Einstein velocity addition formula derivation

WEBVTT
Kind: captions
Language: en

00:00:00.381 --> 00:00:01.944
- [Voiceover] Let's say this is me

00:00:01.944 --> 00:00:04.302
and I am floating in space.

00:00:04.302 --> 00:00:06.415
My coordinate system,
my frame of reference,

00:00:06.415 --> 00:00:07.268
we've seen it before,

00:00:07.268 --> 00:00:09.401
we'll call it the S frame of reference.

00:00:09.401 --> 00:00:12.198
Any point in spacetime,

00:00:12.198 --> 00:00:15.068
we give it an x and y coordinates.

00:00:15.068 --> 00:00:16.804
And let's say that we have my friend.

00:00:16.804 --> 00:00:21.193
We've involved her in
many other videos before.

00:00:21.193 --> 00:00:24.667
She is traveling with a relative velocity

00:00:24.667 --> 00:00:27.450
to me of v.

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So v right over there.

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Her frame of reference,

00:00:31.149 --> 00:00:33.119
we call it the S prime frame of reference,

00:00:33.119 --> 00:00:35.313
and you can denote any event

00:00:35.313 --> 00:00:37.224
in spacetime by the coordinates

00:00:37.224 --> 00:00:40.979
x prime and

00:00:40.979 --> 00:00:43.520
ct prime.

00:00:43.520 --> 00:00:45.734
Now you could also do it
as t prime if you like,

00:00:45.734 --> 00:00:47.480
but we've been doing it as ct prime

00:00:47.480 --> 00:00:49.716
so that we have similar units.

00:00:49.716 --> 00:00:52.947
Now let's say that we have
a third character now.

00:00:52.947 --> 00:00:54.572
This is going to get interesting.

00:00:54.572 --> 00:00:57.416
Let's say this third
character is traveling

00:00:57.416 --> 00:01:01.053
with a velocity u in
my frame of reference.

00:01:01.053 --> 00:01:03.952
So u is equal to change in x

00:01:03.952 --> 00:01:07.666
over change in time.

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If we know all of this information,

00:01:09.571 --> 00:01:11.683
let's see if we can come
up, we can formulate a way

00:01:11.683 --> 00:01:12.800
if we know what the change in x

00:01:12.800 --> 00:01:16.193
and change in time is
and we know what v is,

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if we can figure out what
is this velocity going to be

00:01:18.632 --> 00:01:20.826
in the S prime frame of reference,

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in this purple friend's
frame of reference.

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What we want to figure out is what is,

00:01:26.365 --> 00:01:28.539
let me do it in that purple color,

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we wanna figure out what
is going to be the change

00:01:30.752 --> 00:01:35.384
in x prime over change in t prime.

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If we figure this out then we know

00:01:36.927 --> 00:01:39.669
what will this velocity
look like in the S prime

00:01:39.669 --> 00:01:41.701
frame of reference.

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Well let's just go back to
the Lorentz Transformations.

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Let's first think about what our...

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Let's first think about

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what change in x prime is going to be.

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Well our change in x prime is going to be

00:01:54.589 --> 00:01:58.895
the Lorentz factor times our change in x,

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change, this in white,
times our change in x

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minus beta

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times change in ct.

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Let me close those parentheses.

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Then we wanna divide that
by our change in time.

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We wanna divide it by our change in time.

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So change in time

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or change in t prime I should say.

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Well let me just go back to
the Lorentz Transformation.

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Let me write it the way
that I'm used to writing it.

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I'm used to writing it as

00:02:34.333 --> 00:02:39.067
change in ct prime, which
is the same thing as

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c times the change in t prime,

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is equal to the Lorentz factor

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times

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this is going to be change in ct

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or I could write it as c change in t

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minus beta

00:03:00.398 --> 00:03:04.971
times change in x.

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Once again I could have
written this as change in ct

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or I could write this
as c times change in t

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because the c isn't changing.

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So we know that all ready.

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If we wanted to solve for
just change in t prime,

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we could just divide both sides by c.

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So let's do that.

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If we divide all of this by c,

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what do we get?

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This is a form, we've
seen this in other videos

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that you might recognize
and you might see this

00:03:32.913 --> 00:03:34.234
in some textbooks.

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Our change in t prime is
going to be equal to gamma

00:03:38.805 --> 00:03:41.954
times, well c times delta t divided by c

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is just going to be delta t.

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And then if you take, so
minus, beta divided by c.

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Well let's just remind ourselves

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that beta is equal to v over c.

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So beta divided by c is

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v over c squared.

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I could write this as
minus v over c squared

00:04:05.468 --> 00:04:09.851
change in x.

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Change in t prime, let's write that now.

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That's going to be equal to

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you have your gamma

00:04:19.471 --> 00:04:22.214
times change in time

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in my coordinate system.

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So not the t prime just change in t

00:04:28.013 --> 00:04:32.971
minus v over c squared change in x.

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Immediately there's at
least one simplification

00:04:35.328 --> 00:04:36.364
we can do.

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We can divide both the numerator

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and the denominator by gamma.

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I keep wanting to change...

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So we can divide the numerator
and denominator by gamma

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that simplifies it a little bit.

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We can write this as our change in x prime

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over change in t prime

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is equal to

00:04:56.816 --> 00:04:59.138
in the numerator we have change in x

00:04:59.138 --> 00:05:02.206
and actually let me just
write out what beta is.

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Beta, I'll do it over here,

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minus v over c that's what beta is

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And actually let me take the c out.

00:05:09.951 --> 00:05:13.487
Change in ct is the same
thing as c times change in t.

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So times c delta t.

00:05:17.490 --> 00:05:21.776
v divided by c times c
is just going to be v.

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Let me write it that way.

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This part, these cancel out.

00:05:24.971 --> 00:05:28.628
This is the same thing as
change in x minus v change in t.

00:05:28.628 --> 00:05:31.085
Let me write it that way, simplfy it.

00:05:31.085 --> 00:05:35.616
Minus v change in t, that's our numerator.

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And then our denominator is change in t

00:05:39.558 --> 00:05:43.784
minus v over c squared change in x.

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It looks like we're getting close here.

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But we don't have the
change in x change in t.

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We have how much of a change in x

00:05:51.850 --> 00:05:53.801
we have for a given change in t.

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So what we could do is we
could divide the numerator

00:05:55.650 --> 00:05:59.262
and the denominator by delta t.

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We could multiply the
numerator by one over delta t.

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And the denominator by one over delta t

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which is equivalent to dividing

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the numerator and the
denominator each by delta t.

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And why am I doing that?

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Well, if I do that this
first term right over here

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is going to be delta x over delta t

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which we know.

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Then we have minus v, cuz
delta t divided by delta t

00:06:21.894 --> 00:06:23.460
is just going to be one.

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And then our denominator

00:06:25.126 --> 00:06:27.930
delta t divided by delta t is just one.

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Then we're gonna have minus v

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over c squared

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and then times delta x divided by delta t.

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This is cool.

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We've just been able to
figure out what our velocity

00:06:44.142 --> 00:06:46.032
in the primed coordinate system,

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in the S prime frame of
reference is in terms of

00:06:49.080 --> 00:06:52.350
the relative velocity between
the S prime frame of reference

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and my frame of reference.

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Delta x delta t the velocity
in my frame of reference.

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And well we know all of
this other stuff too,

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this is just delta x
delta t and this is v.

00:07:01.918 --> 00:07:03.035
Or we can even write it like this.

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We said delta x over delta
t is going to be equal to u.

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So it's equal to u minus v

00:07:10.281 --> 00:07:13.281
over one minus,

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delta x over delta t is just u.

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So this is just u here.

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It's u times v

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or v times u,

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all of that over c squared.

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This is a really, really,
really useful deriviation

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which we're going to apply some numbers to

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in the next video so we can appreciate

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kinda how fun this is on some level.


Applying Einstein velocity addition

WEBVTT
Kind: captions
Language: en

00:00:00.132 --> 00:00:02.052
- [Voiceover] Now let's apply the formula

00:00:02.052 --> 00:00:03.400
we came up with the last video

00:00:03.400 --> 00:00:06.929
sometimes known as the Einstein
velocity addition formula.

00:00:06.929 --> 00:00:09.530
And we'll see that it's
a pretty neat thing.

00:00:09.530 --> 00:00:12.034
So let's say, so this is once
again me floating in space,

00:00:12.034 --> 00:00:14.953
my frame of reference is just
the s frame of reference,

00:00:14.953 --> 00:00:16.392
Let's say my friend over here,

00:00:16.392 --> 00:00:18.343
she's flying away from me, her velocity,

00:00:18.343 --> 00:00:19.620
let's give it some numbers.

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Let's say she's flying away from me

00:00:21.176 --> 00:00:23.590
with a velocity of, let's say,

00:00:23.590 --> 00:00:27.770
0.7c, so seven tenths
of the speed of light

00:00:27.770 --> 00:00:29.860
or 70% of the speed of light.

00:00:29.860 --> 00:00:32.646
And let's say this third
character right over here,

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let's say they're flying towards me,

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so let me change the
direction, the velocity vector.

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So let me erase that.

00:00:40.517 --> 00:00:41.933
Gee, that doesn't look like a pure black

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but it looks fairly
erased, so there you go.

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And let me give the velocity
vector in the other direction.

00:00:49.410 --> 00:00:52.538
So the velocity vector,

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this character is flying towards me

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at let's say, so u is
going to be equal to,

00:00:57.490 --> 00:00:59.069
so it's flying towards me

00:00:59.069 --> 00:01:01.701
x is decreasing with time,

00:01:01.701 --> 00:01:03.851
let's say it's negative,

00:01:03.851 --> 00:01:06.871
- 0.5c,

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so flying towards me at
half the speed of light.

00:01:09.331 --> 00:01:11.514
And once again, both of these velocities

00:01:11.514 --> 00:01:14.417
are given in my frame of reference.

00:01:14.417 --> 00:01:16.785
Now if we were in a Newtonian world

00:01:16.785 --> 00:01:19.246
or if we used the
Galilean transformations,

00:01:19.246 --> 00:01:21.568
and this is released as
something on your highway.

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If I'm on a highway and if
I'm going in one direction

00:01:23.890 --> 00:01:26.862
at 70 miles an hour, and
someone is coming towards me

00:01:26.862 --> 00:01:30.554
at 50 miles an hour, it'll look to me

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like they are coming towards
me at 120 miles per hour.

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We would essentially
add these two together.

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And so, if we were in a Newtonian world,

00:01:37.914 --> 00:01:40.608
this person is flying at 0.7c

00:01:40.608 --> 00:01:44.207
and this person is
flying this way at 0.5c,

00:01:44.207 --> 00:01:46.250
you would, in a Newtonian world, say okay

00:01:46.250 --> 00:01:48.456
from this point of view it will look like

00:01:48.456 --> 00:01:50.476
this blue character is approaching

00:01:50.476 --> 00:01:54.041
at 0.7c plus 0.5c

00:01:54.041 --> 00:01:56.061
or at 1.2 times the speed of light

00:01:56.061 --> 00:01:59.415
which we know violates
the laws of the universe.

00:01:59.415 --> 00:02:01.876
And that's where this
formula is really handy.

00:02:01.876 --> 00:02:03.827
Actually let me even
make the spaceship point

00:02:03.827 --> 00:02:05.847
in this direction just
so we don't get confused.

00:02:05.847 --> 00:02:09.421
So the spaceship is going

00:02:09.421 --> 00:02:11.661
in that direction just like that.

00:02:11.661 --> 00:02:13.718
And once again, the
magnitude of the velocity

00:02:13.718 --> 00:02:15.924
coming towards me is
half the speed of light.

00:02:15.924 --> 00:02:17.224
I put the negative there to show

00:02:17.224 --> 00:02:18.339
that it's coming towards me,

00:02:18.339 --> 00:02:22.100
that x is decreasing over time.

00:02:22.100 --> 00:02:25.513
Well, lucky for us in the
last video, we came up with

00:02:25.513 --> 00:02:29.391
this Einstein velocity addition
formula, so let's apply it.

00:02:29.391 --> 00:02:31.527
From this friend's frame of reference,

00:02:31.527 --> 00:02:33.710
the s-prime frame of reference,

00:02:33.710 --> 00:02:34.917
the velocity of this character

00:02:34.917 --> 00:02:37.216
which is change in
x-prime over change in t

00:02:37.216 --> 00:02:38.841
is going to be this.

00:02:38.841 --> 00:02:40.095
So what do we have here?

00:02:40.095 --> 00:02:44.344
So u is the blue velocity
in my frame of reference,

00:02:44.344 --> 00:02:48.918
So that is -0.5c.

00:02:48.918 --> 00:02:51.472
v is my friend's frame of reference,

00:02:51.472 --> 00:02:52.401
the frame of reference

00:02:52.401 --> 00:02:55.210
that we're trying to get the velocity in.

00:02:55.210 --> 00:02:57.945
So that's 0.7c,

00:02:57.945 --> 00:03:01.105
so let me do it in that color.

00:03:01.105 --> 00:03:05.055
So that is 0.7c.

00:03:05.055 --> 00:03:08.561
And then, on top of uv over c-squared,

00:03:08.561 --> 00:03:11.209
so once again, u is

00:03:11.209 --> 00:03:13.759
- 0.5c.

00:03:13.759 --> 00:03:15.782
I know I wrote it really small.

00:03:15.782 --> 00:03:19.854
And then, v is

00:03:19.854 --> 00:03:23.454
0.7c.

00:03:23.454 --> 00:03:24.954
So what is this going to be equal to?

00:03:24.954 --> 00:03:27.414
Well, the numerator -0.5c

00:03:27.414 --> 00:03:30.194
minus 0.7c,

00:03:30.194 --> 00:03:32.222
that's going to be negative.

00:03:32.222 --> 00:03:34.543
Our numerator, right over here

00:03:34.543 --> 00:03:36.706
is going to be

00:03:36.706 --> 00:03:40.686
- 1.2c.

00:03:40.686 --> 00:03:42.902
And this is the velocity
that you would expect

00:03:42.902 --> 00:03:45.642
if we were dealing in a Newtonian world.

00:03:45.642 --> 00:03:47.360
This person says, "Hey, this person looks

00:03:47.360 --> 00:03:48.498
"like they're coming towards me

00:03:48.498 --> 00:03:50.402
"at 1.2 times the speed of light."

00:03:50.402 --> 00:03:53.560
That's what our Galilean
transformations would give us.

00:03:53.560 --> 00:03:56.184
But luckily, we have all
this business at the bottom

00:03:56.184 --> 00:04:00.131
that keeps us from
violating the absoluteness

00:04:00.131 --> 00:04:01.965
of the velocity of light,

00:04:01.965 --> 00:04:04.519
this notion that nothing can travel faster

00:04:04.519 --> 00:04:06.353
than the speed of light.

00:04:06.353 --> 00:04:07.723
Because in our denominator,

00:04:07.723 --> 00:04:12.250
we're going to get 1 minus 0.5c

00:04:12.250 --> 00:04:15.100
times 0.7c.

00:04:15.100 --> 00:04:17.732
Let's see, 0.5 times 0.7

00:04:17.732 --> 00:04:20.712
is going to be 0.3.

00:04:20.712 --> 00:04:24.209
So 0.3, I could say 0.30 if I liked,

00:04:24.209 --> 00:04:28.551
but I'll just write 0.3c squared.

00:04:28.551 --> 00:04:29.619
It'll be a negative.

00:04:29.619 --> 00:04:31.384
I have a negative times a positive.

00:04:31.384 --> 00:04:32.591
So I could put the negative there

00:04:32.591 --> 00:04:34.472
but if I'm subtracting a negative,

00:04:34.472 --> 00:04:36.051
that's just going to be a positive.

00:04:36.051 --> 00:04:38.396
And I'm just going to divide by c-squared.

00:04:38.396 --> 00:04:40.369
And so, that cancels out.

00:04:40.369 --> 00:04:42.622
And so notice, we're going to be dividing

00:04:42.622 --> 00:04:47.149
- 1.2c by something that's
slightly larger than 1.2.

00:04:47.149 --> 00:04:49.224
So this is going to be

00:04:49.224 --> 00:04:52.584
- 1.2c

00:04:52.584 --> 00:04:55.524
over 1.3.

00:04:55.524 --> 00:04:57.434
1.3

00:04:57.434 --> 00:05:01.266
And so, lucky for us, this
is going to be less than c,

00:05:01.266 --> 00:05:03.704
or this absolute value is
going to be less than c.

00:05:03.704 --> 00:05:07.837
So it's going to be, let's
get a calculator out.

00:05:07.837 --> 00:05:11.641
So, if we have

00:05:11.641 --> 00:05:16.441
1.2 divided by 1.3,

00:05:16.441 --> 00:05:19.981
this is equal to approximately 0.92.

00:05:19.981 --> 00:05:23.426
So this is going to be approximately

00:05:23.426 --> 00:05:27.556
- 0.92c

00:05:27.556 --> 00:05:29.245
which is cool.

00:05:29.245 --> 00:05:31.196
It kind of goes with our,

00:05:31.196 --> 00:05:32.635
let me write it over here.

00:05:32.635 --> 00:05:36.545
So this delta x prime over delta t prime

00:05:36.545 --> 00:05:40.705
is -0.92c.

00:05:40.705 --> 00:05:43.943
So it makes sense that from our
friend's frame of reference,

00:05:43.943 --> 00:05:46.567
this ship will look like
it's approaching her

00:05:46.567 --> 00:05:49.005
faster than from my frame of reference.

00:05:49.005 --> 00:05:51.106
It looks like it's approaching
me in my frame of reference,

00:05:51.106 --> 00:05:53.521
it looks like it's coming to
me at half the speed of light.

00:05:53.521 --> 00:05:54.589
In her frame of reference,

00:05:54.589 --> 00:05:58.513
it looks like it's coming at
0.92 times the speed of light.

00:05:58.513 --> 00:06:00.788
The negative is just
specifying the direction.

00:06:00.788 --> 00:06:03.760
But we didn't violate
the fundamental postulate

00:06:03.760 --> 00:06:05.943
of special relativity,
that nothing can go faster

00:06:05.943 --> 00:06:07.220
than the speed of light.

00:06:07.220 --> 00:06:09.217
The speed of light is absolute.

00:06:09.217 --> 00:06:10.215
So this is pretty cool.

00:06:10.215 --> 00:06:12.444
I've often even thought about
making a video game somehow

00:06:12.444 --> 00:06:13.814
where you leverage this,

00:06:13.814 --> 00:06:17.460
where things are flying in
different relative velocities

00:06:17.460 --> 00:06:22.084
but special relativity applies.


Lorentz transformation for change in coordinates

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.468
- We've spent several
videos now getting familiar

00:00:02.468 --> 00:00:04.488
with the Lorentz transformations.

00:00:04.488 --> 00:00:06.828
What I want to do now, instead
of thinking of what X prime

00:00:06.828 --> 00:00:09.108
and C T prime is in terms of X and C T,

00:00:09.108 --> 00:00:11.748
I wanna think about, what
is the change in X prime

00:00:11.748 --> 00:00:14.228
and the change in C T prime
going to be in terms of

00:00:14.228 --> 00:00:16.188
change in X and change in C T.

00:00:16.188 --> 00:00:18.188
And we'll see it's just
going to involve some

00:00:18.188 --> 00:00:21.448
fairly straight-forward
algebraic manipulation.

00:00:21.448 --> 00:00:22.988
So, let's think about it.

00:00:22.988 --> 00:00:25.968
Change in X prime is going to be

00:00:25.968 --> 00:00:29.118
X prime final minus

00:00:29.118 --> 00:00:33.100
X prime initial.

00:00:33.100 --> 00:00:35.110
Well, X prime final,

00:00:35.110 --> 00:00:36.810
let me just pick a
suitable color for that,

00:00:36.810 --> 00:00:39.170
X prime final is going to be

00:00:39.170 --> 00:00:44.170
gamma times X final

00:00:44.307 --> 00:00:49.293
minus beta times C T final.

00:00:49.293 --> 00:00:51.623
All I did is use this formula up here.

00:00:51.623 --> 00:00:53.743
If I want to figure out my final X prime,

00:00:53.743 --> 00:00:57.243
I'm just going to think of
my final X and my final C T.

00:00:57.243 --> 00:00:59.483
So, that's that.

00:00:59.483 --> 00:01:04.163
And from that, I'm going to
subtract the initial X prime.

00:01:04.163 --> 00:01:07.943
Well, X prime initial is just going to be,

00:01:07.943 --> 00:01:09.601
let me get another color here,

00:01:09.601 --> 00:01:14.301
Lorentz factor gamma times X initial

00:01:14.301 --> 00:01:18.941
minus beta times C T initial.

00:01:18.941 --> 00:01:22.901
So, now, let's see, we
can factor out the gamma,

00:01:22.901 --> 00:01:24.861
so this is going to be equal to,

00:01:24.861 --> 00:01:26.561
and I'll do it in my color for gamma.

00:01:26.561 --> 00:01:28.401
If we factor out the
gamma, we're gonna get

00:01:28.401 --> 00:01:32.919
gamma times, we're gonna have X final,

00:01:32.919 --> 00:01:35.059
let me do this in a,

00:01:35.059 --> 00:01:38.299
so we have, do that in white actually.

00:01:38.299 --> 00:01:43.259
We're gonna have X final,
and then if we distribute

00:01:43.259 --> 00:01:47.567
this negative sign, minus X initial,

00:01:47.567 --> 00:01:49.647
and then, let's see,

00:01:49.647 --> 00:01:53.507
if we distribute this negative
sign, well, I don't want

00:01:53.507 --> 00:01:56.087
to skip too many steps
here, so that's that.

00:01:56.087 --> 00:01:58.586
And then we're going to have

00:01:58.586 --> 00:02:03.076
negative beta C T final,

00:02:03.076 --> 00:02:06.941
negative beta C T final,

00:02:06.941 --> 00:02:10.621
and then we have plus, we
distributed this negative,

00:02:10.621 --> 00:02:12.874
plus beta C T initial

00:02:12.874 --> 00:02:17.193
plus beta C T initial

00:02:18.185 --> 00:02:19.580
and so what can we do here,

00:02:19.580 --> 00:02:24.310
well this, that's just
going to be change in X.

00:02:27.130 --> 00:02:29.159
So I can rewrite this as being equal to

00:02:29.159 --> 00:02:34.049
gamma times change in X...

00:02:40.709 --> 00:02:43.860
Let me factor out a negative beta.

00:02:43.860 --> 00:02:47.958
So I'l say, minus beta times,

00:02:50.698 --> 00:02:55.447
well then you're going to have C T final

00:02:57.957 --> 00:03:01.186
minus C T initial.

00:03:01.572 --> 00:03:04.092
And what's C T final minus C T initial?

00:03:04.092 --> 00:03:06.952
I think I'm skipping
too many steps already.

00:03:06.952 --> 00:03:09.532
Well that's just going
to be change in C T.

00:03:09.532 --> 00:03:13.053
So we get, this is all
going to be equal to

00:03:13.053 --> 00:03:16.432
gamma, our Lorentz
factor, times change in X

00:03:20.232 --> 00:03:24.680
minus beta times change in C T.

00:03:28.590 --> 00:03:29.990
And since C isn't changing,
it could also be viewed

00:03:29.990 --> 00:03:32.510
as C times change in T, either way.

00:03:32.510 --> 00:03:34.610
So, there you have it.

00:03:34.610 --> 00:03:36.750
Notice, it takes almost
the exact same form.

00:03:36.750 --> 00:03:40.590
X prime is equal to gamma
times X minus beta C T

00:03:40.590 --> 00:03:43.910
and change in X prime is going to be gamma

00:03:43.910 --> 00:03:48.890
times change in X minus
beta time change in C T.

00:03:48.890 --> 00:03:50.831
And, I'm not going to do it in this video,

00:03:50.831 --> 00:03:52.558
but you can make the exact same algebraic

00:03:52.558 --> 00:03:55.239
argument for your change in C T prime,

00:03:55.239 --> 00:03:57.519
as you'll see and I encourage
you to do this on your own,

00:03:57.519 --> 00:04:02.498
change in C T prime, which
you could also view as,

00:04:02.498 --> 00:04:06.999
since C isn't changing,
C times delta T prime,

00:04:06.999 --> 00:04:09.698
these are equivalent,
is going to be equal to

00:04:13.898 --> 00:04:18.898
gamma times change in C T

00:04:20.448 --> 00:04:25.004
minus beta times change in X,

00:04:25.440 --> 00:04:26.280
and I encourage you,

00:04:26.280 --> 00:04:28.100
right after this video,
actually do this one too.

00:04:28.100 --> 00:04:31.940
Delta X prime is going to be X prime final

00:04:31.940 --> 00:04:34.000
minus X prime initial and
then do what I just did here,

00:04:34.000 --> 00:04:35.300
just a little algebraic manipulation.

00:04:35.300 --> 00:04:37.960
You can make the same
exact argument over here

00:04:37.960 --> 00:04:41.220
to get to the result
that I just wrote down.

00:04:41.220 --> 00:04:43.960
You see, well our change
in C T prime is going to be

00:04:43.960 --> 00:04:47.500
C T prime final minus C T
prime initial and then you can

00:04:47.500 --> 00:04:49.200
substitute with this, do
a little bit of algebraic

00:04:49.200 --> 00:04:51.940
manipulation and you'll
get that right over there.

00:04:51.940 --> 00:04:53.360
And the whole reason I'm doing this is,

00:04:53.360 --> 00:04:54.760
well now, we can think in terms of change

00:04:54.760 --> 00:04:56.360
in the coordinates, which will allow us to

00:04:56.360 --> 00:04:57.920
think about what velocities would be

00:04:57.920 --> 00:05:00.500
in the different frames of reference,

00:05:00.500 --> 00:05:05.087
which is going to be pretty neat.


Origins of the Cold War

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:01.803
- [Voiceover] Hi, Dr. Kutz.

00:00:01.803 --> 00:00:02.879
- [Voiceover] Hello,
David. How are you doing?

00:00:02.879 --> 00:00:05.189
- [Voiceover] I'm doing
well. I'm excited to learn

00:00:05.209 --> 00:00:08.034
about this thing we call "the Cold War."

00:00:08.034 --> 00:00:09.797
What is a "cold war" and what makes it

00:00:09.797 --> 00:00:11.067
different than a "hot war"?

00:00:11.067 --> 00:00:14.142
- [Voiceover] So, a cold
war, and in this case

00:00:14.142 --> 00:00:15.983
it's really,

00:00:16.460 --> 00:00:19.087
it might be a term that we could debate.

00:00:19.867 --> 00:00:23.941
is a war where the two major combatants

00:00:23.941 --> 00:00:27.428
never actually fire bullets at each other

00:00:27.428 --> 00:00:28.977
or drop bombs on each other.

00:00:28.977 --> 00:00:32.935
So never in the course
of the Cold War did the

00:00:32.935 --> 00:00:37.753
US ever meet the ground
troops with the USSR.

00:00:37.753 --> 00:00:40.311
- [Voiceover] But people
still died in combat.

00:00:40.311 --> 00:00:41.383
- [Voiceover] Right.

00:00:41.383 --> 00:00:43.628
- [Voiceover] So the Cold
War is kind of fought

00:00:43.628 --> 00:00:46.638
through proxy wars. And
these are wars that are

00:00:46.638 --> 00:00:49.488
taking place in other
nations, developing nations

00:00:49.488 --> 00:00:53.218
of the world where the US
is supporting one side,

00:00:53.218 --> 00:00:56.142
generally the pro-capitalist side,

00:00:56.142 --> 00:00:57.922
and the Soviet Union is supporting

00:00:57.922 --> 00:01:00.182
the other side, a communist side.

00:01:00.182 --> 00:01:03.310
So this is the case in the Korean War

00:01:03.310 --> 00:01:05.470
in the 1950s, and then definitely

00:01:05.470 --> 00:01:08.450
the case in the Vietnam War in the 1960s.

00:01:08.460 --> 00:01:11.702
- [Voiceover] So the US and
the USSR aren't fighting

00:01:11.702 --> 00:01:15.055
directly but they're kind of
betting on boxing matches,

00:01:15.065 --> 00:01:17.469
betting on different fighters
in the same boxing match.

00:01:17.489 --> 00:01:20.049
- [Voiceover] Yeah, but
they're not just betting,

00:01:20.049 --> 00:01:23.113
they're also putting money and arms

00:01:23.113 --> 00:01:24.422
where their mouth is.

00:01:24.422 --> 00:01:26.899
- [Voiceover] So now if
there are two different

00:01:26.899 --> 00:01:29.508
fighters in the ring, the US is given

00:01:29.508 --> 00:01:31.479
the capitalist fighter...

00:01:31.479 --> 00:01:32.399
feeding him...

00:01:32.399 --> 00:01:34.187
- [Voiceover] So you've
got the capitalist fighter

00:01:34.187 --> 00:01:37.162
in one corner and he's
sweaty and he's beaten,

00:01:37.162 --> 00:01:39.322
but the US is behind
him with a towel and one

00:01:39.322 --> 00:01:41.526
of those water buckets,
splashing water in his face,

00:01:41.526 --> 00:01:42.887
like "Get in this fight! Get in there!'

00:01:42.887 --> 00:01:45.987
- [Voiceover] Right. And if necessary,

00:01:45.987 --> 00:01:47.897
tying up his boxing gloves,

00:01:47.897 --> 00:01:49.225
maybe giving him a new pair of shoes.

00:01:49.225 --> 00:01:50.545
- [Voiceover] Sure.

00:01:50.545 --> 00:01:52.477
- [Voiceover] Doing whatever they can...
- [Voiceover] Paying his rent...

00:01:53.477 --> 00:01:56.350
- [Voiceover] Right. Doing everything
that they can... - [Voiceover]
Buying him meals, probably,

00:01:56.350 --> 00:01:58.017
like at the Marshall Plan.

00:01:58.017 --> 00:01:58.366
- [Voiceover] Exactly!

00:01:58.366 --> 00:01:59.967
- [Voiceover] Okay, so
tell me about these two

00:01:59.967 --> 00:02:01.283
combatants. In this corner...

00:02:01.283 --> 00:02:02.782
- [Voiceover] Laughs.

00:02:02.782 --> 00:02:05.240
- [Voiceover] Wearing a suit...

00:02:05.240 --> 00:02:07.519
- [Voiceover] (laughs) Is Harry Truman.

00:02:07.519 --> 00:02:09.749
And Harry Truman is the President of the

00:02:09.749 --> 00:02:12.677
United States, starting in 1945.

00:02:12.677 --> 00:02:16.141
He was Vice President
to Franklin Roosevelt,

00:02:16.141 --> 00:02:18.969
who had been the US's
president since 1932,

00:02:20.658 --> 00:02:23.292
and who tragically died in 1945.

00:02:25.112 --> 00:02:27.982
So Truman is really in charge of ending

00:02:27.992 --> 00:02:30.788
World War II for the
United States and also

00:02:30.788 --> 00:02:32.472
kind of setting up post-war plans.

00:02:32.472 --> 00:02:34.736
- [Voiceover] So he
prosecutes the end of the war.

00:02:34.736 --> 00:02:36.839
He makes the decision
to drop the atom bomb

00:02:36.839 --> 00:02:39.147
on Nagasaki and Hiroshima,

00:02:39.147 --> 00:02:41.954
and he ends the war in both theaters.

00:02:41.954 --> 00:02:43.290
- [Voiceover] Right.

00:02:43.290 --> 00:02:44.890
- [Voiceover] Okay.

00:02:44.890 --> 00:02:46.145
- [Voiceover] And...

00:02:46.145 --> 00:02:48.670
- [Voiceover] And in the
other corner, wearing a

00:02:48.670 --> 00:02:50.110
very fine mustache...

00:02:50.110 --> 00:02:52.159
- [Voiceover] (laughs) Is Joseph Stalin.

00:02:52.159 --> 00:02:54.130
And he's the Soviet premier.

00:02:54.130 --> 00:02:57.542
He's been in charge since the 1920s.

00:02:59.031 --> 00:03:02.613
And for him, I would
say the most important

00:03:02.613 --> 00:03:05.873
thing that Stalin wanted
after World War II

00:03:05.873 --> 00:03:07.846
was security, shall we say.

00:03:07.846 --> 00:03:10.153
- [Voiceover] Okay.

00:03:10.153 --> 00:03:13.401
- [Voiceover] So if you
remember your history,

00:03:13.401 --> 00:03:15.727
in World War I

00:03:15.727 --> 00:03:18.187
Germany invaded Russia.

00:03:18.187 --> 00:03:20.259
In World War II

00:03:20.259 --> 00:03:21.259
Germany invaded Russia.

00:03:21.259 --> 00:03:22.301
- [Voiceover] Oh, I'm seeing a pattern.

00:03:22.301 --> 00:03:25.984
- [Voiceover] And if there's
anything that Stalin wants

00:03:25.984 --> 00:03:27.724
in the post-1945 era,

00:03:27.724 --> 00:03:30.457
it is not to be invaded
by Germany anymore.

00:03:30.457 --> 00:03:31.785
- [Voiceover] Sure.

00:03:31.785 --> 00:03:35.460
- [Voiceover] So he is
very anxious to make sure

00:03:35.460 --> 00:03:38.676
that the world is safe for communism.

00:03:40.767 --> 00:03:43.634
He thinks that

00:03:43.634 --> 00:03:46.629
the best way to make sure that

00:03:46.629 --> 00:03:49.886
Russians can continue
the experiments and the

00:03:49.886 --> 00:03:52.183
revolution of communism

00:03:52.183 --> 00:03:56.150
is to have a buffer zone,

00:03:56.150 --> 00:03:59.960
shall we say, between Russia
and the rest of Europe.

00:03:59.960 --> 00:04:01.482
- [Voiceover] Okay.

00:04:01.482 --> 00:04:04.650
- [Voiceover] And if he does
that by kind of shoring up

00:04:04.650 --> 00:04:07.965
some puppet governments,
in what we now call the

00:04:07.965 --> 00:04:09.779
Eastern Block,

00:04:09.779 --> 00:04:12.619
these nations that had been

00:04:12.619 --> 00:04:14.792
taken over by Hitler

00:04:14.792 --> 00:04:17.434
and then when the Soviet Union joined the

00:04:17.434 --> 00:04:19.444
war on the side of the Allies,

00:04:19.444 --> 00:04:21.474
then were retaken over

00:04:21.474 --> 00:04:23.322
by Russia as they fought Hitler back.

00:04:23.322 --> 00:04:26.174
- [Voiceover] Okay. So a lot
of those central European

00:04:26.174 --> 00:04:30.477
countries like Hungary, and
Lithuania, and the former

00:04:30.477 --> 00:04:33.320
Czechoslovakia, and the former Yugoslavia.

00:04:33.320 --> 00:04:35.900
- [Voiceover] And he had
the advantage of having

00:04:35.900 --> 00:04:38.932
boots on the ground
there because he'd beaten

00:04:38.942 --> 00:04:41.777
back Hitler's invasion,
eventually White Russian

00:04:41.777 --> 00:04:45.173
troops were fighting
against Hitler, and American

00:04:45.173 --> 00:04:47.687
troops who were fighting against Hitler.

00:04:47.687 --> 00:04:49.581
You know, they meet in Berlin

00:04:49.581 --> 00:04:51.781
at the fall of Hitler,

00:04:51.781 --> 00:04:54.004
and kind of shake hands in Berlin.

00:04:54.004 --> 00:04:56.234
But the advantage that Russia had was

00:04:56.234 --> 00:04:59.352
that they've got a lot more people here.

00:04:59.352 --> 00:05:03.885
They've got most of Europe now,

00:05:05.045 --> 00:05:08.835
at least east of Berlin,

00:05:09.226 --> 00:05:11.307
has Soviet troops on the ground.

00:05:11.307 --> 00:05:13.714
- [Voiceover] Sure. So
you've been telling me

00:05:13.714 --> 00:05:16.651
that the Soviet Union did yeoman's work

00:05:16.651 --> 00:05:20.581
in containing and basically
prosecuting the entire

00:05:20.581 --> 00:05:22.238
Eastern Front during World War II.

00:05:22.238 --> 00:05:24.746
- [Voiceover] Yeah, and
the Soviet Union actually

00:05:24.746 --> 00:05:28.097
lost 20 million people
during World War II.

00:05:28.107 --> 00:05:32.104
That's just a ludicrous number.

00:05:32.104 --> 00:05:34.901
They lost more than anyone except

00:05:34.901 --> 00:05:37.235
for China and Germany.

00:05:37.235 --> 00:05:39.914
So they feel like they have a real stake

00:05:39.914 --> 00:05:41.763
in the outcome of World War II.

00:05:41.763 --> 00:05:45.549
- [Voiceover] So, at the
end of this, what is the

00:05:45.549 --> 00:05:47.492
situation in the USSR?

00:05:47.492 --> 00:05:49.642
Like they've conquered
all of this territory

00:05:49.642 --> 00:05:51.748
but are they strong enough

00:05:51.748 --> 00:05:54.110
economically to hold all of it

00:05:54.110 --> 00:05:55.110
and feed everyone?

00:05:55.110 --> 00:05:57.215
- [Voiceover] No, not really.

00:05:57.215 --> 00:06:01.432
In fact, most of Europe
is in pretty dire straits

00:06:01.432 --> 00:06:03.044
if you think about it.

00:06:03.044 --> 00:06:07.142
All of World War II was
really kind of fought,

00:06:07.142 --> 00:06:08.272
right, in Europe...

00:06:08.272 --> 00:06:09.968
- [Voiceover] Right in
the European bread basket.

00:06:09.968 --> 00:06:13.925
- [Voiceover] And so
there is serious economic

00:06:13.925 --> 00:06:18.716
trouble in the aftermath of World War II.

00:06:18.716 --> 00:06:21.159
People don't have enough to eat.

00:06:21.159 --> 00:06:23.613
They certainly don't have any cash,

00:06:23.613 --> 00:06:25.437
and they don't have any fuel,

00:06:25.437 --> 00:06:28.586
which is very worrisome

00:06:28.586 --> 00:06:32.115
in 1946 because of the terrible winter.

00:06:32.115 --> 00:06:35.760
So people are cold and they are hungry.

00:06:36.749 --> 00:06:39.633
And when people are
cold and hungry there is

00:06:39.633 --> 00:06:43.691
a lot of fuel for a
possible revolution, right?.

00:06:44.991 --> 00:06:48.468
Even in the 1930s, in the
United States, there's

00:06:48.468 --> 00:06:51.405
a lot of different
political ideas that come up

00:06:51.405 --> 00:06:52.935
during the Great Depression.

00:06:52.935 --> 00:06:55.420
Because when your political
system isn't working well,

00:06:55.420 --> 00:06:57.714
you consider other kinds
of political systems.

00:06:57.714 --> 00:07:00.289
- [Voiceover] So the United
States is worried that,

00:07:00.289 --> 00:07:03.101
because of the cold winter
of 1946, and scarcity

00:07:03.101 --> 00:07:05.734
across western Europe,
this blue part of the

00:07:05.744 --> 00:07:07.378
map will turn much redder.

00:07:07.378 --> 00:07:10.307
- [Voiceover] Right. So for
the United States, they're

00:07:10.307 --> 00:07:14.048
worried that communism
is kind of the child

00:07:14.048 --> 00:07:18.315
of hunger and poverty.

00:07:18.315 --> 00:07:22.684
And they're afraid that
because Stalin has so much

00:07:22.684 --> 00:07:27.074
territory in Europe, that
he is really well poised

00:07:27.077 --> 00:07:28.773
to become "Hitler, part two."

00:07:28.773 --> 00:07:30.527
- [Voiceover] Okay. And
that is a sequel the

00:07:30.527 --> 00:07:31.854
United States does not want to see.

00:07:31.854 --> 00:07:33.342
- [Voiceover] No. Absolutely not.

00:07:33.342 --> 00:07:35.198
And if they really learned
anything from World War II,

00:07:35.198 --> 00:07:38.630
it's that appeasement doesn't work, right?

00:07:38.630 --> 00:07:42.885
During the 1930s, many people in the West,

00:07:44.576 --> 00:07:48.719
the prime minister of
England, Neville Chamberlain,

00:07:48.719 --> 00:07:52.015
kind of felt like they
didn't want to go back to war

00:07:52.015 --> 00:07:53.957
because World War I is still very much on

00:07:53.957 --> 00:07:56.472
people's minds during the 1930s.

00:07:56.472 --> 00:08:00.176
And so they figured,
let's not confront Hitler

00:08:00.176 --> 00:08:03.538
head on because we're not
up for that right now.

00:08:03.538 --> 00:08:06.945
We're also in the middle
of worldwide depression.

00:08:06.945 --> 00:08:09.745
And that helped nothing
because it just meant that

00:08:09.745 --> 00:08:12.450
Hitler could gain a whole lot of territory

00:08:12.450 --> 00:08:15.226
and World War II was
much worse than it might

00:08:15.226 --> 00:08:18.040
have been if they hadn't
gone after Hitler earlier.

00:08:18.040 --> 00:08:19.708
- [Voiceover] And stopped the Anschluss.

00:08:19.708 --> 00:08:20.870
- [Voiceover] Yes.

00:08:20.870 --> 00:08:21.640
- [Voiceover] If you stop
the Anschluss, you stop

00:08:21.640 --> 00:08:23.475
the "onschlaught."

00:08:23.475 --> 00:08:25.845
- [Voiceover] (laughs) Exactly.

00:08:25.845 --> 00:08:28.585
So they're really trying

00:08:28.585 --> 00:08:31.495
to say, all right, Stalin, if
he wants to, could probably

00:08:31.495 --> 00:08:35.270
just run his way through
the rest of Europe, right,

00:08:36.640 --> 00:08:38.659
with very little resistance,

00:08:38.659 --> 00:08:40.899
because the only nation in the world

00:08:40.899 --> 00:08:42.306
that has the military and economic

00:08:42.306 --> 00:08:45.044
power to stop the Soviet
Union is the United States.

00:08:45.044 --> 00:08:47.160
- [Voiceover] Because
their factories and fields

00:08:47.160 --> 00:08:49.403
were not bombed to cinders during the

00:08:49.403 --> 00:08:50.500
European theater.

00:08:50.500 --> 00:08:52.640
- [Voiceover] Right. So if they wanna stop

00:08:52.640 --> 00:08:55.854
"Hitler, part two, the Stalin years,"

00:08:55.854 --> 00:08:58.464
then they're going to have to really

00:08:58.464 --> 00:09:01.154
stand up for capitalism

00:09:01.154 --> 00:09:05.900
and also for the kind
of material comforts and

00:09:05.900 --> 00:09:09.179
democratic, what we
call, self-determination.

00:09:09.179 --> 00:09:12.901
This is one of the most
important ideas to come

00:09:12.901 --> 00:09:16.531
out of the alliance
between the United States

00:09:16.531 --> 00:09:20.503
and Britain, which is that
the citizens of a region

00:09:20.503 --> 00:09:22.500
should have the right to decide their own

00:09:22.500 --> 00:09:23.674
form of government.

00:09:23.674 --> 00:09:27.662
- [Voiceover] And they think
of the Soviets as being

00:09:27.662 --> 00:09:31.136
totalitarians. And that's
not a wrong assessment

00:09:31.136 --> 00:09:33.983
because there is a very
strong totalitarian control

00:09:33.983 --> 00:09:36.823
coming out of Moscow and the Soviet Union.

00:09:36.823 --> 00:09:39.815
So they say, If we're going to keep Europe

00:09:39.815 --> 00:09:42.999
from turning all red, all communist,

00:09:42.999 --> 00:09:46.988
then we're going to need
to kind of shore up Europe.

00:09:46.988 --> 00:09:49.471
They think of communism
as kind of being a little

00:09:49.471 --> 00:09:51.559
bit like a flood, shall we say,

00:09:51.559 --> 00:09:54.899
that you gotta, you gotta put sandbags

00:09:54.899 --> 00:09:57.762
around the edges of communism.

00:09:57.772 --> 00:10:00.676
Otherwise, it's gonna leak out.

00:10:00.676 --> 00:10:03.642
- [Voiceover] So is the
United States and their allies

00:10:03.642 --> 00:10:06.022
also interested in
creating their own kind of

00:10:06.022 --> 00:10:09.428
light blue buffer zone, also
next to the Eastern Block?

00:10:09.428 --> 00:10:10.838
Or are they interested in...

00:10:10.848 --> 00:10:13.265
Is this when we get into
the creation of NATO?

00:10:13.265 --> 00:10:15.355
- [Voiceover] NATO really comes out of an

00:10:15.355 --> 00:10:19.642
understanding that World
War II has not created peace

00:10:19.642 --> 00:10:23.660
and so the US is going to have to

00:10:23.660 --> 00:10:26.871
forego their more than a century-long

00:10:26.871 --> 00:10:30.733
commitment to being isolationist

00:10:30.733 --> 00:10:33.893
and take a stronger role in the world.

00:10:33.893 --> 00:10:35.899
- [Voiceover] So, okay,
for the folks at home, what

00:10:35.899 --> 00:10:38.652
does NATO stand for? North
Atlantic Treaty Organization?

00:10:38.652 --> 00:10:41.540
- [Voiceover] North Atlantic
Treaty Organization.

00:10:43.050 --> 00:10:46.011
And this is a defensive alliance

00:10:46.021 --> 00:10:49.047
between, at first, 12 nations,

00:10:51.167 --> 00:10:55.197
which say that an attack on one

00:10:55.197 --> 00:10:57.100
will be treated as an attack on all.

00:10:57.100 --> 00:10:59.832
- [Voiceover] Gotcha! Well,
that sounds like they're

00:10:59.832 --> 00:11:02.142
maneuvering their boxers into position

00:11:02.142 --> 00:11:05.064
and rubbing the shoulders
and getting them ready.

00:11:05.064 --> 00:11:06.478
- [Voiceover] Yes. Very much so.

00:11:06.478 --> 00:11:08.660
And I think one of the tragedies

00:11:08.660 --> 00:11:11.660
of the post-war era

00:11:11.660 --> 00:11:16.100
is that maybe things didn't
have to be like this?

00:11:16.100 --> 00:11:18.990
Right after the US and the USSR had worked

00:11:18.990 --> 00:11:21.169
together to defeat Hitler,

00:11:21.169 --> 00:11:23.699
it might have possible for them

00:11:23.699 --> 00:11:25.488
to coexist peacefully?

00:11:25.488 --> 00:11:30.151
But I think they both had
the idea that the other

00:11:30.151 --> 00:11:32.503
economic system, and we're talking about

00:11:32.503 --> 00:11:35.203
communism and capitalism,
were just kind of

00:11:35.203 --> 00:11:38.402
riddled with internal
inconsistencies and that

00:11:38.402 --> 00:11:41.676
eventually the world
would be all capitalist

00:11:41.676 --> 00:11:43.880
or all communist.

00:11:43.880 --> 00:11:46.159
And they were going to have to really

00:11:46.159 --> 00:11:49.188
marshal all of their resources behind

00:11:49.188 --> 00:11:52.677
their chosen boxer or
they were gonna lose.

00:11:52.677 --> 00:11:53.971
- [Voiceover] Sounds
like a fight that's gonna

00:11:53.971 --> 00:11:55.124
take a long time.

00:11:55.124 --> 00:11:56.340
- [Voiceover] And it did.

00:11:56.340 --> 00:11:57.450
- [Voiceover] End round one!


Irregular plural nouns

WEBVTT
Kind: captions
Language: en

00:00:00.374 --> 00:00:01.521
- [Voiceover] Hello grammarians.

00:00:01.521 --> 00:00:04.717
Today we're going to be
discussing more irregular plurals.

00:00:04.717 --> 00:00:09.551
So, this is the irregular
plural part II, the two-ening.

00:00:12.061 --> 00:00:16.105
Because we really want to
focus on this en particle.

00:00:16.105 --> 00:00:17.313
I'll explain in a minute.

00:00:17.313 --> 00:00:21.704
So, previously we discussed
the idea that there's one

00:00:21.704 --> 00:00:23.301
regular plural.

00:00:25.041 --> 00:00:27.111
And then, there are a
couple of different kinds

00:00:27.111 --> 00:00:29.039
of irregular plurals.

00:00:31.119 --> 00:00:34.542
So, we have the word dog.

00:00:34.542 --> 00:00:36.549
Then it becomes dogs.

00:00:36.549 --> 00:00:39.578
We have the word leaf.

00:00:39.578 --> 00:00:43.283
Then it becomes leaves.

00:00:43.283 --> 00:00:48.052
I'd like to introduce something
else too, the word child

00:00:48.052 --> 00:00:51.235
is also an irregular plural
because its plural form

00:00:51.235 --> 00:00:53.599
is not childs, but in fact, you ready,

00:00:53.599 --> 00:00:56.017
say it with me, children.

00:00:56.017 --> 00:00:56.639
Wow.

00:00:56.639 --> 00:00:57.632
That's weird.

00:00:57.632 --> 00:01:00.374
What is that about?

00:01:00.374 --> 00:01:03.946
Like I said, English is
not a predictable language.

00:01:03.946 --> 00:01:05.248
Few languages are.

00:01:05.248 --> 00:01:07.894
When we think of English
as one solid block of wood,

00:01:07.894 --> 00:01:11.168
what we're actually seeing
is this lashed together raft

00:01:11.168 --> 00:01:13.981
made of spit, hope, and history.

00:01:13.981 --> 00:01:17.399
So, it's your job as a writer
and a speaker of English

00:01:17.399 --> 00:01:20.311
to make of this crooked
timber something straight,

00:01:20.311 --> 00:01:23.055
to borrow a phrase and then mangle it.

00:01:23.055 --> 00:01:25.551
So, the question is what is the deal?

00:01:25.551 --> 00:01:26.584
Right.

00:01:26.584 --> 00:01:29.627
And the answer is that it has
to do with English language

00:01:29.627 --> 00:01:32.034
history and heritage.

00:01:32.034 --> 00:01:35.142
English comes to us from Old English,

00:01:38.452 --> 00:01:41.503
which, if you look at it now,
it kinda looks like German

00:01:41.503 --> 00:01:44.714
because English is a Germanic language.

00:01:44.714 --> 00:01:46.758
That's the term linguists use.

00:01:49.388 --> 00:01:51.892
So, in Old English, there
were a bunch of different ways

00:01:51.892 --> 00:01:53.492
to make something plural.

00:01:53.492 --> 00:01:57.070
Some nouns would become
plural by adding an e n.

00:01:57.070 --> 00:01:58.527
Some you could add an s.

00:01:59.587 --> 00:02:01.466
Some you would add an r.

00:02:02.486 --> 00:02:05.021
And part of this depended on
what part of England you lived

00:02:05.021 --> 00:02:06.484
in at the time.

00:02:06.484 --> 00:02:09.238
There were a bunch of
different regional variations.

00:02:09.238 --> 00:02:13.999
Today, this e n ending only
applies to basically three

00:02:13.999 --> 00:02:15.140
or four words.

00:02:15.140 --> 00:02:18.279
These are all you need to
remember for this one weird

00:02:18.279 --> 00:02:19.395
e n thing.

00:02:19.395 --> 00:02:22.770
So, the e n ending plural only applies to

00:02:23.760 --> 00:02:24.944
child,

00:02:25.994 --> 00:02:26.874
ox,

00:02:28.424 --> 00:02:29.964
brother, or sister.

00:02:31.714 --> 00:02:34.265
And even those, these are really rare.

00:02:34.265 --> 00:02:36.180
I bet you don't even know
what I'm talking about.

00:02:36.180 --> 00:02:39.605
The plural of brother
originally was brethren.

00:02:39.605 --> 00:02:42.640
The plural of sister
is sistren, or sistren.

00:02:42.640 --> 00:02:46.685
But these are so rare, and
brothers and sisters is

00:02:46.685 --> 00:02:49.484
so common that we don't
even need to consider them.

00:02:49.484 --> 00:02:53.582
So, really, the only weird e
n plurals that are gonna be

00:02:53.582 --> 00:02:58.175
in common usage are children, and oxen.

00:02:59.215 --> 00:03:03.101
And unless your family
raises musk oxen or you're

00:03:03.101 --> 00:03:06.398
an ox driver, you're probably
not gonna be using the word

00:03:06.398 --> 00:03:10.369
ox very often, and if oxen
are a big part of your life,

00:03:10.369 --> 00:03:12.981
you probably knew how to
pluralize them already

00:03:12.981 --> 00:03:14.618
just from hearing people
talk about them your

00:03:14.618 --> 00:03:15.721
entire childhood.

00:03:15.721 --> 00:03:17.000
So, I'm going to cross that off too.

00:03:18.130 --> 00:03:21.764
The only thing you need
to worry about is the e n

00:03:21.764 --> 00:03:23.245
in children.

00:03:23.245 --> 00:03:27.702
Child becomes children.

00:03:29.132 --> 00:03:33.261
Now, you're probably wondering
David if there's the word ox,

00:03:33.261 --> 00:03:35.352
and ox becomes oxen.

00:03:35.352 --> 00:03:37.336
What about box?

00:03:37.336 --> 00:03:41.092
Why is the plural of
box boxes and not boxen?

00:03:42.182 --> 00:03:45.173
To borrow a gag from Brian Regan.

00:03:45.173 --> 00:03:48.332
The simple truth is that it has to do with

00:03:48.332 --> 00:03:50.419
this word's origin.

00:03:50.419 --> 00:03:53.322
Ox is a native Old English word.

00:03:53.322 --> 00:03:57.679
Box comes to us originally
from Greek, and then Latin.

00:03:57.679 --> 00:03:58.600
And then French.

00:03:58.600 --> 00:04:00.491
That's how it came to English.

00:04:01.531 --> 00:04:04.861
I'm sorry if that's not a very
satisfying answer, but that's

00:04:04.861 --> 00:04:06.370
how English works.

00:04:06.370 --> 00:04:08.832
English is this language
that is composed of a bunch

00:04:08.832 --> 00:04:12.256
of different puzzle pieces
all mashed together by time,

00:04:12.256 --> 00:04:15.442
history, and repeated invasions,
which are totally metal.

00:04:16.472 --> 00:04:17.835
But I'm getting away from myself.

00:04:17.835 --> 00:04:19.766
The important thing to
remember is that for this

00:04:19.766 --> 00:04:23.553
irregular plural, this e n
ending, really, unless you work

00:04:23.553 --> 00:04:28.006
with oxen, or unless you are
leading some sort of old-timey

00:04:28.006 --> 00:04:30.352
folk revival where you
have to refer to brethren

00:04:30.352 --> 00:04:33.723
and sistren, child to
children is really the only

00:04:33.723 --> 00:04:36.157
change you need concern yourself with.

00:04:37.227 --> 00:04:40.490
All this can be summed up just like that.

00:04:41.520 --> 00:04:43.102
You can learn anything.

00:04:43.102 --> 00:04:44.310
David out.

00:04:45.690 --> 00:04:47.998
Now, if you're still with
me, I would like to relate

00:04:47.998 --> 00:04:51.471
a humorous, historical
anecdote that comes to us from

00:04:51.471 --> 00:04:55.735
the 15th century English
printer William Caxton.

00:04:56.755 --> 00:05:01.108
Now, this story comes to us
from about the year 1490.

00:05:01.108 --> 00:05:04.473
And I just want to use this
story to illustrate just

00:05:04.473 --> 00:05:08.085
how different, how dependent
on what part of the country

00:05:08.085 --> 00:05:11.149
you were living in was on
the language that you spoke.

00:05:11.149 --> 00:05:14.404
Because before mass transit,
and mass communication

00:05:14.404 --> 00:05:16.794
media, which William Caxton
was a part of because he was

00:05:16.794 --> 00:05:19.100
one of the first English
language printers.

00:05:19.100 --> 00:05:22.107
Before any of this technology
allowed us all to communicate

00:05:22.107 --> 00:05:24.981
with each other, we basically
just had letters, and most

00:05:24.981 --> 00:05:26.128
people were illiterate.

00:05:26.128 --> 00:05:27.866
And talking to each other.

00:05:27.866 --> 00:05:30.431
And since it wasn't really
easy to get from place to place

00:05:30.431 --> 00:05:33.810
to place before the advent
of cars most people didn't

00:05:33.810 --> 00:05:35.452
really go very far.

00:05:36.482 --> 00:05:38.978
So, here is the story, which
I will translate for you

00:05:38.978 --> 00:05:40.261
from Middle English.

00:05:40.261 --> 00:05:41.041
It goes like this.

00:05:41.041 --> 00:05:43.584
So, these merchants were
traveling from the Thames river,

00:05:43.584 --> 00:05:45.263
which flows through London,
and they're trying to get

00:05:45.263 --> 00:05:48.845
to Zeeland, here, which is in
the modern day Netherlands.

00:05:49.875 --> 00:05:53.487
This is, incidentally, this is England.

00:05:53.487 --> 00:05:55.809
This is the southern
southeast coast of England.

00:05:55.809 --> 00:05:57.608
They're trying to get to
Zeeland, and this is a distance

00:05:57.608 --> 00:05:59.963
of about 160 miles.

00:05:59.963 --> 00:06:01.278
Give or take.

00:06:03.498 --> 00:06:05.751
But they can't make it
there because of bad winds.

00:06:05.751 --> 00:06:09.964
They stop here at Foreland
to refresh their stores.

00:06:09.964 --> 00:06:11.707
They stop for groceries basically.

00:06:11.707 --> 00:06:14.128
And then, they realize that
their London English and the

00:06:14.128 --> 00:06:15.568
English of the place where they are,

00:06:15.568 --> 00:06:19.735
which is maybe 60 miles
out of London, maybe,

00:06:19.735 --> 00:06:21.987
is mutually incomprehensible.

00:06:21.987 --> 00:06:23.682
A man goes into a house.

00:06:23.682 --> 00:06:25.608
Asks the woman of the house for eggs.

00:06:25.608 --> 00:06:27.712
She doesn't understand the
word eggs because the word

00:06:27.712 --> 00:06:30.653
she uses for eggs is eyren.

00:06:30.653 --> 00:06:33.328
The word egg is Norse.

00:06:33.328 --> 00:06:35.163
And he says I would like some egges.

00:06:35.163 --> 00:06:36.945
And she says I don't have any egges.

00:06:36.945 --> 00:06:38.176
I don't speak French.

00:06:38.176 --> 00:06:39.655
And he gets angry because
he's like I don't speak

00:06:39.655 --> 00:06:40.298
French either.

00:06:40.298 --> 00:06:41.107
I'm English.

00:06:41.107 --> 00:06:42.160
He says do you have any eggs.

00:06:42.160 --> 00:06:43.320
She says I don't know what that is.

00:06:43.320 --> 00:06:45.709
And finally, someone says
I think he means eyren.

00:06:45.709 --> 00:06:46.797
Do you have any eyren?

00:06:46.797 --> 00:06:48.363
And she said oh eyren.

00:06:48.363 --> 00:06:49.419
Of course, here.

00:06:49.419 --> 00:06:50.486
Buy some.

00:06:50.486 --> 00:06:52.542
And it just goes to show
you that before the advent

00:06:52.542 --> 00:06:56.938
of mass communication, we
could have had variations

00:06:56.938 --> 00:07:01.005
from, as Caxton put it,
shire to shire to shire.

00:07:01.005 --> 00:07:03.919
Each little town had its own
variant of language, and it

00:07:03.919 --> 00:07:07.215
wasn't until everything was
united by this technology

00:07:07.215 --> 00:07:10.106
that Caxton worked with, the
printing press, that people

00:07:10.106 --> 00:07:12.521
started to become literate
and converge toward what

00:07:12.521 --> 00:07:15.840
we would consider a more
unified standard of English.

00:07:15.840 --> 00:07:16.556
Alright.

00:07:16.556 --> 00:07:18.477
That was just a little bonus, silly video.

00:07:18.477 --> 00:07:20.545
You can learn anything.

00:07:20.545 --> 00:07:21.509
David out.


Introduction to singular and plural nouns

WEBVTT
Kind: captions
Language: en

00:00:00.146 --> 00:00:01.423
- Hello grammarians!

00:00:01.423 --> 00:00:03.942
Let's talk about singular
and plural nouns.

00:00:03.942 --> 00:00:06.527
Nouns, as we discussed previously,

00:00:06.527 --> 00:00:08.969
are a type of word.

00:00:08.969 --> 00:00:10.579
They are a part of speech.

00:00:10.579 --> 00:00:15.157
A noun is any word that is a person,

00:00:15.157 --> 00:00:17.246
a place,

00:00:17.246 --> 00:00:19.893
a thing,

00:00:19.893 --> 00:00:22.424
or an idea.

00:00:22.424 --> 00:00:25.453
In English, we can figure
out just by looking at a noun

00:00:25.453 --> 00:00:27.379
whether or not there is one of something,

00:00:27.379 --> 00:00:28.575
whether it's a singular,

00:00:28.575 --> 00:00:31.523
or whether or not there is
more than one of something.

00:00:31.523 --> 00:00:33.311
There's an easy way to
tell the difference between

00:00:33.311 --> 00:00:34.634
singular and plural.

00:00:34.634 --> 00:00:36.419
If you write the words down,

00:00:36.419 --> 00:00:41.409
singular contains the word single.

00:00:41.409 --> 00:00:43.511
Single, means there's only one of it.

00:00:43.511 --> 00:00:45.243
Plural is maybe a little bit less obvious,

00:00:45.243 --> 00:00:46.854
but it comes to us from Latin.

00:00:46.854 --> 00:00:49.198
It comes to us from this word plus,

00:00:49.198 --> 00:00:52.580
which means more,

00:00:52.580 --> 00:00:53.633
which you might recognize

00:00:53.633 --> 00:00:56.759
plus, as we call it in
English, from mathematics,

00:00:56.759 --> 00:00:58.083
from arithmetic.

00:00:58.083 --> 00:01:00.633
We usually says it looks
like this little plus symbol.

00:01:00.633 --> 00:01:02.982
So, whenever you think,
whenever you see plural,

00:01:02.982 --> 00:01:05.532
just think more; just think plus.

00:01:05.532 --> 00:01:07.463
There is more than one.

00:01:07.463 --> 00:01:09.298
Singular is one thing.

00:01:09.298 --> 00:01:13.395
Plural, more than one
thing; there is more.

00:01:13.395 --> 00:01:14.208
Let's go through it.

00:01:14.208 --> 00:01:15.960
Let's do some examples.

00:01:15.960 --> 00:01:18.852
I'll show you how you make
the plural in English,

00:01:18.852 --> 00:01:20.952
how you indicate using your language

00:01:20.952 --> 00:01:22.822
that there is more than one thing.

00:01:22.822 --> 00:01:26.247
So let's just throw out a couple of words.

00:01:29.880 --> 00:01:31.364
Dog,

00:01:31.364 --> 00:01:32.769
cat,

00:01:34.849 --> 00:01:36.318
dinosaur,

00:01:36.318 --> 00:01:38.837
and whale.

00:01:38.837 --> 00:01:40.927
All you need to do to make
it plural is very simply

00:01:40.927 --> 00:01:44.673
just take an s and you add
it onto the end like so.

00:01:44.673 --> 00:01:46.298
Dogs, cats,

00:01:46.298 --> 00:01:49.402
dinosaurs, whales.

00:01:49.402 --> 00:01:51.364
If you want to make something plural,

00:01:51.364 --> 00:01:53.454
think about plus, more.

00:01:53.454 --> 00:01:56.530
All you have to do is add an s

00:01:56.530 --> 00:01:59.048
like that: add an s.

00:01:59.048 --> 00:02:02.573
This is what we call the regular plural.

00:02:02.573 --> 00:02:04.295
This is the regular plural.

00:02:04.295 --> 00:02:06.988
What that means is it obeys this one rule.

00:02:06.988 --> 00:02:08.890
All you have to do to say
that there's more than one dog

00:02:08.890 --> 00:02:10.980
is throw on an s,

00:02:10.980 --> 00:02:13.278
and we're lucky because most English nouns

00:02:13.278 --> 00:02:14.760
behave that way.

00:02:14.760 --> 00:02:17.280
Most nouns are regular.

00:02:17.280 --> 00:02:19.164
However, here's the bad news.

00:02:19.164 --> 00:02:22.580
There are some irregular plurals.

00:02:24.220 --> 00:02:26.223
They are not regular,

00:02:26.223 --> 00:02:29.856
thus irregular, not.

00:02:29.856 --> 00:02:32.968
Now we have words like leaf,

00:02:32.968 --> 00:02:36.299
child, and fungus,

00:02:36.299 --> 00:02:39.584
which is like a mushroom,

00:02:39.584 --> 00:02:42.441
mouse,

00:02:42.441 --> 00:02:45.389
and sheep.

00:02:45.389 --> 00:02:48.036
How would you, you know you
can't just add an s to these?

00:02:48.036 --> 00:02:51.020
That's unfortunately not how
these nouns work in English.

00:02:51.020 --> 00:02:53.713
You can't say leafs, childs, and funguses,

00:02:53.713 --> 00:02:56.569
and mouses, and sheeps.

00:02:56.569 --> 00:02:58.481
This is how you do it.

00:02:58.481 --> 00:03:01.525
Each one of these words
corresponds to a class of words

00:03:01.525 --> 00:03:04.591
that has its own unique
pluralization standards.

00:03:04.591 --> 00:03:07.748
So, leaf becomes leaves.

00:03:07.748 --> 00:03:11.320
Child becomes children.

00:03:11.320 --> 00:03:14.365
Fungus becomes fungi.

00:03:14.365 --> 00:03:16.664
Mouse becomes mice.

00:03:16.664 --> 00:03:20.205
And sheep stays sheep,

00:03:20.205 --> 00:03:21.656
believe it or not.

00:03:21.656 --> 00:03:23.061
These are the irregular plurals,

00:03:23.061 --> 00:03:26.346
and we'll be covering each of
these in turn in later videos,

00:03:26.346 --> 00:03:29.805
but for now I just want you to
focus on the regular plural,

00:03:29.805 --> 00:03:31.593
which again we can sum up in this way.

00:03:31.593 --> 00:03:33.729
All you have to do is add an s.

00:03:33.729 --> 00:03:35.517
Here's a good example, right?

00:03:35.517 --> 00:03:38.466
We have one elephant here.

00:03:38.466 --> 00:03:40.939
Down here we have two elephants.

00:03:40.939 --> 00:03:44.638
The only difference between
this word and this word

00:03:44.638 --> 00:03:46.952
is that this one has
an s on the end of it.

00:03:46.952 --> 00:03:49.216
So if we wanted to say that
this elephant here was not,

00:03:49.216 --> 00:03:52.291
in fact, one elephant,
and was two elephants,

00:03:52.291 --> 00:03:55.991
all we have to do is add an s,

00:03:55.991 --> 00:03:58.351
changing it from singular to plural.

00:03:58.351 --> 00:04:00.778
Remember, plural comes from plus.

00:04:00.778 --> 00:04:02.796
Add an s.

00:04:02.796 --> 00:04:05.944
So one elephant becomes

00:04:05.944 --> 00:04:08.285
two elephants.

00:04:08.285 --> 00:04:11.749
(humming)

00:04:11.749 --> 00:04:14.043
World's fastest elephant drawing, go!

00:04:14.043 --> 00:04:17.967
(humming)

00:04:17.967 --> 00:04:19.341
It's kind of an elephant monkey,

00:04:19.341 --> 00:04:23.223
but you get the vague idea.

00:04:24.199 --> 00:04:27.418
If you're ever in need
of more than one thing,

00:04:27.418 --> 00:04:31.633
for the regular plural, just add an s.

00:04:31.633 --> 00:04:33.212
You can learn anything.

00:04:33.212 --> 00:04:34.622
David, out!


Introduction to nouns

WEBVTT
Kind: captions
Language: en

00:00:00.674 --> 00:00:01.988
- [Voiceover] Hello grammarians.

00:00:01.988 --> 00:00:04.082
Welcome to the English parts of speech.

00:00:04.082 --> 00:00:06.630
We're gonna begin with the noun,

00:00:06.630 --> 00:00:09.626
the lovely wonderful noun,
your friend and mine.

00:00:09.626 --> 00:00:12.660
They're mostly what you're
gonna encounter in sentences.

00:00:12.660 --> 00:00:15.728
Most sentences in English
contain at least one noun

00:00:15.728 --> 00:00:18.848
or a pronoun, but we'll
get to pronouns later.

00:00:18.848 --> 00:00:22.827
A noun is basically anything.

00:00:24.417 --> 00:00:28.506
And I know that's not an
especially helpful definition,

00:00:28.506 --> 00:00:31.574
but we'll get more specific in a minute.

00:00:31.574 --> 00:00:34.070
A noun is basically anything at all.

00:00:34.070 --> 00:00:36.410
Now the way this is taught
in traditional grammar

00:00:36.410 --> 00:00:39.973
is to say that a noun is
a person, place or thing,

00:00:39.973 --> 00:00:44.522
which is fine, I think we can
make that a little bit sharper

00:00:44.522 --> 00:00:47.077
and expand that out by
saying that they are,

00:00:47.077 --> 00:00:51.152
that nouns are people or living things,

00:00:51.152 --> 00:00:52.725
places,

00:00:55.455 --> 00:00:59.069
things, or ideas.

00:01:00.579 --> 00:01:03.063
I think ideas is the one
that usually gets left out.

00:01:03.063 --> 00:01:07.532
So nouns can be people,
places, things and ideas.

00:01:07.532 --> 00:01:09.533
Let's put that in action.

00:01:10.993 --> 00:01:13.648
So, this is Raul.

00:01:13.648 --> 00:01:15.596
He is from Argentina.

00:01:15.596 --> 00:01:17.416
He is a penguin.

00:01:17.416 --> 00:01:19.870
Raul has big dreams.

00:01:19.870 --> 00:01:20.930
Now, okay.

00:01:20.930 --> 00:01:22.824
So, I wanna take these three sentences

00:01:22.824 --> 00:01:26.074
and find the noun in them using the test.

00:01:26.074 --> 00:01:30.070
So the test is, is it a person

00:01:30.070 --> 00:01:33.702
or a living thing, a place,

00:01:33.702 --> 00:01:37.620
a thing, or an idea?

00:01:37.620 --> 00:01:39.366
And if it's any of those things,

00:01:39.366 --> 00:01:42.054
then it falls into the
category of words in English

00:01:42.054 --> 00:01:44.127
which we call nouns.

00:01:44.127 --> 00:01:45.823
So, sentence number one.

00:01:45.823 --> 00:01:47.573
This is Raul.

00:01:48.873 --> 00:01:50.069
What is Raul?

00:01:50.069 --> 00:01:52.781
Well, Raul is a person or a living thing

00:01:52.781 --> 00:01:56.151
so we're just gonna say person so noun.

00:01:57.917 --> 00:01:59.069
Next sentence.

00:01:59.069 --> 00:02:01.708
He is from Argentina.

00:02:01.708 --> 00:02:04.612
Now Argentina happens to be a place

00:02:04.612 --> 00:02:06.436
so therefore, it is also a noun.

00:02:06.436 --> 00:02:08.356
It's a kind of noun called a proper noun,

00:02:08.356 --> 00:02:10.900
just like Raul is but
we'll get to that later.

00:02:10.900 --> 00:02:12.868
So, Argentina is a noun.

00:02:12.868 --> 00:02:15.640
Argentina incidentally is a country

00:02:15.640 --> 00:02:19.024
and the word country is also a noun

00:02:19.024 --> 00:02:21.962
because it is a thing.

00:02:23.488 --> 00:02:24.951
So, sentence the third.

00:02:24.951 --> 00:02:27.313
He is a penguin.

00:02:27.313 --> 00:02:30.951
Now, a penguin is a
living being or a thing

00:02:30.951 --> 00:02:35.223
so we can say oh yes, penguin,
that is a noun as well.

00:02:35.223 --> 00:02:39.903
So, you've noticed I'm
not circling he or this.

00:02:39.903 --> 00:02:42.903
These words are pronouns,
relative pronouns,

00:02:42.903 --> 00:02:45.807
and they can sometimes behave like nouns,

00:02:45.807 --> 00:02:48.256
but I wouldn't call them nouns.

00:02:48.256 --> 00:02:49.408
That'll just get confusing.

00:02:49.408 --> 00:02:51.652
So, these are pronouns and
we'll get to them later.

00:02:53.562 --> 00:02:54.763
Sentence number four.

00:02:54.763 --> 00:02:56.850
Raul has big dreams.

00:02:56.850 --> 00:02:58.266
So, here we have Raul again.

00:02:58.266 --> 00:03:01.074
We know from the first sentence
that it's a person's name.

00:03:01.074 --> 00:03:04.289
So, we're just gonna say
this is also a noun again.

00:03:04.289 --> 00:03:06.233
And dreams.

00:03:06.233 --> 00:03:08.657
Now, dreams isn't a person, a place.

00:03:08.657 --> 00:03:11.153
It's a thing, sure.

00:03:12.253 --> 00:03:14.921
The reason I put in idea
as a fourth category is

00:03:14.921 --> 00:03:17.986
to get it stuff that you can't pick up.

00:03:17.986 --> 00:03:21.078
Like for example, so dream,
yes, dream is a noun.

00:03:23.058 --> 00:03:24.859
Dream is maybe a little tangible

00:03:24.859 --> 00:03:27.023
'cause it's something you can imagine,

00:03:27.023 --> 00:03:30.526
but the idea of like a word like bigness.

00:03:32.686 --> 00:03:35.438
Or if you prefer, you know, size.

00:03:40.238 --> 00:03:45.052
The size of Raul's
plumage was astonishing.

00:03:46.202 --> 00:03:48.255
Look at that gorgeous plumage.

00:03:48.255 --> 00:03:50.473
It's a little penguin mohawk.

00:03:50.473 --> 00:03:52.455
(high-pitched murmurs)

00:03:52.455 --> 00:03:55.801
The size of Raul's plumage was astounding.

00:03:55.801 --> 00:03:58.392
Now in that sentence, size is a noun,

00:03:58.392 --> 00:04:00.860
but it's not a physical thing,

00:04:00.860 --> 00:04:02.290
it's not a person, it's not a place,

00:04:02.290 --> 00:04:04.578
it's not something you can pick up.

00:04:04.578 --> 00:04:05.644
It's an idea.

00:04:05.644 --> 00:04:08.010
So, that's why I include
this fourth category.

00:04:08.010 --> 00:04:10.844
So, if you're trying to figure out

00:04:10.844 --> 00:04:12.950
whether or not a word is a noun,

00:04:12.950 --> 00:04:14.614
just apply this test.

00:04:14.614 --> 00:04:17.760
Ask yourself, is it a person, a place,

00:04:17.760 --> 00:04:19.753
a thing, or an idea?

00:04:21.556 --> 00:04:24.262
And you, my friend, will be golden.

00:04:24.262 --> 00:04:25.944
You can learn anything.

00:04:25.944 --> 00:04:27.084
David out.


Lorentz transformation derivation part 3

WEBVTT
Kind: captions
Language: en

00:00:00.445 --> 00:00:01.988
- [Voiceover] We've
made some good progress

00:00:01.988 --> 00:00:05.017
in our derivation of parts of
the Lorentz transformation.

00:00:05.017 --> 00:00:08.010
We've been able to
express x prime in terms

00:00:08.010 --> 00:00:12.102
of our Lorentz factor and x and v and t.

00:00:12.102 --> 00:00:14.297
And we've been able to
switch things around

00:00:14.297 --> 00:00:16.457
and represent x in terms
of the Lorentz factor

00:00:16.457 --> 00:00:18.834
and x prime and v and t prime.

00:00:18.834 --> 00:00:21.440
And we were able to solve
for the Lorentz factor.

00:00:21.760 --> 00:00:23.680
Now, the final missing
piece in order for us

00:00:23.680 --> 00:00:27.885
to have the full transformation
is to express t prime

00:00:27.885 --> 00:00:30.971
in terms of x and t.

00:00:31.222 --> 00:00:32.365
So, how can do we do that?

00:00:32.822 --> 00:00:34.868
Well, the way I'm going to
tackle it is I'm just going

00:00:34.868 --> 00:00:37.348
to take this equation right over here,

00:00:37.348 --> 00:00:38.400
let me underline it.

00:00:38.765 --> 00:00:41.337
I'm just gonna take that
equation right over there

00:00:41.622 --> 00:00:43.931
and solve for t prime.

00:00:44.297 --> 00:00:46.582
And the parts that have an x prime in it,

00:00:46.582 --> 00:00:48.640
I'm gonna substitute with this.

00:00:48.948 --> 00:00:52.468
So let's do that, let's
solve this for t prime.

00:00:52.582 --> 00:00:54.662
The first thing that I wanna do is

00:00:54.902 --> 00:00:56.914
I wanna divide both sides of this equation

00:00:56.914 --> 00:00:58.982
by the Lorentz factor or by gamma.

00:00:59.508 --> 00:01:01.245
So if I do that, I'm going to get,

00:01:01.245 --> 00:01:03.600
I'm gonna move it over to the
left so I have more space,.

00:01:04.068 --> 00:01:07.508
It's going to be x over gamma,

00:01:08.260 --> 00:01:09.702
x over gamma

00:01:12.171 --> 00:01:14.160
is equal to x prime.

00:01:15.210 --> 00:01:18.845
x prime plus v

00:01:21.177 --> 00:01:22.640
times t prime.

00:01:23.417 --> 00:01:26.708
t prime, let me do it
in that same blue color.

00:01:27.508 --> 00:01:28.537
Now everything I'm gonna do here

00:01:28.537 --> 00:01:30.457
is pretty straightforward algebra,

00:01:30.674 --> 00:01:31.680
but it's gonna get a little hairy,

00:01:31.680 --> 00:01:35.074
so that's why I wanna take some caution

00:01:35.074 --> 00:01:37.154
with the colors and progress slowly.

00:01:37.542 --> 00:01:38.857
Now, since I wanna solve for t prime,

00:01:38.857 --> 00:01:41.028
let me subtract x prime from both sides.

00:01:41.485 --> 00:01:46.034
So the left-hand side is
going to be x over gamma.

00:01:47.702 --> 00:01:49.062
I think that looks like a v too much.

00:01:49.382 --> 00:01:52.377
x over gamma minus x prime.

00:01:53.130 --> 00:01:55.268
Minus x prime

00:01:57.817 --> 00:02:01.462
is equal to v times t prime.

00:02:02.285 --> 00:02:05.257
v times t prime.

00:02:05.634 --> 00:02:07.165
And now, to solve for t prime,

00:02:07.165 --> 00:02:09.657
let's just divide both sides by v.

00:02:10.045 --> 00:02:12.971
And so, we are going to get x,

00:02:13.794 --> 00:02:15.085
let me do that white color.

00:02:15.862 --> 00:02:17.851
We're gonna get x over,

00:02:18.274 --> 00:02:20.422
well, now it's going to be gamma v.

00:02:21.051 --> 00:02:23.542
So, gamma v,

00:02:24.034 --> 00:02:25.508
v's in that orange color.

00:02:25.840 --> 00:02:26.720
Gamma v

00:02:28.731 --> 00:02:31.668
minus x prime

00:02:35.234 --> 00:02:36.457
over v

00:02:36.777 --> 00:02:38.628
is going to be equal to,

00:02:39.085 --> 00:02:41.771
is equal to t prime.

00:02:42.948 --> 00:02:44.937
So now we'll do what I said before.

00:02:45.931 --> 00:02:48.857
We've solved for t prime
in terms of now gamma v,

00:02:48.857 --> 00:02:51.850
x, and x prime, but now
we can take that x prime

00:02:51.850 --> 00:02:54.514
and replace it with gamma
and all of this business

00:02:54.514 --> 00:02:55.897
right over here, so let's do that.

00:02:57.337 --> 00:02:59.314
If we take this

00:02:59.314 --> 00:03:01.954
and substitute it in for x prime,

00:03:02.217 --> 00:03:03.862
actually, let me swap sides too,

00:03:04.057 --> 00:03:06.205
we are going to get t prime

00:03:06.500 --> 00:03:10.662
is equal to x

00:03:10.948 --> 00:03:13.188
over gamma.

00:03:14.194 --> 00:03:16.102
Let me do the gamma in that red color.

00:03:17.120 --> 00:03:18.457
Over gamma

00:03:18.845 --> 00:03:21.817
times v

00:03:24.354 --> 00:03:27.657
minus this stuff.

00:03:27.908 --> 00:03:32.171
So we're gonna have gamma times,

00:03:32.994 --> 00:03:35.142
it's a little bit tedious,
but we'll power through it.

00:03:39.748 --> 00:03:42.868
x minus vt .

00:03:42.868 --> 00:03:44.350
And if at any point you get inspired,

00:03:44.350 --> 00:03:45.600
I encourage you to run with it.

00:03:46.982 --> 00:03:49.382
So we just replaced x prime
with this stuff over here

00:03:49.382 --> 00:03:51.154
and then we're gonna
have all of that over v.

00:03:51.960 --> 00:03:53.531
So, all of that

00:03:55.485 --> 00:03:57.234
over v.

00:03:57.920 --> 00:04:00.262
And now what we can do, let's see.

00:04:05.782 --> 00:04:09.440
Let's factor out a
gamma out of everything.

00:04:10.011 --> 00:04:12.388
So we will get t prime

00:04:13.291 --> 00:04:14.697
is equal to

00:04:16.080 --> 00:04:18.628
gamma times,

00:04:18.960 --> 00:04:20.830
and we're gonna have, it's
gonna get pretty hairy now.

00:04:20.830 --> 00:04:23.291
Gamma times x

00:04:24.297 --> 00:04:25.851
over gamma squared.

00:04:27.051 --> 00:04:30.034
x over gamma squared, you
factor out a gamma here,

00:04:30.285 --> 00:04:32.011
or another way to think
about gamma times x

00:04:32.011 --> 00:04:33.725
over gamma squared is
gonna be x over gamma.

00:04:33.725 --> 00:04:35.188
We still have that v over there.

00:04:35.920 --> 00:04:38.617
v and then minus.

00:04:40.228 --> 00:04:41.348
Minus.

00:04:41.760 --> 00:04:44.537
And actually, let me just
distribute the minus sign,

00:04:44.537 --> 00:04:45.405
the negative sign.

00:04:45.405 --> 00:04:47.222
So, minus x over v.

00:04:48.502 --> 00:04:52.137
Minus x over v.

00:04:52.685 --> 00:04:55.577
x over v.

00:04:55.577 --> 00:04:57.565
Looking forward to this
getting a little bit simpler.

00:04:57.862 --> 00:05:00.731
And then a negative times
a negative is a positive,

00:05:01.142 --> 00:05:04.925
so it's gonna be plus vt divided by v.

00:05:05.154 --> 00:05:06.640
Well, that's just going to be plus t.

00:05:07.691 --> 00:05:08.605
Plus t.

00:05:08.605 --> 00:05:09.794
So, simplify it a little bit.

00:05:10.445 --> 00:05:12.022
Plus t.

00:05:13.062 --> 00:05:14.617
We're making some progress here.

00:05:15.782 --> 00:05:18.731
And so that is going to be
equal to, actually, let me just,

00:05:19.360 --> 00:05:21.131
so I don't have to keep
rewriting everything,

00:05:21.131 --> 00:05:23.382
let me just focus on this
part right over here,

00:05:23.382 --> 00:05:25.634
try to simplify it and
actually, even better,

00:05:25.840 --> 00:05:29.862
let me just focus on that
part right over there.

00:05:30.662 --> 00:05:34.342
That part, we can factor out an x.

00:05:35.348 --> 00:05:37.405
If we factor out an x,

00:05:40.765 --> 00:05:42.217
it is going to be equal to

00:05:42.365 --> 00:05:46.388
x times one over gamma squared, v.

00:05:47.600 --> 00:05:48.857
So let me write that down.

00:05:49.257 --> 00:05:53.554
One over gamma squared, v.

00:05:54.057 --> 00:05:55.177
Get the colors right.

00:05:55.794 --> 00:05:58.160
Gamma squared, v.

00:05:59.880 --> 00:06:01.005
Gamma squared, v,

00:06:01.394 --> 00:06:04.891
and then minus one over v.

00:06:06.660 --> 00:06:10.480
W have the minus one over v.

00:06:11.702 --> 00:06:14.365
One over v.

00:06:15.211 --> 00:06:17.600
Now, if we want to
subtract these two things,

00:06:17.600 --> 00:06:19.348
and let me put the close parentheses.

00:06:19.782 --> 00:06:21.970
If we wanna subtract these
two things, it's nice to have

00:06:21.970 --> 00:06:25.348
a common denominator, so
let's multiply the numerator

00:06:25.348 --> 00:06:27.440
and denominator here by gamma squared.

00:06:30.685 --> 00:06:33.154
This is gamma squared,

00:06:33.645 --> 00:06:35.280
gamma squared.

00:06:35.580 --> 00:06:38.240
And so, now I'm going to focus on

00:06:41.954 --> 00:06:43.382
this part right over here

00:06:43.382 --> 00:06:45.485
and hopefully this will simplify nicely.

00:06:45.780 --> 00:06:47.531
This is the same thing as

00:06:48.182 --> 00:06:51.337
one minus gamma squared,

00:06:55.508 --> 00:06:58.068
over gamma squared v.

00:06:58.960 --> 00:07:02.400
Over gamma, I picked a different color.

00:07:02.868 --> 00:07:06.308
Over gam, (chuckles) I'm having
trouble switching colors.

00:07:06.990 --> 00:07:07.828
Over

00:07:10.685 --> 00:07:12.788
gamma squared, v.

00:07:13.234 --> 00:07:14.811
Now what does this simplify to?

00:07:15.074 --> 00:07:16.582
Well, it seems like it
will be useful to have

00:07:16.582 --> 00:07:17.771
a different way of writing,

00:07:18.000 --> 00:07:19.577
well, let's just think
about what gamma squared is.

00:07:19.577 --> 00:07:22.777
Gamma is this business right
over here, which we could,

00:07:22.777 --> 00:07:25.988
if we were to square it, gamma squared.

00:07:26.960 --> 00:07:28.765
Gamma, that looks like a v again.

00:07:29.680 --> 00:07:34.020
Gamma squared is going
to be equal to one over

00:07:34.020 --> 00:07:37.154
one minus v squared over c squared.

00:07:37.154 --> 00:07:39.108
I just squared the numerator,
one squared is one.

00:07:39.371 --> 00:07:40.651
Take the square of the square root,

00:07:40.651 --> 00:07:41.485
you're just gonna get that.

00:07:41.771 --> 00:07:43.234
And if I wanted to
simplify it a little bit,

00:07:46.720 --> 00:07:49.130
I could multiply the numerator
and denominator by c squared.

00:07:49.130 --> 00:07:50.982
And so then that's going to be equal to

00:07:51.794 --> 00:07:55.028
c squared over, you
multiply the denominator

00:07:55.028 --> 00:07:57.222
by c squared, you're gonna get c squared

00:07:57.725 --> 00:07:59.405
minus v squared.

00:08:00.011 --> 00:08:01.531
So how does that help us?

00:08:01.908 --> 00:08:03.234
Well, this business.

00:08:08.354 --> 00:08:10.525
One, we can write as

00:08:10.800 --> 00:08:12.331
c squared minus v squared

00:08:12.331 --> 00:08:13.908
over c squared minus v squared.

00:08:13.908 --> 00:08:14.560
So, let's do that.

00:08:14.685 --> 00:08:17.291
c squared minus v squared

00:08:17.291 --> 00:08:19.805
over c squared minus v squared.

00:08:19.805 --> 00:08:22.274
I just did that so it
has the same denominator

00:08:22.274 --> 00:08:23.565
as gamma squared.

00:08:23.965 --> 00:08:26.868
And then, we're going to
subtract gamma squared.

00:08:26.868 --> 00:08:28.297
So we're going to subtract,

00:08:29.371 --> 00:08:33.154
we're going to subtract c squared

00:08:33.520 --> 00:08:37.177
over c squared minus v squared.

00:08:37.497 --> 00:08:40.160
And it looks like this is
going to simplify nicely.

00:08:40.377 --> 00:08:42.297
Well, let me just do one step at a time.

00:08:42.800 --> 00:08:45.965
And we're gonna have all of that over,

00:08:46.100 --> 00:08:47.440
gamma squared

00:08:49.074 --> 00:08:51.417
is once again c squared

00:08:52.060 --> 00:08:55.097
over c squared minus v squared.

00:08:55.097 --> 00:08:56.754
And we're gonna multiply that times v.

00:08:57.702 --> 00:09:00.830
Times, the v in that same color,

00:09:00.830 --> 00:09:04.148
all of that times v.

00:09:04.765 --> 00:09:06.811
Now, let's see, up here in the numerator,

00:09:07.051 --> 00:09:10.914
I have c squared minus v
squared minus c squared.

00:09:12.194 --> 00:09:14.137
So, what's going to happen is

00:09:15.394 --> 00:09:16.890
this, and we have the same denominator,

00:09:16.890 --> 00:09:19.508
so this and that are going to cancel.

00:09:19.770 --> 00:09:22.480
And so this whole expression
is going to simplify to

00:09:24.240 --> 00:09:28.217
negative v, and let me do
it in that same v color.

00:09:28.605 --> 00:09:30.914
Negative v squared

00:09:32.937 --> 00:09:36.022
over c squared minus v squared.

00:09:37.291 --> 00:09:39.920
c squared minus v squared.

00:09:40.994 --> 00:09:42.320
And we're dividing by this

00:09:42.320 --> 00:09:44.560
and that's the same thing as
multiplying by the reciprocal,

00:09:44.560 --> 00:09:46.620
so let's multiply b the reciprocal.

00:09:46.620 --> 00:09:49.325
So, times c squared minus v squared

00:09:53.131 --> 00:09:56.308
over c squared times v.

00:09:56.308 --> 00:09:58.434
And we are now in the home stretch.

00:09:59.120 --> 00:10:00.457
c squared times v.

00:10:00.697 --> 00:10:02.891
And we could do some simplification now.

00:10:02.891 --> 00:10:04.731
That's going to cancel with that.

00:10:05.131 --> 00:10:07.817
And then, v squared divided by v,

00:10:07.817 --> 00:10:10.171
you're just gonna be left
with a v right over there.

00:10:10.171 --> 00:10:12.365
So, all of this crazy business

00:10:12.628 --> 00:10:16.057
has simplified to a negative v,

00:10:16.285 --> 00:10:17.942
and I'll just write it
in that orange color.

00:10:18.217 --> 00:10:21.062
It has all simplified to negative v

00:10:21.440 --> 00:10:23.588
over c squared.

00:10:23.862 --> 00:10:26.251
So all of this

00:10:26.377 --> 00:10:30.445
has simplified to
negative v over c squared

00:10:30.594 --> 00:10:31.874
and so we're in the home stretch.

00:10:32.130 --> 00:10:33.622
This expression, t prime,

00:10:38.491 --> 00:10:41.020
it's going to be equal to gamma times,

00:10:41.020 --> 00:10:42.445
let's write this t first.

00:10:43.150 --> 00:10:44.102
So, t .

00:10:44.960 --> 00:10:48.170
And then this expression has simplified

00:10:48.171 --> 00:10:50.530
to negative v over c squared times x.

00:10:50.530 --> 00:10:51.748
So we can write this as

00:10:54.274 --> 00:10:57.737
minus v over c squared

00:11:00.594 --> 00:11:02.491
times x.

00:11:02.800 --> 00:11:03.497
And we're done.

00:11:03.862 --> 00:11:06.685
We have just completed our
Lorentz transformation.

00:11:06.685 --> 00:11:08.440
We started with this, that
we've been able to show

00:11:08.440 --> 00:11:10.948
in the last few videos, and
we did a little bit of hairy,

00:11:10.948 --> 00:11:12.800
carefully, did a little
bit of hairy algebra

00:11:12.914 --> 00:11:15.702
to get this result: t
prime is equal to gamma

00:11:15.702 --> 00:11:19.805
times t minus v over c squared times x.