

00:01:55.250 --> 00:01:58.350
So this is equal to 5 times 5.

00:01:58.350 --> 00:02:04.470
And then c to the third,
that's times c times c times c.

00:02:04.470 --> 00:02:07.840
And then d squared,
times d squared.

00:02:07.840 --> 00:02:11.470
d squared is times d times d.

00:02:11.470 --> 00:02:15.770
So what's their greatest
common factor in this context?

00:02:15.770 --> 00:02:18.330
Well, they both
have at least one 5.

00:02:21.070 --> 00:02:23.700
Then they both have at
least one c over here.

00:02:23.700 --> 00:02:26.400
So let's just take up one
of the c's right over there.

00:02:26.400 --> 00:02:28.030
And then they both have two d's.

00:02:31.210 --> 00:02:34.510
So the greatest common factor
in this context, the greatest

00:02:34.510 --> 00:02:36.890
common factor of
these two monomials

00:02:36.890 --> 00:02:39.620
is going to be the factors
that they have in common.

00:02:39.620 --> 00:02:44.190
So it's going to be equal to
this 5 times-- we only have one

00:02:44.190 --> 00:02:46.240
c in common, times--
and we have two

00:02:46.240 --> 00:02:49.020
d's in common, times d times d.

00:02:49.020 --> 00:02:54.060
So this is equal to 5cd squared.

00:02:54.060 --> 00:02:56.745
And so 5d squared, we can kind
of view it as the greatest.

00:02:56.745 --> 00:02:59.060
But I'll put that
in quotes depending

00:02:59.060 --> 00:03:00.920
on whether c is
negative or positive

00:03:00.920 --> 00:03:03.360
and d is greater
than or less than 0.

00:03:03.360 --> 00:03:06.920
But this is the greatest common
factor of these two monomials.

00:03:06.920 --> 00:03:08.950
It's divisible
into both of them,

00:03:08.950 --> 00:03:12.410
and it uses the most
factors possible.


How we know about the earth's core

WEBVTT
Kind: captions
Language: en

00:00:00.730 --> 00:00:02.860
So how do we know
that there is a core,

00:00:02.860 --> 00:00:05.470
and that the core is made
up of a liquid outer core

00:00:05.470 --> 00:00:07.220
and a solid inner core?

00:00:07.220 --> 00:00:09.610
And the answer there comes
from the same technique

00:00:09.610 --> 00:00:14.780
that we saw Mohorovicic use
in 1909 to essentially see

00:00:14.780 --> 00:00:17.430
the behavior, or when you
measure the seismic waves,

00:00:17.430 --> 00:00:19.780
or whether you can even
measure the seismic waves,

00:00:19.780 --> 00:00:23.100
the different distances
from an earthquake.

00:00:23.100 --> 00:00:25.120
So if there's an
earthquake right here.

00:00:25.120 --> 00:00:28.040
We're calling that zero degrees.

00:00:28.040 --> 00:00:30.090
Let's remember a
couple of things here.

00:00:30.090 --> 00:00:34.550
Let's remember that P-waves
can travel through anything.

00:00:34.550 --> 00:00:37.316
They can travel through solid or
liquid or air for that matter.

00:00:37.316 --> 00:00:38.815
So they can travel
through anything.

00:00:41.680 --> 00:00:47.770
But S-waves, S for secondary,
these are the transverse waves,

00:00:47.770 --> 00:00:52.460
these can only travel
through solids.

00:00:52.460 --> 00:00:54.600
So it turns out that if
an earthquake happens

00:00:54.600 --> 00:00:57.420
at zero degrees, and you
had seismograph stations

00:00:57.420 --> 00:01:00.620
all over the world, and
these are extremely sensitive

00:01:00.620 --> 00:01:03.180
in order to be able to measure
earthquakes that are happening

00:01:03.180 --> 00:01:06.140
thousands of kilometers
away, it turns out

00:01:06.140 --> 00:01:11.582
that there's something called
an S shadow, an S-wave shadow.

00:01:11.582 --> 00:01:13.540
If these are S-waves you
can measure them here.

00:01:13.540 --> 00:01:14.580
You can measure them here.

00:01:14.580 --> 00:01:15.996
They can go all
the way over here.

00:01:15.996 --> 00:01:17.180
They can go over here.

00:01:17.180 --> 00:01:18.930
They can go over there.

00:01:18.930 --> 00:01:20.600
You can measure them over here.

00:01:20.600 --> 00:01:22.790
So you could measure them
at all of these points,

00:01:22.790 --> 00:01:26.514
but then all of a
sudden at 105 degrees,

00:01:26.514 --> 00:01:28.180
and so we're measuring
zero degrees here

00:01:28.180 --> 00:01:29.930
and we're going
outwards like that,

00:01:29.930 --> 00:01:33.140
all of a sudden at 105
degrees and further

00:01:33.140 --> 00:01:35.930
you stop measuring S-waves.

00:01:35.930 --> 00:01:38.830
For some reason you would
think that some of the S-waves

00:01:38.830 --> 00:01:41.980
would get over here, maybe they
would be a little bit weaker,

00:01:41.980 --> 00:01:44.190
but they would be able to
get all the way over here.

00:01:44.190 --> 00:01:46.050
But they just abruptly stop.

00:01:46.050 --> 00:01:47.540
No more S-waves.

00:01:47.540 --> 00:01:53.500
So in this whole area right
over here you get no S-waves.

00:01:53.500 --> 00:01:55.400
And obviously I could
flip this picture over

00:01:55.400 --> 00:01:56.858
and you would see
a symmetric thing

00:01:56.858 --> 00:02:00.860
on the other side of the globe
that all of this area over here

00:02:00.860 --> 00:02:04.470
you also would not see S-waves.

00:02:04.470 --> 00:02:08.699
You'd only see them from 105
degrees in this direction

00:02:08.699 --> 00:02:10.979
and 105 degrees
in that direction.

00:02:10.979 --> 00:02:15.160
And the only reasonable
explanation that we can give is

00:02:15.160 --> 00:02:18.290
that there must be some material
that an S-wave cannot travel

00:02:18.290 --> 00:02:21.200
through that it would have to
travel through to get to these

00:02:21.200 --> 00:02:23.680
points beyond 105 degrees.

00:02:23.680 --> 00:02:28.090
And we know that S-waves
only travel in solids.

00:02:28.090 --> 00:02:31.880
So the assumption there is
that at some point beyond 105

00:02:31.880 --> 00:02:34.100
degrees it's hitting liquid.

00:02:34.100 --> 00:02:36.050
So that's what tells
us that this right here

00:02:36.050 --> 00:02:41.880
is probably a liquid.

00:02:41.880 --> 00:02:43.919
It's hitting some
layer that is liquid.

00:02:43.919 --> 00:02:45.460
So that tells us
that there's a core,

00:02:45.460 --> 00:02:48.260
and at least the outer part
of that core is liquid,

00:02:48.260 --> 00:02:51.360
enough to stop S-waves.

00:02:51.360 --> 00:02:54.190
So the S-waves, because
it only travels in solids

00:02:54.190 --> 00:02:58.330
it leads to this S-wave shadow.

00:02:58.330 --> 00:03:02.560
And this tells us
that we have a core.

00:03:02.560 --> 00:03:05.240
And that core, at least
the outer part, is liquid.

00:03:05.240 --> 00:03:10.024
We don't know yet whether the
inner part is liquid or solid.

00:03:10.024 --> 00:03:11.440
Now, the next point
of evidence is

00:03:11.440 --> 00:03:13.190
how do we know that
there's an inner core?

00:03:13.190 --> 00:03:16.140
And we can use P-waves for that.

00:03:16.140 --> 00:03:18.480
A P-wave can travel
through anything,

00:03:18.480 --> 00:03:23.870
but remember, in general for
the same type of material

00:03:23.870 --> 00:03:27.206
if you get denser material
it's going to move faster,

00:03:27.206 --> 00:03:28.580
so it's going to
refract outwards

00:03:28.580 --> 00:03:30.010
like we've seen over here.

00:03:30.010 --> 00:03:32.840
But if it goes into a liquid,
in general, sound waves,

00:03:32.840 --> 00:03:39.550
or I should say P-waves, seismic
waves move slower in liquids.

00:03:39.550 --> 00:03:41.330
And so the refraction
patterns we

00:03:41.330 --> 00:03:44.760
get when we do measure
from seismograph stations

00:03:44.760 --> 00:03:49.090
around the world is that it
looks like the P-waves are kind

00:03:49.090 --> 00:03:51.230
of doing what you would
expect in the mantle,

00:03:51.230 --> 00:03:53.030
but then they're
getting refracted

00:03:53.030 --> 00:03:54.800
as if they're going
to a slower medium

00:03:54.800 --> 00:03:57.660
as they go through
the outer core.

00:03:57.660 --> 00:03:59.780
And we see that right over here.

00:03:59.780 --> 00:04:01.440
And then they get
refracted again

00:04:01.440 --> 00:04:03.730
to get to some point
on the other side.

00:04:03.730 --> 00:04:06.450
Now, that is just what you would
expect if it was all liquid,

00:04:06.450 --> 00:04:10.180
but if you go to stations
that are even further out

00:04:10.180 --> 00:04:12.680
it looks like, if you just look
at the refraction patterns,

00:04:12.680 --> 00:04:14.638
and you can now model
this with fancy computers

00:04:14.638 --> 00:04:16.680
and get all the data
points, but you could say,

00:04:16.680 --> 00:04:20.019
well, the only way that reality
can fit the data that we

00:04:20.019 --> 00:04:22.060
get based on when
things reach here

00:04:22.060 --> 00:04:25.140
is if the P-waves are
being first refracted

00:04:25.140 --> 00:04:27.656
through the outer core, but
then they're refracted in a way

00:04:27.656 --> 00:04:30.030
that they're going through
denser material, significantly

00:04:30.030 --> 00:04:33.670
denser material
than the inner core.

00:04:33.670 --> 00:04:36.040
And then they're just
continuing to refract

00:04:36.040 --> 00:04:37.170
the way you would expect.

00:04:37.170 --> 00:04:41.284
So it's really the refraction
pattern of the P-waves.

00:04:41.284 --> 00:04:42.700
And frankly, the
fact that there's

00:04:42.700 --> 00:04:45.800
this what you call
a P-wave shadow.

00:04:45.800 --> 00:04:48.060
The P-wave shadow
by itself, all that

00:04:48.060 --> 00:04:50.720
tells you is that kind
of roughly crazy things

00:04:50.720 --> 00:04:53.020
are happening
someplace in the core.

00:04:53.020 --> 00:04:56.010
But the real way to know that
we have an inner core that's

00:04:56.010 --> 00:04:58.990
solid, as opposed to the
whole thing being liquid,

00:04:58.990 --> 00:05:00.870
is that the P-waves
is the pattern

00:05:00.870 --> 00:05:03.520
of when and how
the P-waves reach

00:05:03.520 --> 00:05:05.220
essentially the other
side of the globe.

00:05:05.220 --> 00:05:08.060
And then you can kind
of, based on modeling

00:05:08.060 --> 00:05:11.054
how waves would travel
through different densities

00:05:11.054 --> 00:05:12.970
and different types of
mediums, you could say,

00:05:12.970 --> 00:05:15.867
well, there's got to be an
inner core right over here.

00:05:15.867 --> 00:05:18.200
And obviously, it's a lot
more math than I'm going into.

00:05:18.200 --> 00:05:20.146
But if you do the math
based on the shadow,

00:05:20.146 --> 00:05:21.770
and you know the
speed of the material,

00:05:21.770 --> 00:05:23.394
and all of that type
of thing, then you

00:05:23.394 --> 00:05:28.340
can figure out the depth at
which these transitions occur.

00:05:28.340 --> 00:05:30.400
We know that we have a
transition from mantle

00:05:30.400 --> 00:05:31.590
to outer core here.

00:05:31.590 --> 00:05:34.380
And then a transition from
outer core to core there.

00:05:34.380 --> 00:05:37.650
So hopefully that
satiates your questions

00:05:37.650 --> 00:05:40.450
about how do we know what
the composition of the earth

00:05:40.450 --> 00:05:45.080
is without ever having
to dig down there,

00:05:45.080 --> 00:05:48.472
because we've never even
gotten below our crust.


The mohorovicic seismic discontinuity

WEBVTT
Kind: captions
Language: en

00:00:00.574 --> 00:00:02.490
The first realization
that there were actually

00:00:02.490 --> 00:00:05.470
distinct layers of the earth
came from this guy right

00:00:05.470 --> 00:00:08.029
over here, Andrija Mohorovicic.

00:00:08.029 --> 00:00:10.800
And I apologize ahead
of time to any Croatians

00:00:10.800 --> 00:00:13.660
for butchering any
of the pronunciation.

00:00:13.660 --> 00:00:16.420
And he was a meteorologist
and a seismologist.

00:00:16.420 --> 00:00:18.130
And he was the
first one to notice,

00:00:18.130 --> 00:00:22.040
in 1909, when there
was an earthquake.

00:00:22.040 --> 00:00:24.760
There was an
earthquake in Croatia,

00:00:24.760 --> 00:00:27.120
a little bit
southeast of Zagreb.

00:00:27.120 --> 00:00:29.600
So the earthquake was
roughly over here.

00:00:29.600 --> 00:00:32.089
And lucky for him and
lucky for us, before that

00:00:32.089 --> 00:00:33.630
earthquake there
was actually a bunch

00:00:33.630 --> 00:00:36.700
of seismographic stations
already in the area.

00:00:36.700 --> 00:00:38.765
And all these
seismographic stations are

00:00:38.765 --> 00:00:41.200
are, essentially,
instruments were installed

00:00:41.200 --> 00:00:43.310
so that if there
was any essentially

00:00:43.310 --> 00:00:45.170
seismic waves
passing, they would

00:00:45.170 --> 00:00:47.880
be able to measure it
when the waves got there.

00:00:47.880 --> 00:00:49.600
And what was
interesting about this,

00:00:49.600 --> 00:00:54.170
Andrija realized that if
the entire earth was just

00:00:54.170 --> 00:00:58.470
kind of a uniform materials--
let's draw that scenario--

00:00:58.470 --> 00:01:00.920
it would get denser
as you go down.

00:01:00.920 --> 00:01:03.490
And so you would have
kind of this refraction,

00:01:03.490 --> 00:01:06.710
this continuous refraction, or
these curved pats, happening.

00:01:06.710 --> 00:01:09.630
But he realized that, let's
say we had an earthquake

00:01:09.630 --> 00:01:12.390
right over here, so this
is the uniform case.

00:01:12.390 --> 00:01:13.760
Uniform.

00:01:13.760 --> 00:01:17.050
Uniform layer, only one layer,
although it does get denser.

00:01:17.050 --> 00:01:20.710
Then the closer you
are to the earthquake--

00:01:20.710 --> 00:01:23.360
so waves would get
there first, then waves

00:01:23.360 --> 00:01:27.310
would get over there, then
waves would you get over there--

00:01:27.310 --> 00:01:29.329
and these are the body waves.

00:01:29.329 --> 00:01:30.870
These are the ones
that are traveling

00:01:30.870 --> 00:01:32.630
through the earth's crust.

00:01:32.630 --> 00:01:35.840
But in general, the further you
are away from the earthquake,

00:01:35.840 --> 00:01:39.580
or the time it takes for
the waves to get to a point,

00:01:39.580 --> 00:01:41.640
is going to be proportional
to the distance

00:01:41.640 --> 00:01:43.830
that point is away
from the earthquake.

00:01:43.830 --> 00:01:46.300
So you would expect to
see something like this.

00:01:46.300 --> 00:01:49.440
So if you were to plot
on the horizontal axis,

00:01:49.440 --> 00:01:53.490
if you were to plot distance,
and on the vertical axis

00:01:53.490 --> 00:01:59.030
you were to plot time, you
should see something like this.

00:01:59.030 --> 00:02:01.780
You should see a straight line.

00:02:01.780 --> 00:02:04.140
And that's just
because it's traveling

00:02:04.140 --> 00:02:07.396
roughly the same velocity
along any of these arcs.

00:02:07.396 --> 00:02:09.979
It's maybe getting a little bit
faster as it's getting deeper.

00:02:09.979 --> 00:02:11.520
But roughly the same
velocity as it's

00:02:11.520 --> 00:02:12.780
traveling along these arcs.

00:02:12.780 --> 00:02:14.200
And the distance
of these arcs are

00:02:14.200 --> 00:02:16.850
proportional to the
distance along the surface,

00:02:16.850 --> 00:02:18.340
along the distance
of the surface.

00:02:18.340 --> 00:02:20.764
So essentially, the time is,
they're all traveling roughly

00:02:20.764 --> 00:02:22.680
at the same velocity,
and their just traveling

00:02:22.680 --> 00:02:24.800
different distances,
so the time it takes

00:02:24.800 --> 00:02:27.720
is just going to be
proportional to the distance.

00:02:27.720 --> 00:02:30.000
But he noticed
something interesting.

00:02:30.000 --> 00:02:35.080
When he actually measured when
the waves from that earthquake

00:02:35.080 --> 00:02:37.640
reached different
seismographic stations,

00:02:37.640 --> 00:02:40.230
he saw something interesting.

00:02:40.230 --> 00:02:41.800
So this is in the
theoretical, if we

00:02:41.800 --> 00:02:44.260
had a kind of this
uniform one-layered earth.

00:02:44.260 --> 00:02:46.020
But he saw something
interesting.

00:02:46.020 --> 00:02:48.650
So once again, this
is the distance,

00:02:48.650 --> 00:02:51.550
and this right
over here is time.

00:02:51.550 --> 00:02:54.930
And at 200 kilometers,
at 200 kilometers away

00:02:54.930 --> 00:02:56.980
from the earthquake--
so until 200 kilometers,

00:02:56.980 --> 00:03:00.650
he saw exactly what you would
expect from a uniform earth.

00:03:00.650 --> 00:03:04.270
It was just the time took was
proportional to the distance.

00:03:04.270 --> 00:03:07.010
But at 200 kilometers, he
saw something interesting.

00:03:07.010 --> 00:03:10.190
All of a sudden, the waves
were reaching there faster.

00:03:10.190 --> 00:03:12.030
The slope of this line changed.

00:03:12.030 --> 00:03:15.670
It took less time for
each incremental distance.

00:03:15.670 --> 00:03:17.377
So for some reason,
the waves that we're

00:03:17.377 --> 00:03:19.460
going at these farther
stations, the stations that

00:03:19.460 --> 00:03:23.184
were more than 200
kilometers away, somehow

00:03:23.184 --> 00:03:24.100
they were accelerated.

00:03:24.100 --> 00:03:26.830
Somehow they were
able to move faster.

00:03:26.830 --> 00:03:29.120
And he's the one that
realized that this

00:03:29.120 --> 00:03:31.180
was because the waves
that were getting

00:03:31.180 --> 00:03:35.130
to these further stations must
have traveled through a more

00:03:35.130 --> 00:03:37.070
dense layer of the earth.

00:03:37.070 --> 00:03:38.850
So let's just think about it.

00:03:38.850 --> 00:03:40.660
So if we have a
more dense layer,

00:03:40.660 --> 00:03:44.110
it will fit this
information right over here.

00:03:44.110 --> 00:03:47.190
So if we have a layer
like this, which we now

00:03:47.190 --> 00:03:50.640
know to be the crust, and then
you have a denser layer, which

00:03:50.640 --> 00:03:54.760
we now know to be the mantle,
then what you would have is--

00:03:54.760 --> 00:03:57.750
so you have your earthquake
right over here, closer

00:03:57.750 --> 00:03:59.860
by, while you're still
within the crust,

00:03:59.860 --> 00:04:01.520
it would be proportional.

00:04:01.520 --> 00:04:03.200
It would be proportional.

00:04:03.200 --> 00:04:04.970
And then let's say
that this is exactly,

00:04:04.970 --> 00:04:08.450
this right here is
200 kilometers away.

00:04:08.450 --> 00:04:12.360
But then if you go any further,
the waves would have to travel.

00:04:12.360 --> 00:04:14.800
They would travel, so
they would go like this.

00:04:14.800 --> 00:04:17.050
And then they would get
refracted even harder.

00:04:17.050 --> 00:04:18.560
So they would get refracted.

00:04:18.560 --> 00:04:20.709
So they would be a little
bit curved ahead of time.

00:04:20.709 --> 00:04:22.750
But then they're going to
a much denser material.

00:04:22.750 --> 00:04:25.955
Or it's not gradually dense,
it's actually kind of a all

00:04:25.955 --> 00:04:27.913
of a sudden a considerably
more dense material,

00:04:27.913 --> 00:04:30.480
so it will get
refracted even more.

00:04:30.480 --> 00:04:32.072
And then it'll go over here.

00:04:32.072 --> 00:04:33.530
And since it was
able to travel all

00:04:33.530 --> 00:04:35.113
of this distance in
a denser material,

00:04:35.113 --> 00:04:37.790
it would have traveled
faster along this path.

00:04:37.790 --> 00:04:39.670
And so it would get
to this distance

00:04:39.670 --> 00:04:42.010
on the surface that's more
than 200 kilometers away,

00:04:42.010 --> 00:04:45.630
it would get there faster.

00:04:45.630 --> 00:04:50.840
And so he said that there
must be a denser layer

00:04:50.840 --> 00:04:52.710
that those waves are
traveling through,

00:04:52.710 --> 00:04:55.420
which we now know
to be the mantle.

00:04:55.420 --> 00:04:58.710
And the boundary
between what we now

00:04:58.710 --> 00:05:01.870
know to be the crust
and this denser layer,

00:05:01.870 --> 00:05:05.230
which we now to be the mantle,
is actually named after him.

00:05:05.230 --> 00:05:10.370
It's called the
Mohorovicic discontinuity.

00:05:10.370 --> 00:05:13.320
And sometimes this is
called the Moho for short.

00:05:13.320 --> 00:05:15.410
So that boundary between
the crust and the mantle

00:05:15.410 --> 00:05:16.460
is now named for him.

00:05:16.460 --> 00:05:19.020
But this was a huge discovery,
because not only was he

00:05:19.020 --> 00:05:21.855
able to tell us,
based on the data--

00:05:21.855 --> 00:05:25.195
based on, kind of, indirect
data, just based on earthquakes

00:05:25.195 --> 00:05:27.320
happening, and measuring
when the earthquakes reach

00:05:27.320 --> 00:05:29.444
different points of the
earth-- that there probably

00:05:29.444 --> 00:05:30.590
is a denser layer.

00:05:30.590 --> 00:05:33.740
And if you do the math,
under continental crust

00:05:33.740 --> 00:05:36.571
that denser layer is
about 35 kilometers down.

00:05:36.571 --> 00:05:38.570
He was able to tell us
that there is that layer.

00:05:38.570 --> 00:05:40.210
But even more
importantly, he was

00:05:40.210 --> 00:05:42.390
able to give the clue that
just using information

00:05:42.390 --> 00:05:44.780
from earthquakes,
we could essentially

00:05:44.780 --> 00:05:47.890
figure out the actual
composition of the earth.

00:05:47.890 --> 00:05:49.660
Because no one has
ever dug that deep.

00:05:49.660 --> 00:05:51.990
No one has ever dug into
the mantle, much less

00:05:51.990 --> 00:05:53.592
the outer core or
the inner core.

00:05:53.592 --> 00:05:55.050
In the next few
videos, we're going

00:05:55.050 --> 00:05:57.425
to kind of take this insight,
that we can use information

00:05:57.425 --> 00:05:59.010
from earthquakes,
to actually think

00:05:59.010 --> 00:06:02.860
about how we know that there
is an outer liquid core

00:06:02.860 --> 00:06:04.760
and that there's an
inner core, as well.

00:06:04.760 --> 00:06:06.160
And then, obviously,
you could keep going

00:06:06.160 --> 00:06:07.984
and think about all
the different densities

00:06:07.984 --> 00:06:09.400
within the mantle
and all of that.

00:06:09.400 --> 00:06:10.775
I won't go into
that much detail,

00:06:10.775 --> 00:06:13.120
but I'll see you
in the next video.


Refraction of seismic waves

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:04.610
I want to do a quick
primer on refraction.

00:00:04.610 --> 00:00:07.490
And our focus here is going
to be on the seismic waves,

00:00:07.490 --> 00:00:09.560
but the principles,
how things refract

00:00:09.560 --> 00:00:11.380
when they go from a
fast to a slow medium

00:00:11.380 --> 00:00:13.240
or a slow to a fast
medium, It's actually

00:00:13.240 --> 00:00:16.710
the same as you would see when
you're studying light waves

00:00:16.710 --> 00:00:18.610
or actually any type of wave.

00:00:18.610 --> 00:00:20.430
So let's think about
it a little bit.

00:00:20.430 --> 00:00:24.810
So let's say I have a slow
medium right over here.

00:00:24.810 --> 00:00:28.130
And let's say I have a fast
medium right over here.

00:00:28.130 --> 00:00:30.269
And let's say, just
so we can travel

00:00:30.269 --> 00:00:32.060
through both solid and
liquids, let's think

00:00:32.060 --> 00:00:33.920
about maybe P-waves.

00:00:33.920 --> 00:00:36.620
And a slow medium could be
maybe some type of liquid,

00:00:36.620 --> 00:00:40.280
and our fast medium could
be some type of solid.

00:00:40.280 --> 00:00:44.330
So let me draw the
boundary right over here.

00:00:44.330 --> 00:00:48.306
So if I have a P-wave.

00:00:48.306 --> 00:00:49.930
Let's say it's going
through the water,

00:00:49.930 --> 00:00:53.100
and it's going right
perpendicular to the boundary.

00:00:53.100 --> 00:00:56.150
It will then just continue
to travel in the faster

00:00:56.150 --> 00:00:57.740
medium in the same
direction if it's

00:00:57.740 --> 00:01:01.980
going right at the boundary.

00:01:01.980 --> 00:01:04.449
And it'll just travel
faster in the faster medium.

00:01:04.449 --> 00:01:05.990
And that's because
that faster medium

00:01:05.990 --> 00:01:08.030
is going to be more
dense, and the molecules

00:01:08.030 --> 00:01:09.980
are going to bump into
each other faster.

00:01:09.980 --> 00:01:12.800
In the same amount of time
kind of the chain reaction

00:01:12.800 --> 00:01:14.660
is going to be
able travel further

00:01:14.660 --> 00:01:16.240
because they are
more closely packed

00:01:16.240 --> 00:01:19.550
and they rebound faster than
it would in the slow medium.

00:01:19.550 --> 00:01:22.030
So that's obviously no
refraction is going on.

00:01:22.030 --> 00:01:24.350
It has not been deflected.

00:01:24.350 --> 00:01:27.070
And just as a bit of reminder,
in general, refraction

00:01:27.070 --> 00:01:29.500
is when a wave gets deflected.

00:01:29.500 --> 00:01:31.257
Reflection is when
it bounces back.

00:01:31.257 --> 00:01:33.340
Refraction is when it gets
deflected a little bit.

00:01:33.340 --> 00:01:35.180
Let me just make that clear.

00:01:35.180 --> 00:01:37.270
So if I have some
type of boundary here,

00:01:37.270 --> 00:01:40.180
and I have a wave that bounces
off, that's reflection.

00:01:40.180 --> 00:01:42.370
But if the wave goes
through the boundary

00:01:42.370 --> 00:01:44.870
and just gets bent a little
bit, its direction changes,

00:01:44.870 --> 00:01:46.100
that is refraction.

00:01:46.100 --> 00:01:47.440
That's what we're talking about.

00:01:47.440 --> 00:01:50.830
So clearly so far this P-wave
has not been refracted.

00:01:50.830 --> 00:01:54.060
But if this P-wave comes in
at an angle-- so let's make

00:01:54.060 --> 00:01:57.560
this P-wave wave come in at an
angle-- what's going to happen

00:01:57.560 --> 00:01:59.490
is, and the way you
should think about it--

00:01:59.490 --> 00:02:01.610
and it's the easiest way to
think about which direction

00:02:01.610 --> 00:02:03.943
will be refracted, or at least
the way I think about it,

00:02:03.943 --> 00:02:10.560
is literally I imagine some type
of vehicle with wheels on it.

00:02:10.560 --> 00:02:13.700
So this is the top
view of my vehicle.

00:02:13.700 --> 00:02:16.660
So if I have some type of
vehicle, and the wheels

00:02:16.660 --> 00:02:18.520
will be able to move
slowly in this medium.

00:02:18.520 --> 00:02:19.860
You could kind of view
it as it's kind of on mud

00:02:19.860 --> 00:02:21.220
so it doesn't get good
traction, and then

00:02:21.220 --> 00:02:23.678
the fast medium maybe it's a
road so it gets good traction,

00:02:23.678 --> 00:02:25.230
it could move faster.

00:02:25.230 --> 00:02:27.640
So what's going to happen
when the vehicle gets

00:02:27.640 --> 00:02:29.320
to the boundary?

00:02:29.320 --> 00:02:32.450
Well, this bottom right wheel is
going to go on the fast medium

00:02:32.450 --> 00:02:34.470
before any of the
other wheels do.

00:02:34.470 --> 00:02:36.320
So it's going to get
the traction first.

00:02:36.320 --> 00:02:39.235
These wheels on the left side
of the vehicle, these wheels

00:02:39.235 --> 00:02:41.610
right here, these are still
going to be stuck in the mud.

00:02:41.610 --> 00:02:44.010
So what's going to happen is
this wheel right over here

00:02:44.010 --> 00:02:45.650
is moving faster,
so it's essentially

00:02:45.650 --> 00:02:47.560
going to be able to
turn the vehicle.

00:02:47.560 --> 00:02:49.590
These guys are still
stuck in the mud.

00:02:49.590 --> 00:02:51.260
And so you fast
forward a little bit,

00:02:51.260 --> 00:02:53.740
the direction of the
vehicle will change.

00:02:53.740 --> 00:02:56.220
And so the vehicle will now
move in a direction something

00:02:56.220 --> 00:02:57.370
like this.

00:02:57.370 --> 00:02:59.580
The same thing would
happen in a wave.

00:02:59.580 --> 00:03:02.780
If the P-wave is approaching
the boundary like this,

00:03:02.780 --> 00:03:04.280
and something
analogous to this is

00:03:04.280 --> 00:03:05.710
happening at the
molecular level.

00:03:05.710 --> 00:03:08.155
You can kind of view it
as even billiard balls,

00:03:08.155 --> 00:03:10.030
and maybe they're kind
of hitting each other.

00:03:10.030 --> 00:03:12.570
Well, I won't go into that,
because that can kind of get

00:03:12.570 --> 00:03:14.550
confusing depending
on the different cases

00:03:14.550 --> 00:03:16.034
and the different boundaries.

00:03:16.034 --> 00:03:17.950
But this is the easiest
way to think about it,

00:03:17.950 --> 00:03:19.449
and which direction
it will refract.

00:03:19.449 --> 00:03:22.450
And hopefully it makes a
little bit of intuitive sense.

00:03:22.450 --> 00:03:25.990
And so when you go from a slow
to a fast medium our P-wave

00:03:25.990 --> 00:03:28.520
its angle would accentuate
in that direction.

00:03:28.520 --> 00:03:33.030
If you went from the fast medium
to the slow medium, once again,

00:03:33.030 --> 00:03:35.630
you can just go through the
same thought experiment.

00:03:35.630 --> 00:03:38.400
So let's say you have our
wave coming in like that.

00:03:38.400 --> 00:03:39.540
Draw the car.

00:03:39.540 --> 00:03:41.910
Visualize the car here.

00:03:41.910 --> 00:03:43.887
Visualize the car right here.

00:03:43.887 --> 00:03:45.470
And you'd say, well,
look, this tire's

00:03:45.470 --> 00:03:48.540
going to get stuck in the mud,
because it was on the road.

00:03:48.540 --> 00:03:52.470
Now, this top tire right over
here is get stuck in the mud

00:03:52.470 --> 00:03:53.150
first.

00:03:53.150 --> 00:03:54.640
So it's going to
be moving slower.

00:03:54.640 --> 00:03:57.000
So these tires are going
to be able to move faster.

00:03:57.000 --> 00:03:58.610
So the vehicle is going to turn.

00:03:58.610 --> 00:04:02.110
So you'll be refracted in a
direction like that when you're

00:04:02.110 --> 00:04:04.400
going from the fast
to the slow medium.

00:04:04.400 --> 00:04:07.015
So that's just a primer
on refraction generally.

00:04:07.015 --> 00:04:08.390
Now, let's think
about what would

00:04:08.390 --> 00:04:11.810
happen when sound waves are
traveling through the Earth.

00:04:11.810 --> 00:04:14.420
And this will help inform
us of, essentially,

00:04:14.420 --> 00:04:17.910
how do we figure out what the
actual structure of the Earth

00:04:17.910 --> 00:04:19.500
is.

00:04:19.500 --> 00:04:23.190
So if the Earth was just made
up of some uniform material

00:04:23.190 --> 00:04:27.990
and you had an earthquake right
here on Earth, maybe a little

00:04:27.990 --> 00:04:29.090
bit below the surface.

00:04:29.090 --> 00:04:31.600
So it's happening in the
crust, but a little bit

00:04:31.600 --> 00:04:33.080
below the surface of the Earth.

00:04:33.080 --> 00:04:35.110
If Earth was of
uniform density, if it

00:04:35.110 --> 00:04:40.050
was all the same material, let's
just think about the P-waves.

00:04:40.050 --> 00:04:41.750
Because P-waves can
travel in anything.

00:04:41.750 --> 00:04:43.600
Let's think about how
those P-waves would travel.

00:04:43.600 --> 00:04:45.391
Well, they would just
go in straight lines.

00:04:45.391 --> 00:04:47.710
There's nothing that
would refract the P-waves.

00:04:47.710 --> 00:04:52.180
They would just go in straight
lines radially outward

00:04:52.180 --> 00:04:53.855
from where the
earthquake occurred.

00:04:58.000 --> 00:05:00.480
Now, at a first
approximation, we

00:05:00.480 --> 00:05:02.880
know that as we go deeper
and deeper into Earth

00:05:02.880 --> 00:05:04.510
there's more and
more rock above that.

00:05:04.510 --> 00:05:06.730
The weight of that rock is
kind of compressing the rock

00:05:06.730 --> 00:05:07.240
below it.

00:05:07.240 --> 00:05:09.220
So you get higher
and higher pressures

00:05:09.220 --> 00:05:11.280
and higher and higher densities.

00:05:11.280 --> 00:05:13.110
So this is a uniform Earth.

00:05:15.940 --> 00:05:17.530
But let's imagine
an Earth that's

00:05:17.530 --> 00:05:21.690
made up of uniform material,
that's all solid, a completely

00:05:21.690 --> 00:05:25.160
solid Earth, but one where
the density is constantly

00:05:25.160 --> 00:05:27.290
increasing as you go down.

00:05:27.290 --> 00:05:29.240
So let's just think
about it before we

00:05:29.240 --> 00:05:30.576
go into the continuous case.

00:05:30.576 --> 00:05:32.950
Because we're talking about
the density as you go deeper,

00:05:32.950 --> 00:05:35.010
it's just getting
continuously more dense,

00:05:35.010 --> 00:05:36.600
let's think about
the discrete case,

00:05:36.600 --> 00:05:40.214
where we have the
least dense layer.

00:05:40.214 --> 00:05:41.630
So let me draw it
right over here.

00:05:41.630 --> 00:05:43.546
So let's say this is the
surface of the Earth.

00:05:43.546 --> 00:05:44.680
This is least dense.

00:05:47.810 --> 00:05:49.980
Then let's say you have
another layer over here that

00:05:49.980 --> 00:05:51.330
is more dense.

00:05:51.330 --> 00:05:52.915
So this is more dense.

00:05:56.410 --> 00:06:00.080
Let's say you have another
layer that's even more dense.

00:06:00.080 --> 00:06:07.660
So you have another layer over
here that's even more dense.

00:06:07.660 --> 00:06:10.180
And then let's do
one more layer.

00:06:10.180 --> 00:06:15.350
Let's do this layer here,
this is the densest layer.

00:06:15.350 --> 00:06:18.860
So in general, your
P-wave, your seismic wave,

00:06:18.860 --> 00:06:21.072
is going to travel faster
in denser material.

00:06:21.072 --> 00:06:22.780
So it's going to travel
the fastest here,

00:06:22.780 --> 00:06:23.655
then here, then here.

00:06:23.655 --> 00:06:26.280
It's going to travel the slowest
in this least dense material.

00:06:26.280 --> 00:06:30.320
So if you're coming
in at an angle

00:06:30.320 --> 00:06:32.140
let's think about
what's going to happen.

00:06:32.140 --> 00:06:34.710
So let's say you have
your P-wave coming

00:06:34.710 --> 00:06:36.300
in at an angle like this.

00:06:36.300 --> 00:06:40.110
So it's going straight through
the least dense material.

00:06:40.110 --> 00:06:41.790
Let me do a slightly
shallower angle.

00:06:41.790 --> 00:06:43.430
So let's say it's like that.

00:06:43.430 --> 00:06:44.888
What's going to
happen when it goes

00:06:44.888 --> 00:06:46.240
into the more dense material?

00:06:46.240 --> 00:06:49.160
So once again, let's
imagine our little car.

00:06:49.160 --> 00:06:51.250
So this tire's going
to be able to go faster

00:06:51.250 --> 00:06:53.250
before the tires
on the other side.

00:06:53.250 --> 00:06:56.310
So the car is going to
be deflected to the left,

00:06:56.310 --> 00:06:57.480
to the down left.

00:06:57.480 --> 00:06:59.810
So now it's going
to travel like this.

00:06:59.810 --> 00:07:02.296
So it's now going to
travel something like this.

00:07:02.296 --> 00:07:04.170
Now, what's going to
happen at this boundary?

00:07:04.170 --> 00:07:05.796
Once again, imagine the car.

00:07:05.796 --> 00:07:08.170
This tire right here is going
to be able to travel faster

00:07:08.170 --> 00:07:10.430
before the other tire, so
it'll be deflected even more

00:07:10.430 --> 00:07:11.594
in that direction.

00:07:11.594 --> 00:07:13.510
Then we go, and we go
to the densest material.

00:07:13.510 --> 00:07:16.252
Once again, the tires on
kind of the bottom side when

00:07:16.252 --> 00:07:17.710
we look at this
way are going to be

00:07:17.710 --> 00:07:20.420
able to move faster
before the other tire.

00:07:20.420 --> 00:07:23.009
So we're going to get
deflected even more.

00:07:23.009 --> 00:07:25.300
So you see, as you go from
least dense material to more

00:07:25.300 --> 00:07:27.650
dense material you're
kind of curving outward.

00:07:27.650 --> 00:07:29.640
So if this was
continuous, if you

00:07:29.640 --> 00:07:34.360
had a kind of a continuous
structure, where as you go down

00:07:34.360 --> 00:07:37.370
it just gets more and
more dense as you go.

00:07:37.370 --> 00:07:43.590
So this is less dense, and
then it just continuously

00:07:43.590 --> 00:07:44.580
gets more dense.

00:07:44.580 --> 00:07:47.650
So this is the most
dense down here.

00:07:47.650 --> 00:07:49.132
How would the refraction look?

00:07:49.132 --> 00:07:51.090
Well, then it would just
be a continuous curve.

00:07:51.090 --> 00:07:52.130
It would look like this.

00:07:52.130 --> 00:07:56.770
Your P-wave would constantly
be refracted out like that.

00:07:56.770 --> 00:07:58.630
It would curve outwards.

00:07:58.630 --> 00:08:02.280
So this was the simplest
example where Earth is uniform.

00:08:02.280 --> 00:08:04.030
And that's pretty
easy to dismiss,

00:08:04.030 --> 00:08:06.880
that obviously things will get
denser because of more pressure

00:08:06.880 --> 00:08:07.380
down.

00:08:07.380 --> 00:08:08.921
So let's say we
assume another thing.

00:08:08.921 --> 00:08:11.470
We have a uniform Earth
in terms of composition,

00:08:11.470 --> 00:08:13.610
but let's say it gets denser.

00:08:13.610 --> 00:08:17.540
So denser at the center.

00:08:17.540 --> 00:08:22.170
Then how would the
P-waves travel,

00:08:22.170 --> 00:08:24.600
or how would any
seismic waves travel?

00:08:24.600 --> 00:08:27.009
Well, then if you have your
earthquake right over here,

00:08:27.009 --> 00:08:28.550
the ones that are
going straight down

00:08:28.550 --> 00:08:29.759
still would go straight down.

00:08:29.759 --> 00:08:31.133
Because we know
that we won't get

00:08:31.133 --> 00:08:33.070
refracted if we're kind
of going perpendicular

00:08:33.070 --> 00:08:37.440
to the change in medium, or
the change in boundaries.

00:08:37.440 --> 00:08:39.746
But things that are
coming at a slight angle,

00:08:39.746 --> 00:08:42.370
as they get deeper they're going
to get deflected more and more

00:08:42.370 --> 00:08:44.161
and more, and they're
going to be refracted

00:08:44.161 --> 00:08:46.078
outward just like we saw
in this example here.

00:08:46.078 --> 00:08:47.660
If they go on this
angle they're going

00:08:47.660 --> 00:08:49.140
to be refracted
outward like that.

00:08:49.140 --> 00:08:51.806
If they go here they're going to
be refracted outward like that.

00:08:51.806 --> 00:08:54.237
They're going to be
refracted outward like that.

00:08:54.237 --> 00:08:56.820
If you're here you're going to
be refracted outward like that.

00:08:56.820 --> 00:08:59.527
If you're here you're going to
be refracted outward like this.

00:08:59.527 --> 00:09:01.610
Now, what we're going to
do in the next few videos

00:09:01.610 --> 00:09:04.350
is use what we just
learned about refraction

00:09:04.350 --> 00:09:06.890
in the case of seismic waves--
and hopefully we learned it

00:09:06.890 --> 00:09:09.030
in this video-- and
how it would refract

00:09:09.030 --> 00:09:11.790
as we're going through ever
increasing denser material.

00:09:11.790 --> 00:09:14.520
We're going to use
that information

00:09:14.520 --> 00:09:17.650
to essentially try to figure
out the composition of the Earth

00:09:17.650 --> 00:09:21.259
based on what we've
actually observed.


Why S-waves only travel in solids

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:03.750
In the last video I gave a
little bit of a hand wavy

00:00:03.750 --> 00:00:08.289
explanation about why S-waves
don't travel in liquid or air.

00:00:08.289 --> 00:00:10.830
What I want to do in this video
is give you a little bit more

00:00:10.830 --> 00:00:12.780
intuitive understanding
of that, and really

00:00:12.780 --> 00:00:14.530
go down to the molecular level.

00:00:14.530 --> 00:00:17.790
So let's draw a solid.

00:00:17.790 --> 00:00:22.170
And it has nice covalent
bonds, strong bonds

00:00:22.170 --> 00:00:24.020
between the different molecules.

00:00:24.020 --> 00:00:26.950
And the bonds are drawn
by these lines in between.

00:00:26.950 --> 00:00:28.979
So if I were to
hit this solid, you

00:00:28.979 --> 00:00:31.020
know I have this really
small hammer where I just

00:00:31.020 --> 00:00:32.860
hit at a molecular
level, but if I

00:00:32.860 --> 00:00:38.580
were to hit these molecules
hard enough so that they move

00:00:38.580 --> 00:00:41.260
but not so hard enough
that it breaks the bonds,

00:00:41.260 --> 00:00:44.020
then essentially what
it's going to look

00:00:44.020 --> 00:00:46.260
like is this kind
of row of molecules

00:00:46.260 --> 00:00:48.100
is going to move to the left.

00:00:48.100 --> 00:00:50.220
So you're going to have
that row of molecules

00:00:50.220 --> 00:00:51.630
moving to the left.

00:00:51.630 --> 00:00:54.690
And then the row above it
won't fully move to the left

00:00:54.690 --> 00:00:56.730
just yet, but it will
start to get pulled.

00:00:56.730 --> 00:00:59.040
So let me just draw
all of the bonds.

00:00:59.040 --> 00:01:00.810
I'm just drawing all
of the same bonds.

00:01:00.810 --> 00:01:06.250
Because these are strong bonds
that we have in a solid--

00:01:06.250 --> 00:01:09.090
Actually, they could
be ionic bonds as well.

00:01:09.090 --> 00:01:12.260
Because they are strong bonds
that we have in this solid,

00:01:12.260 --> 00:01:15.560
they'll essentially be pulled.

00:01:15.560 --> 00:01:17.380
The top row will be
pulled in the direction

00:01:17.380 --> 00:01:20.450
of the bottom row.

00:01:20.450 --> 00:01:23.280
And so they'll start kind
of moving in that direction.

00:01:23.280 --> 00:01:25.940
And then the bottom row will
essentially recoil back.

00:01:25.940 --> 00:01:27.700
And then you fast
forward a little bit.

00:01:27.700 --> 00:01:30.180
And so then the top row
will have moved to the left.

00:01:33.150 --> 00:01:35.220
And now the bottom
row will start

00:01:35.220 --> 00:01:43.670
to move back, especially
because, remember,

00:01:43.670 --> 00:01:47.030
it's bonded to other
things down here.

00:01:47.030 --> 00:01:49.194
It's bonded to more of
the solid down here.

00:01:49.194 --> 00:01:50.110
So it would move back.

00:01:50.110 --> 00:01:52.140
And you can see this
transverse wave,

00:01:52.140 --> 00:01:54.740
you can see this
S-wave propagating.

00:01:54.740 --> 00:01:57.790
Essentially right over here
the kind of peak of the S-wave

00:01:57.790 --> 00:01:58.570
is here.

00:01:58.570 --> 00:02:00.602
Now it has moved up.

00:02:00.602 --> 00:02:02.560
Now, let's think about
the exact same situation

00:02:02.560 --> 00:02:03.590
with the liquids.

00:02:03.590 --> 00:02:06.920
In liquids you don't have these
strong ionic or covalent bonds

00:02:06.920 --> 00:02:08.900
between the different molecules.

00:02:08.900 --> 00:02:10.900
You just have these
weak kind of bonds,

00:02:10.900 --> 00:02:13.240
usually formed due to polarity.

00:02:13.240 --> 00:02:17.600
So in a liquid,
water's a good example,

00:02:17.600 --> 00:02:20.260
you just have these kind
of weaker bonds formed

00:02:20.260 --> 00:02:24.290
because water is
a polar molecule.

00:02:24.290 --> 00:02:26.250
So the kind of
half-way polar sides

00:02:26.250 --> 00:02:28.460
or the half-way positive
sides are somewhat

00:02:28.460 --> 00:02:30.500
attracted to the
half-way negative sides.

00:02:30.500 --> 00:02:32.840
So they kind of flow
past each other.

00:02:32.840 --> 00:02:36.330
But if I were to hit these
water molecules right here

00:02:36.330 --> 00:02:39.030
with my hammer,
what would happen?

00:02:39.030 --> 00:02:41.722
Well, they're definitely going
to start moving to the left.

00:02:41.722 --> 00:02:44.180
And actually, this one's going
to bump into that one, which

00:02:44.180 --> 00:02:45.170
is going to bump
into that, which

00:02:45.170 --> 00:02:46.050
is going to bump into that one.

00:02:46.050 --> 00:02:47.466
They're going to
move to the left.

00:02:50.980 --> 00:02:53.454
But these molecules aren't
going to move with them.

00:02:53.454 --> 00:02:55.120
You could view it as
it's going to break

00:02:55.120 --> 00:02:57.540
that very weak bond
due to polarity.

00:02:57.540 --> 00:02:59.420
They're going to move
away from each other.

00:02:59.420 --> 00:03:02.440
Let me draw these top
molecules in green.

00:03:02.440 --> 00:03:05.410
They're essentially just
going to flow past each other.

00:03:05.410 --> 00:03:07.270
They're going to
flow past each other.

00:03:07.270 --> 00:03:10.130
And this guy might have had also
weak bonds with stuff below it,

00:03:10.130 --> 00:03:10.630
too.

00:03:10.630 --> 00:03:12.004
I should draw it
as dotted lines.

00:03:15.243 --> 00:03:19.422
But because of the
impact here, these guys

00:03:19.422 --> 00:03:20.380
are just going to flow.

00:03:20.380 --> 00:03:22.588
They're actually going to
compress in this direction.

00:03:22.588 --> 00:03:24.870
You're going to have a
P-wave, a compression wave,

00:03:24.870 --> 00:03:26.840
go in this direction, where
this one bumps into that one,

00:03:26.840 --> 00:03:28.840
and then goes back, and then
this one bumps into that one

00:03:28.840 --> 00:03:31.048
and goes back, and then this
one bumps into that one.

00:03:31.048 --> 00:03:33.900
But the bonds aren't
strong enough,

00:03:33.900 --> 00:03:35.770
and it's even more
the case with air,

00:03:35.770 --> 00:03:38.530
but the bonds aren't strong
enough for these blue guys

00:03:38.530 --> 00:03:40.720
to take these green
guys for a ride.

00:03:40.720 --> 00:03:42.570
And the bonds are
also not strong enough

00:03:42.570 --> 00:03:47.040
for the adjacent
molecules to kind of help

00:03:47.040 --> 00:03:50.870
these blue guys to retract
to their original position.

00:03:50.870 --> 00:03:53.840
So when I talked about the
elasticity in the last video

00:03:53.840 --> 00:03:55.314
that's what I was talking about.

00:03:55.314 --> 00:03:57.605
The bonds aren't strong enough
to cause the things that

00:03:57.605 --> 00:04:00.330
have deformed to kind of
move back to where they were,

00:04:00.330 --> 00:04:02.120
and also the bonds
aren't strong enough

00:04:02.120 --> 00:04:03.780
to allow the things
that are deformed

00:04:03.780 --> 00:04:06.280
to pull other things with it.

00:04:06.280 --> 00:04:10.530
And so that's why, in general,
S-waves only travel in solid,

00:04:10.530 --> 00:04:14.190
and they won't travel
in liquid or air.


Seismic waves

WEBVTT
Kind: captions
Language: en

00:00:00.164 --> 00:00:02.330
What I want to do in this
video is talk a little bit

00:00:02.330 --> 00:00:03.860
about seismic waves.

00:00:06.760 --> 00:00:09.330
One, because they're
interesting by themselves,

00:00:09.330 --> 00:00:11.580
but they're also really
useful for figuring out

00:00:11.580 --> 00:00:13.690
what the actual composition
of the Earth is.

00:00:13.690 --> 00:00:17.070
You've seen my video on the
actual layers of the Earth,

00:00:17.070 --> 00:00:20.940
and seismic waves are
crucial to actually realizing

00:00:20.940 --> 00:00:24.190
how people figured out what the
different layers of the Earth

00:00:24.190 --> 00:00:24.810
are.

00:00:24.810 --> 00:00:26.546
And just to be
clear, seismic waves,

00:00:26.546 --> 00:00:28.420
they're normally associated
with earthquakes,

00:00:28.420 --> 00:00:32.619
but they're any waves that
travel through the Earth.

00:00:32.619 --> 00:00:37.890
They could be due to
an earthquake, or just

00:00:37.890 --> 00:00:41.890
really any kind of a large
explosion, or anything

00:00:41.890 --> 00:00:44.130
that really essentially
starts sending energy

00:00:44.130 --> 00:00:48.890
through the rock on Earth,
really through Earth itself.

00:00:48.890 --> 00:00:51.980
Now, there's two
fundamentally different types

00:00:51.980 --> 00:00:54.174
of the seismic waves.

00:00:54.174 --> 00:00:56.340
And we're going to focus
on one more than the other.

00:00:56.340 --> 00:00:57.500
One is surface waves.

00:01:00.200 --> 00:01:03.251
And the other is body waves.

00:01:03.251 --> 00:01:05.000
Now, surface waves are
ones that literally

00:01:05.000 --> 00:01:07.050
travel across the
surface of something.

00:01:07.050 --> 00:01:08.925
In this case, we're
talking about the surface

00:01:08.925 --> 00:01:09.680
of the ground.

00:01:09.680 --> 00:01:12.330
And this right here is a
depiction of surface waves.

00:01:12.330 --> 00:01:14.790
And these really are more
analogous to the type

00:01:14.790 --> 00:01:18.430
of waves we normally associate
with the surface of water.

00:01:18.430 --> 00:01:21.110
And there's two types of
surface waves, rally waves,

00:01:21.110 --> 00:01:21.770
and love waves.

00:01:21.770 --> 00:01:23.010
We won't go into
a lot of detail,

00:01:23.010 --> 00:01:24.509
but you can see
that rally waves are

00:01:24.509 --> 00:01:26.260
kind of the ground
moving up and down.

00:01:26.260 --> 00:01:27.980
Right here the
ground is moving up.

00:01:27.980 --> 00:01:29.090
Here it's moving down.

00:01:29.090 --> 00:01:30.080
Here it's moving up.

00:01:30.080 --> 00:01:31.840
Here it's moving down.

00:01:31.840 --> 00:01:34.210
So you can kind of view it
as kind of a ground roll.

00:01:34.210 --> 00:01:37.360
The love waves are
essentially the ground

00:01:37.360 --> 00:01:39.300
shifting left and right.

00:01:39.300 --> 00:01:40.870
So here it's not
moving up and down,

00:01:40.870 --> 00:01:43.470
but here it's moving,
if you're facing

00:01:43.470 --> 00:01:46.640
the direction of the wave
movement, to the left here.

00:01:46.640 --> 00:01:47.990
Here it's moving to the right.

00:01:47.990 --> 00:01:49.090
Here it's moving to the left.

00:01:49.090 --> 00:01:50.340
Here it's moving to the right.

00:01:50.340 --> 00:01:53.250
In both cases, the movement
of the surface wave

00:01:53.250 --> 00:01:56.080
is perpendicular to the
direction of motion.

00:01:56.080 --> 00:01:58.505
So we sometimes call
these transverse waves.

00:01:58.505 --> 00:02:00.130
And these are
essentially analogous to,

00:02:00.130 --> 00:02:02.680
as I said, kind of what
we see in water waves.

00:02:02.680 --> 00:02:05.460
Now, the more interesting
thing are the body waves,

00:02:05.460 --> 00:02:07.700
because the body waves,
first of all, they're

00:02:07.700 --> 00:02:09.210
the fastest moving waves.

00:02:09.210 --> 00:02:10.610
And these are also
the waves that

00:02:10.610 --> 00:02:13.910
are used to figure out the
structure of the Earth.

00:02:13.910 --> 00:02:16.560
So the body ways come
in two varieties.

00:02:16.560 --> 00:02:19.740
You have your P-waves,
or Primary waves.

00:02:22.520 --> 00:02:27.290
And you have your S-waves,
or Secondary waves.

00:02:27.290 --> 00:02:29.470
And they're depicted
right over here.

00:02:29.470 --> 00:02:32.200
And this is actually
energy that's

00:02:32.200 --> 00:02:34.120
being transferred
through a body.

00:02:34.120 --> 00:02:37.680
So it's not just moving
along the surface of one.

00:02:37.680 --> 00:02:39.130
And so here in
this diagram that I

00:02:39.130 --> 00:02:42.550
got from Wikipedia, which I
think Wikipedia got from the US

00:02:42.550 --> 00:02:46.750
Geological Survey, we have a
hammer being hit on some rock

00:02:46.750 --> 00:02:47.274
or whatever.

00:02:47.274 --> 00:02:49.190
And what you see is right
when the hammer gets

00:02:49.190 --> 00:02:52.480
hit at this end of the rock,
and I can zoom in a little bit--

00:02:52.480 --> 00:02:55.260
so let's say I have
this rock over here

00:02:55.260 --> 00:02:59.030
and I hit it right over here
with a hammer or something.

00:02:59.030 --> 00:03:01.060
What that's
immediately going to do

00:03:01.060 --> 00:03:04.240
is it's going to compress the
rock that the hammer comes

00:03:04.240 --> 00:03:05.220
in touch with.

00:03:05.220 --> 00:03:07.260
It's going to
compress that rock.

00:03:07.260 --> 00:03:10.990
But then that energy,
essentially the molecules

00:03:10.990 --> 00:03:13.400
are going to bump into
the adjacent molecules.

00:03:13.400 --> 00:03:14.982
And then those
adjacent molecules

00:03:14.982 --> 00:03:17.440
are then going to bump into
the molecules right next to it,

00:03:17.440 --> 00:03:18.970
and then they're going to
bump into the molecules

00:03:18.970 --> 00:03:19.761
right next to it.

00:03:19.761 --> 00:03:22.260
So you're going to have this
kind of compressed part of rock

00:03:22.260 --> 00:03:23.680
moving through the wave.

00:03:23.680 --> 00:03:25.590
So these are compressed,
and those molecules

00:03:25.590 --> 00:03:28.330
are going to go bump into
the adjacent molecules.

00:03:28.330 --> 00:03:31.980
So kind of immediately
after that the rock will be

00:03:31.980 --> 00:03:34.084
denser right over here.

00:03:34.084 --> 00:03:35.500
The first things
that were bumped,

00:03:35.500 --> 00:03:38.120
those will essentially bump
into the ones right above them,

00:03:38.120 --> 00:03:40.690
and then they will kind of
move back to where they were.

00:03:40.690 --> 00:03:42.670
And so now the compression will
have moved, and if you fast

00:03:42.670 --> 00:03:44.720
forward it will have moved
a little bit forward.

00:03:44.720 --> 00:03:46.360
So you essentially have
this compression wave.

00:03:46.360 --> 00:03:48.340
You hit the hammer here,
and you essentially

00:03:48.340 --> 00:03:51.550
have a changing
density that is moving

00:03:51.550 --> 00:03:53.800
in the same direction
of the wave.

00:03:53.800 --> 00:04:00.080
In this situation that is
the direction of the wave,

00:04:00.080 --> 00:04:01.620
and you see that
the molecules are

00:04:01.620 --> 00:04:04.840
kind of going back and
forth along that same axis.

00:04:04.840 --> 00:04:07.840
They're going along the
same direction as the wave.

00:04:07.840 --> 00:04:10.830
So those are P-waves.

00:04:10.830 --> 00:04:15.080
And P waves can
travel through air.

00:04:15.080 --> 00:04:17.279
Essentially sound waves
are compression waves.

00:04:17.279 --> 00:04:18.680
They can travel through liquid.

00:04:21.579 --> 00:04:24.710
And they can obviously
travel through solids.

00:04:24.710 --> 00:04:27.630
And, depending, in air
they'll travel the slowest.

00:04:27.630 --> 00:04:29.171
They'll essentially
essentially, move

00:04:29.171 --> 00:04:31.380
at the speed of sound, 330
metres per second, which

00:04:31.380 --> 00:04:34.100
isn't really slow by
every day human standards.

00:04:34.100 --> 00:04:37.480
In a liquid they'll move
about 1,500 meters per second.

00:04:37.480 --> 00:04:39.610
And then in granite,
which is most

00:04:39.610 --> 00:04:42.580
of the crustal
material of the Earth,

00:04:42.580 --> 00:04:45.102
they'll move at around
5,000 meters per second.

00:04:45.102 --> 00:04:46.060
Let me write that down.

00:04:46.060 --> 00:04:48.940
So 5,000 meters per
second, or essentially

00:04:48.940 --> 00:04:53.700
5 kilometers per second if
they're moving through granite.

00:04:53.700 --> 00:04:57.440
Now, S-waves are
essentially-- if you

00:04:57.440 --> 00:04:59.470
were to hit a hammer on
the side of this rock--

00:04:59.470 --> 00:05:04.840
so let me draw another diagram
since this is pretty small.

00:05:04.840 --> 00:05:07.920
If you were to hit a
hammer right over here what

00:05:07.920 --> 00:05:09.860
it would do is it would
temporarily kind of

00:05:09.860 --> 00:05:11.462
push all the rock over here.

00:05:11.462 --> 00:05:13.170
It would deform it a
little bit, and that

00:05:13.170 --> 00:05:15.660
would pull a little bit
of the rock back with it.

00:05:15.660 --> 00:05:18.200
And then this rock that's
right above it would slowly

00:05:18.200 --> 00:05:20.500
be pulled down, while this
rock that was initially hit

00:05:20.500 --> 00:05:21.620
will be moved back up.

00:05:21.620 --> 00:05:24.630
So you fast forward maybe
a millisecond, and now

00:05:24.630 --> 00:05:26.950
the next layer of
rock right above

00:05:26.950 --> 00:05:35.490
that will be kind of
deformed to the right.

00:05:35.490 --> 00:05:37.080
And if you keep
fast forwarding it

00:05:37.080 --> 00:05:39.490
the deformation
will move upwards.

00:05:39.490 --> 00:05:44.560
And notice, over here, once
again, the movement of the wave

00:05:44.560 --> 00:05:46.600
is upwards.

00:05:46.600 --> 00:05:48.530
But now the movement
of the material

00:05:48.530 --> 00:05:51.720
is not going along the same axis
that we saw with the P-waves,

00:05:51.720 --> 00:05:52.980
or the compression waves.

00:05:52.980 --> 00:05:54.750
It's now moving perpendicular.

00:05:54.750 --> 00:05:58.010
It's now moving along
a perpendicular axis,

00:05:58.010 --> 00:06:00.660
or you could call this
a transverse wave.

00:06:00.660 --> 00:06:02.710
The movement of the
particles is now

00:06:02.710 --> 00:06:06.860
on a perpendicular axis to the
actual movement of the waves.

00:06:06.860 --> 00:06:08.570
And so that's what an S-wave is.

00:06:08.570 --> 00:06:10.865
And they move a little bit
slower than the P-waves.

00:06:10.865 --> 00:06:12.740
So if an earthquake that
were to happen you'd

00:06:12.740 --> 00:06:14.650
see the P-waves first.

00:06:14.650 --> 00:06:17.870
And then at about 60% of
the speed of the P-waves

00:06:17.870 --> 00:06:19.460
you would see the S-waves.

00:06:19.460 --> 00:06:22.287
Now, the most important thing
to think about, especially

00:06:22.287 --> 00:06:23.870
from the point of
view of figuring out

00:06:23.870 --> 00:06:29.930
the composition of the Earth,
is that the S-waves can only

00:06:29.930 --> 00:06:31.455
travel through solid.

00:06:34.356 --> 00:06:36.480
And you might say, wait,
I've seen transverse waves

00:06:36.480 --> 00:06:38.050
on water that look like this.

00:06:38.050 --> 00:06:39.890
But remember, that
is a surface wave.

00:06:39.890 --> 00:06:41.224
We are talking about body waves.

00:06:41.224 --> 00:06:43.056
We're talking about
things that are actually

00:06:43.056 --> 00:06:44.570
going through the body of water.

00:06:44.570 --> 00:06:46.400
And one way to
think about this is

00:06:46.400 --> 00:06:50.280
if I had some water
over here-- so let's say

00:06:50.280 --> 00:06:51.500
that this is a pool.

00:06:51.500 --> 00:06:53.245
I'll draw a cross
section of water.

00:06:56.860 --> 00:06:59.880
I could have drawn
it better than that.

00:06:59.880 --> 00:07:04.932
If I have a cross section
of water right over here,

00:07:04.932 --> 00:07:06.640
let's think about it,
and hopefully it'll

00:07:06.640 --> 00:07:08.584
make intuitive sense to you.

00:07:08.584 --> 00:07:11.000
If I were to compress some of
the water, if I were to kind

00:07:11.000 --> 00:07:13.512
of slam some part of the
water here with like a big,

00:07:13.512 --> 00:07:15.220
I don't know, some
type of-- I would just

00:07:15.220 --> 00:07:17.950
compress it really fast.

00:07:17.950 --> 00:07:21.202
A P-wave could transmit, because
those water molecules would

00:07:21.202 --> 00:07:23.160
bump into the water
molecules next to it, which

00:07:23.160 --> 00:07:25.201
would bump into the water
molecules next to that.

00:07:25.201 --> 00:07:27.770
And so you would have a
compression wave, or a P-wave,

00:07:27.770 --> 00:07:29.426
moving in the
direction of my bump.

00:07:29.426 --> 00:07:31.550
So P-waves waves it makes
sense, and the same thing

00:07:31.550 --> 00:07:34.057
is true with air or sound
waves, that it make sense

00:07:34.057 --> 00:07:35.640
that it could travel
through a liquid.

00:07:38.601 --> 00:07:40.100
And remember, we're
under the water.

00:07:40.100 --> 00:07:41.260
We're not thinking
about the surface.

00:07:41.260 --> 00:07:43.676
We're thinking about moving
through the body of the water.

00:07:43.676 --> 00:07:46.620
Let's say that you
were to kind of take

00:07:46.620 --> 00:07:52.070
that hammer and kind of slapped
the side of this little volume

00:07:52.070 --> 00:07:52.750
of water here.

00:07:52.750 --> 00:07:54.570
Well, essentially
all that would do

00:07:54.570 --> 00:07:58.730
is it would send a compression
wave in that direction.

00:07:58.730 --> 00:08:00.350
It really wouldn't do anything.

00:08:00.350 --> 00:08:04.400
It wouldn't allow a transverse
wave to go that way,

00:08:04.400 --> 00:08:13.050
because the water doesn't have
this elastic property where

00:08:13.050 --> 00:08:14.830
if something bounces
that way it's

00:08:14.830 --> 00:08:16.760
going to immediately
bounce back that way.

00:08:16.760 --> 00:08:19.830
It's being pulled back
like a solid would.

00:08:19.830 --> 00:08:25.490
So S-waves only
travel through solids.

00:08:25.490 --> 00:08:26.940
So we're going to
use essentially

00:08:26.940 --> 00:08:29.255
our understanding of P-waves,
which travel through air,

00:08:29.255 --> 00:08:31.600
liquid, or solid, and our
understanding of S-waves

00:08:31.600 --> 00:08:37.071
to essentially figure out what
the composition of Earth is.