Introduction to special relativity and Minkowski spacetime diagrams

WEBVTT
Kind: captions
Language: en

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- [Voiceover] In the last
videos, we constructed

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a little bit of a conundrum for us.

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We had the situation where
I'm drifting through space,

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and right at time equal zero,

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one of my friends, she passes me by

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in a spaceship going
half the speed of light

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in the positive x
direction, relative to me.

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And at time equal zero,

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is exactly where she is at my position.

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And then she keeps
traveling, and so I draw

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the path that she is taking.

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So after one second, she would be here,

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after two seconds, she would be there,

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and we set up our scales

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so that on our time axis,

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one second is the same length

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as on our space axis, or on our path axis,

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three times ten to the eighth meters.

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And we did that so that at
least in my frame of reference,

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the speed of light would
be at a 45 degree angle,

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or would have a slope of one.

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But the conundrum we hit was

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is that the speed of
light would be perceived,

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based on this model I set up,

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would be perceived differently

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based on which frame
of reference you're in.

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The photon that I emit from my flashlight,

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right at time equals zero,

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well sure I'm going to see that as moving

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at the speed of light in
the positive x direction.

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But my friend, since she's already moving

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in that positive x direction
at half of the speed of light,

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if we assumed a Newtonian world,

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she would see that photon going
at half the speed of light.

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And likewise, if she emitted
a photon from her flashlight,

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to her, it would look like it's
going at the speed of light,

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but to me, it would look
like it's going faster

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than the speed of light.

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And the reason why that is a conundrum

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is we know from observation
of the universe around us,

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and this is profound, this
is a little mind-blowing,

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this is counterintuitive
to our everyday experience,

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but we know from observation

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that the speed of light is absolute,

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that it doesn't matter your
inertial frame of reference.

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As long as you're in an
inertial frame of reference,

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it doesn't matter your relative velocity

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relative to another
inertial frame of reference.

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You will always measure the speed of light

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at three times ten to the
eighth meters per second.

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But that's at a direct contradiction

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with what our model just set up.

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So it really forces us to
question all of our assumptions.

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So what are the assumptions that we made

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in this Newtonian world?

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Well, we assumed that time is absolute.

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Time is absolute.

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And what do I mean by that?

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Well, we assumed that
one second passing for me

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is going to be one second
passing for my friend.

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That's just our everyday notion.

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If I'm in a car and you're not in a car,

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and we both have watches, and
we synchronize our watches,

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they seem to say synchronized.

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But maybe we need to
loosen this assumption.

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We also assumed that space is absolute.

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Space is absolute.

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And what do I mean by that?

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Well in our everyday experience,

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regardless of what our relative
frames of reference are,

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we seem to agree that, well if
that meter stick over there,

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that's riding on that train,

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that is a meter stick, whether
I'm sitting on the train

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or whether I'm not.

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But maybe things start to
break down a little bit

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as we think about higher velocities.

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And maybe they break down
even at the slower velocities,

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but we just don't notice them

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because it's a very small error.

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And maybe there's something
even more interesting.

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In our everyday experience,

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we just assume that time is somehow

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something very different than space,

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that you can move in the time direction

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without moving in the space direction,

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or you can move in the space direction

00:03:32.032 --> 00:03:33.495
without moving in the time direction,

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and that they're independent regardless

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of which inertial frame
of reference you're in.

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But maybe they aren't separate things.

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In fact, maybe they're all
one continuum of spacetime,

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and I don't mean space dash time,

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I mean spacetime.

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I mean the word spacetime,

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where space and time really
aren't different things.

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It's just one continuum, spacetime.

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And I keep saying it fast like that,

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because it's not space dash time,

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like thinking about the
different dimensions,

00:04:05.504 --> 00:04:07.756
or thinking about two different things.

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We're renaming one thing called spacetime.

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So these are the things
that we're going to start

00:04:13.538 --> 00:04:14.652
to hold into question.

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So what if we loosen these,
and we assume the thing

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that we have observed in the universe,

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that the speed of light is absolute,

00:04:22.709 --> 00:04:26.146
regardless of your inertial
frame of reference.

00:04:26.146 --> 00:04:28.677
So let's word from there, do
a little thought experiment,

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and then think about what type
of a model we can construct,

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in which case the speed
of light is absolute.

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And to do that, we're still going to use

00:04:36.882 --> 00:04:38.275
our little model here,

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and we're going to focus on my
friend's frame of reference,

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my friend who's on a spaceship,

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who right at time equal
zero, is passing me by

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at half the speed of light.

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And we assumed that she's
in this train of spaceships.

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So we assumed already that she's,

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there are these spaceships
that are all moving

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at half the speed of light
in the positive x direction,

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relative to me,

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and they're all three times
ten to the eighth meters apart.

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So they're like that.

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So let's say, and I
did these axes in blue,

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because we're assuming this is my friend's

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frame of reference, so you could say

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this is the S prime frame of reference,

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this is Sally's frame of reference

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from a couple of videos ago.

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Let's say a second before she gets to me,

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and I'm also assuming, the
way I've drawn these axes,

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and we're going to
modify them in the future

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where we're actually going
to use the same units

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for space and time.

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Right now, I'm still sticking
to what we've classically done

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where we use seconds for time

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and meters for our path or our space,

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but in the future, we'll
actually use meters for both.

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But we'll get to that, I don't want to do

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too many things at once.

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But the way I've drawn it

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along my space axis,

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three times ten to the eighth meters

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is going to be the exact
same length as one second,

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and I'm doing that so
that the path of light

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on my diagram

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can be a 45 degree angle.

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So this is three times
ten to the eighth meters

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right over there.

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So let's say a second
before Sally gets to me,

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she releases a photon from her spaceship

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in the direction of the spaceship
that is in front of her.

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How far is it in front of her?

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It's three times ten to the
eighth meters in front of her.

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And let's say on the
back of that spaceship,

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there is a mirror.

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So she's pointing her
flashlight at that mirror.

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So what's going to happen
from her frame of reference

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after one second?

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So for one second, from
her frame of reference,

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she is stationary, so
this is Sally still here.

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She is going to be still
at x prime equals zero

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in her frame of reference,

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and that ship is still going to be

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three times ten to the eighth
meters in front of her.

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They're all moving, relative
to me, at the same velocity,

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but relative to each other,
they seem to be stationary.

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And so what would be
the path of that photon?

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Well that photon will have
gone from Sally's flashlight

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from her headlamp, or whatever it is,

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to that spaceship in front of her

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would've just reached that mirror

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on the back of that spaceship.

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And so we can draw the path of light.

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It will be, so let me,

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so that path of light

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will look like that on this diagram.

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And then right at this moment,

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right at t prime equals zero seconds,

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the photon will be
reflected back to Sally.

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Well how long will it
take to get back to Sally?

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Well Sally's gonna receive
the reflection of that photon

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after one second, because that's how long

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it's going to take it to go

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three times ten to the eighth meters.

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So the path of that very first photon,

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the path of that very first photon

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is going to look like that.

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All right, well hopefully this is pretty

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straightforward here.

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This is what would happen from
Sally's frame of reference.

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A second before she reaches me,

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at t prime equals negative one seconds,

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emits a photon at t equals zero seconds,

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and gets to the spaceship

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that's three times ten to the
eighth meters in front of her,

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essentially one light-second
in front of her,

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and then a second later,

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it's reflecting back, a second later,

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she gets the reflection.

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And so that's what this
diagram is describing.

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But now let's draw it projected on top

00:08:03.482 --> 00:08:05.015
of my frame of reference.

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And this is when things are going to get

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really, really, really,
really interesting.

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So I've drawn my frame of reference here,

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and I've intentionally
not marked off the seconds

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or the meters on my frame of reference,

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because once again,
I'm not going to assume

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that a second in my frame of reference

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is a second in hers, or a
meter in my frame of reference

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is a meter in hers.

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I have drawn her t prime
axis at the same angle

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as I did before,

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because for every second

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we move into the future,

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she's going to move half a
light-second in distance,

00:08:37.944 --> 00:08:39.225
in the positive x direction.

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So this slope right here,
one way to think about it,

00:08:41.896 --> 00:08:42.965
the way I've drawn it,

00:08:42.965 --> 00:08:45.276
this is a slope of two.

00:08:47.506 --> 00:08:50.628
For every unit she moves
in the x direction,

00:08:50.628 --> 00:08:54.413
she will move two in the time direction.

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And what we're going to do again

00:08:55.713 --> 00:08:58.030
is assume that on her axis,

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I haven't drawn the x prime axis.

00:08:59.377 --> 00:09:00.626
In fact, this is an
exercise to think about

00:09:00.626 --> 00:09:03.064
where should the x prime axis be.

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Should it be coincident with the x axis

00:09:04.875 --> 00:09:05.757
like we assumed before?

00:09:05.757 --> 00:09:07.518
Or is it going to be in a different place?

00:09:07.518 --> 00:09:10.258
But we're going to assume
that the lengths I draw

00:09:10.258 --> 00:09:12.231
for one second on, let's say,

00:09:12.231 --> 00:09:15.389
in the S prime frame of reference,

00:09:15.389 --> 00:09:17.757
is going to be the same
as the length I would draw

00:09:17.757 --> 00:09:20.126
for three times ten to the eighth meters.

00:09:20.126 --> 00:09:21.240
And we're also going to assume

00:09:21.240 --> 00:09:22.564
that the speed of light is absolute,

00:09:22.564 --> 00:09:25.443
so it's always going to
moving at a 45 degree angle

00:09:25.443 --> 00:09:29.297
with respect to either frame of reference.

00:09:29.297 --> 00:09:31.015
So that's where things are going
to get a little bit whacky,

00:09:31.015 --> 00:09:32.989
but let's see what's going to happen.

00:09:32.989 --> 00:09:35.288
So at negative one seconds,

00:09:35.288 --> 00:09:38.585
we still have Sally
turning on her flashlight.

00:09:38.585 --> 00:09:41.487
She wants to bounce it off of
the spaceship in front of her.

00:09:41.487 --> 00:09:44.598
And so that photon's going to
move with the speed of light

00:09:44.598 --> 00:09:46.688
in either frame of reference.

00:09:46.688 --> 00:09:48.453
And so let me draw that.

00:09:48.453 --> 00:09:49.747
Let me draw that,

00:09:49.747 --> 00:09:53.467
so it's going to look like this.

00:09:53.467 --> 00:09:55.960
I'm drawing at a 45 degree angle.

00:09:55.960 --> 00:09:57.701
So actually, I don't know
where it gets reflected.

00:09:57.701 --> 00:09:59.861
It's gonna get reflected where
it hits that x prime axis,

00:09:59.861 --> 00:10:01.231
but I don't know where that is.

00:10:01.231 --> 00:10:03.460
But we do know that it
then gets reflected,

00:10:03.460 --> 00:10:06.339
and then it gets back to Sally
at one second in the future.

00:10:06.339 --> 00:10:09.195
So the return path of that photon

00:10:09.195 --> 00:10:11.401
is going to look something

00:10:11.401 --> 00:10:13.397
is going to look something like this.

00:10:13.397 --> 00:10:14.930
And that point that it changes direction,

00:10:14.930 --> 00:10:16.068
I could have,

00:10:16.068 --> 00:10:16.857
let me,

00:10:16.857 --> 00:10:18.357
I could have done it like this,

00:10:18.357 --> 00:10:19.332
whoops,

00:10:19.332 --> 00:10:20.461
I could've done it like this.

00:10:20.461 --> 00:10:22.991
But the interesting point is
where it changes direction

00:10:22.991 --> 00:10:25.308
because that's where that
spaceship in front of her

00:10:25.308 --> 00:10:28.513
must be at that point in spacetime.

00:10:28.513 --> 00:10:30.393
Because now we're going to
start thinking of mixing up

00:10:30.393 --> 00:10:31.601
space and time,

00:10:31.601 --> 00:10:33.783
but I'm not going to get
too much involved in that.

00:10:33.783 --> 00:10:35.641
Now why is this interesting?

00:10:35.641 --> 00:10:38.102
Because from Sally's point of view,

00:10:38.102 --> 00:10:39.704
from Sally's point of view,

00:10:39.704 --> 00:10:44.116
this point here where the
light changes direction,

00:10:44.116 --> 00:10:45.904
from Sally's frame of reference,

00:10:45.904 --> 00:10:50.037
is happening simultaneously
with when she reaches me.

00:10:50.037 --> 00:10:52.928
This is happening at t prime

00:10:52.928 --> 00:10:55.528
is equal to zero for Sally.

00:10:55.528 --> 00:10:59.476
So anything that is t prime
is equal to zero for Sally

00:10:59.476 --> 00:11:02.007
must be on the x prime axis.

00:11:02.007 --> 00:11:05.118
So this must sit on the x prime axis.

00:11:05.118 --> 00:11:06.743
Once again, why do I know that?

00:11:06.743 --> 00:11:08.879
Because everything on the x prime axis,

00:11:08.879 --> 00:11:11.341
any event on the x prime axis,

00:11:11.341 --> 00:11:13.407
let me do it in a different color,

00:11:13.407 --> 00:11:15.984
I keep doing black.

00:11:15.984 --> 00:11:18.771
Any event on the x prime axis

00:11:18.771 --> 00:11:22.184
is going to be at t prime
is equal to zero seconds,

00:11:22.184 --> 00:11:24.088
and from Sally's frame of reference

00:11:24.088 --> 00:11:27.013
is going to be simultaneous
with when she passes me up.

00:11:27.013 --> 00:11:29.591
So based on that, we know
that this is going to be,

00:11:32.467 --> 00:11:34.603
we know that this point,

00:11:34.603 --> 00:11:36.275
which is where Sally is,

00:11:36.275 --> 00:11:38.876
that's going to be the origin
from her frame of reference.

00:11:38.876 --> 00:11:43.078
And we know that this point
sits on the x prime axis,

00:11:43.078 --> 00:11:45.934
so based on that, we can
draw the x prime axis.

00:11:45.934 --> 00:11:47.968
You just need two points to define a line,

00:11:47.968 --> 00:11:50.220
and so let me try to do it.

00:11:50.220 --> 00:11:52.727
So let me try to draw.

00:11:52.727 --> 00:11:54.074
I can do a better job than that.

00:11:54.074 --> 00:11:55.792
Let me try to draw

00:11:55.792 --> 00:11:57.186
the x prime axis.

00:11:57.186 --> 00:12:00.204
It's going to look like,

00:12:00.204 --> 00:12:02.108
it's going to look like this.

00:12:02.108 --> 00:12:04.708
That right over there is the x prime axis.

00:12:04.708 --> 00:12:07.704
Now at this point, you should
find this mildly mind-blowing.

00:12:07.704 --> 00:12:10.281
Actually, even more than
mildly mind-blowing.

00:12:10.281 --> 00:12:12.533
'Cause it's saying some
pretty, pretty crazy things.

00:12:12.533 --> 00:12:15.714
First of all, let's just make
sure we know how to read this.

00:12:15.714 --> 00:12:17.270
So for any event,

00:12:17.270 --> 00:12:18.508
and now we're going to start thinking

00:12:18.508 --> 00:12:20.033
in terms of spacetime,

00:12:20.033 --> 00:12:23.385
although I'm still using different
units for space and time,

00:12:23.385 --> 00:12:25.126
but we'll address that in the future.

00:12:25.126 --> 00:12:27.727
If I want to read its coordinates
in my frame of reference,

00:12:27.727 --> 00:12:29.492
well if I want to read its x coordinate,

00:12:29.492 --> 00:12:31.558
I go parallel to the t axis,

00:12:31.558 --> 00:12:33.648
and if I wanna read its t coordinate,

00:12:33.648 --> 00:12:35.645
I go parallel to the x axis.

00:12:35.645 --> 00:12:36.945
But for Sally's frame of reference,

00:12:36.945 --> 00:12:38.617
well I essentially do the same thing.

00:12:38.617 --> 00:12:40.567
If I want its x prime coordinates,

00:12:40.567 --> 00:12:43.585
I go parallel to the t prime axis,

00:12:43.585 --> 00:12:46.163
and if I want the t prime coordinates,

00:12:46.163 --> 00:12:49.971
I go parallel to the x prime axis.

00:12:49.971 --> 00:12:51.805
But what's really interesting,

00:12:51.805 --> 00:12:55.265
and I'll go even deeper into
this into the next videos,

00:12:55.265 --> 00:12:56.621
that moment,

00:12:56.621 --> 00:12:58.920
that moment right over here,

00:12:58.920 --> 00:13:01.312
that from Sally's frame of reference,

00:13:01.312 --> 00:13:04.887
it looks like it's simultaneous
with her passing me up.

00:13:04.887 --> 00:13:07.116
It looks like it's happening
at t prime equals zero.

00:13:07.116 --> 00:13:08.742
In fact, it is happening
at t prime equals zero.

00:13:08.742 --> 00:13:09.991
From her frame of reference,

00:13:09.991 --> 00:13:12.057
it's happening at t prime equals zero.

00:13:12.057 --> 00:13:14.054
From our frame of reference,

00:13:14.054 --> 00:13:17.398
it is happening after Sally passes us up.

00:13:17.398 --> 00:13:20.671
Notice, it is happening at t
equals some positive value.

00:13:20.671 --> 00:13:23.239
It's not happening at t equals zero.

00:13:23.239 --> 00:13:25.434
So this is starting to
get a little bit whacky.

00:13:25.434 --> 00:13:30.378
One second, and simultaneous,
time, and space,

00:13:30.378 --> 00:13:33.391
and things being simultaneous,

00:13:33.391 --> 00:13:35.829
we're not going to agree on those,

00:13:35.829 --> 00:13:37.895
depending on which frame
of reference we're in.

00:13:37.895 --> 00:13:41.656
The thing we will agree
on is the speed of light.


Galilean transformation and contradictions with light

WEBVTT
Kind: captions
Language: en

00:00:00.333 --> 00:00:01.856
- [Voiceover] Things are
starting to get interesting.

00:00:01.856 --> 00:00:04.404
In the first video, we set
up a space-time diagram

00:00:04.404 --> 00:00:07.173
from my frame of reference
and started to plot things

00:00:07.173 --> 00:00:09.609
past in that space-time diagram.

00:00:09.609 --> 00:00:12.612
And then we thought about
our friend, Sally, who

00:00:12.612 --> 00:00:15.615
right at time equals
zero is at x equals zero,

00:00:15.615 --> 00:00:18.618
but she's passing me up
at a relative velocity

00:00:18.618 --> 00:00:21.321
of half the speed of light
in the positive x direction.

00:00:21.321 --> 00:00:24.791
So after one second, she
has gone 1 1/2 times 10

00:00:24.791 --> 00:00:27.060
to the 8th meters in the
positive x direction.

00:00:27.060 --> 00:00:29.596
After two seconds, she's gone
three times 10 to the 8th.

00:00:29.596 --> 00:00:31.032
And then we assume that
she was part of a train

00:00:31.032 --> 00:00:31.994
of spaceships.

00:00:31.994 --> 00:00:34.240
So three times 10 to the
8th meters in front of her

00:00:34.240 --> 00:00:36.402
was another spaceship and
it's moving with the same

00:00:36.402 --> 00:00:39.160
relative velocity relative
to me, or in my frame

00:00:39.160 --> 00:00:40.084
of reference.

00:00:40.084 --> 00:00:42.844
So at time equals zero is three
times 10 to the 8th meters

00:00:42.844 --> 00:00:43.924
in front of me.

00:00:43.924 --> 00:00:46.398
But then at time equals
one, if we take this point

00:00:46.398 --> 00:00:50.268
in space-time, in my frame of reference,

00:00:50.268 --> 00:00:52.908
that ship is now, it now has,

00:00:52.908 --> 00:00:54.760
so t, let me write this.

00:00:54.760 --> 00:00:57.175
If we look at my space-time coordinates,

00:00:57.175 --> 00:00:59.555
t is equal to one second

00:00:59.555 --> 00:01:03.797
and x is equal to 4.5

00:01:03.797 --> 00:01:06.399
times 10 to the 8th meters.

00:01:06.399 --> 00:01:09.035
That's if you look at my
space-time coordinates.

00:01:09.035 --> 00:01:11.805
But what about Sally's
space-time coordinates?

00:01:11.805 --> 00:01:13.807
Well, to figure out Sally's
space-time coordinates,

00:01:13.807 --> 00:01:16.643
we go parallel to the x
prime axis and see where we

00:01:16.643 --> 00:01:19.446
intersect the t prime
axis, but that's still t

00:01:19.446 --> 00:01:22.081
is equal to one second, this
is t is equal to two seconds,

00:01:22.081 --> 00:01:24.317
this is t is equal to three seconds.

00:01:24.317 --> 00:01:27.220
So t, or I should say t
prime is equal to one second,

00:01:27.220 --> 00:01:30.089
t prime is two seconds,
t prime is three seconds.

00:01:30.089 --> 00:01:33.560
So t prime is equal to one second still.

00:01:33.560 --> 00:01:36.162
And so in general, we can say that t prime

00:01:36.162 --> 00:01:40.433
is going to be equal to
t, but what is x prime

00:01:40.433 --> 00:01:42.335
going to be equal to?

00:01:42.335 --> 00:01:44.771
So x prime, well, to figure out x prime,

00:01:44.771 --> 00:01:46.773
you go horizontal to the t prime axis

00:01:46.773 --> 00:01:49.008
and see where you
intersect the x prime axis,

00:01:49.008 --> 00:01:52.778
and you see that it's three times 10

00:01:52.778 --> 00:01:55.855
to the 8th meters.

00:01:55.855 --> 00:01:57.675
And hopefully, this makes intuitive sense.

00:01:57.675 --> 00:01:59.985
If it doesn't, pause the video
and really think about it

00:01:59.985 --> 00:02:03.482
because in her frame of
reference, that spaceship

00:02:03.482 --> 00:02:06.247
looks stationary because it's
moving with the exact same

00:02:06.247 --> 00:02:08.627
relative velocity to me.

00:02:08.627 --> 00:02:10.964
It's going to continue
to stay three times 10

00:02:10.964 --> 00:02:12.959
to the 8th meters in front
of her, which is exactly

00:02:12.959 --> 00:02:13.800
what we see.

00:02:13.800 --> 00:02:16.102
So that's why its x prime coordinates stay

00:02:16.102 --> 00:02:18.104
three times 10 to the 8th meters.

00:02:18.104 --> 00:02:20.106
From my point of view, it's
getting further and further

00:02:20.106 --> 00:02:22.976
away from me at the relative velocity,

00:02:22.976 --> 00:02:25.845
at 1.5 times 10 to the
8th meters per second.

00:02:25.845 --> 00:02:29.381
So how do we translate between our,

00:02:30.671 --> 00:02:33.152
between our x coordinates?

00:02:33.152 --> 00:02:35.592
Let me do this in a different color.

00:02:35.592 --> 00:02:40.432
Between our x coordinates
and our x prime coordinates?

00:02:40.432 --> 00:02:42.795
Well, you see for these examples,

00:02:42.795 --> 00:02:46.932
we see x prime is going to be less than x,

00:02:47.400 --> 00:02:50.403
and that should also make sense because

00:02:51.063 --> 00:02:53.373
especially for this case, it's stationary

00:02:53.373 --> 00:02:56.075
from the s prime point of view,

00:02:56.075 --> 00:03:00.046
but its x is continuously
increasing as time passes on

00:03:00.046 --> 00:03:03.259
from my frame of reference,
from Sally's frame of reference,

00:03:03.259 --> 00:03:05.084
from the S frame of reference.

00:03:05.084 --> 00:03:08.054
So if we start with x, we
should subtract something,

00:03:08.054 --> 00:03:09.525
and the difference between the two,

00:03:09.525 --> 00:03:12.058
the discrepancy between
the two is going to be

00:03:12.058 --> 00:03:14.903
the relative velocity times time.

00:03:14.903 --> 00:03:17.397
And so for this particular example,

00:03:17.397 --> 00:03:20.999
we saw that three times
10 to the 8th meters

00:03:22.368 --> 00:03:26.639
was equal to 4.5 times 10 to the 8th,

00:03:26.639 --> 00:03:30.109
minus the relative velocity, 1.5 times 10

00:03:30.109 --> 00:03:33.479
to the 8th meters per second times time,

00:03:33.479 --> 00:03:34.714
which was one second.

00:03:34.714 --> 00:03:36.077
And I didn't write the units
but if you write the units,

00:03:36.077 --> 00:03:39.087
it all works out and you get exactly this.

00:03:39.485 --> 00:03:41.621
So hopefully that's starting
to get you comfortable

00:03:41.621 --> 00:03:44.724
with having these two coordinate planes

00:03:46.144 --> 00:03:49.729
or two space-time diagrams
over on top of each other.

00:03:49.729 --> 00:03:52.065
And the reason why the
blue one is distorted

00:03:52.065 --> 00:03:54.434
is because it's on top,
they're moving with a relative

00:03:54.434 --> 00:03:57.070
velocity relative to what I'm considering

00:03:57.070 --> 00:04:01.074
to be a stationary frame of
reference, which is mine.

00:04:01.074 --> 00:04:03.109
Obviously, there's no
such thing as an absolute

00:04:03.109 --> 00:04:05.111
stationary frame of
reference, and we'll talk more

00:04:05.111 --> 00:04:06.112
about that in the future.

00:04:06.112 --> 00:04:09.315
But what I now want to
focus on is that photon

00:04:09.315 --> 00:04:12.485
that I emitted at time equal zero.

00:04:12.485 --> 00:04:14.754
Because we saw it moves
with the speed of light

00:04:14.754 --> 00:04:16.221
in my frame of reference.

00:04:16.221 --> 00:04:18.458
After one second, it has moved,

00:04:18.458 --> 00:04:21.427
its x coordinate is three
times 10 to the 8th meters.

00:04:21.427 --> 00:04:25.794
After two seconds, after two seconds,

00:04:25.794 --> 00:04:29.068
the photon is at six times
10 to the 8th meters.

00:04:29.068 --> 00:04:31.170
But let's see what that photon looks like

00:04:31.170 --> 00:04:33.940
from the s prime frame of
reference, from Sally's

00:04:33.940 --> 00:04:35.441
frame of reference.

00:04:35.441 --> 00:04:37.610
Well, from Sally's frame of reference,

00:04:37.610 --> 00:04:40.413
let's think about that
photon after two seconds.

00:04:40.413 --> 00:04:42.415
So the photon is right over there.

00:04:42.415 --> 00:04:46.786
So t prime is equal to
two seconds, two seconds,

00:04:46.786 --> 00:04:48.354
but what is x prime?

00:04:48.354 --> 00:04:50.356
What is x prime going to be equal to?

00:04:50.356 --> 00:04:53.693
Well, x prime, we go
parallel to the t prime axis,

00:04:53.693 --> 00:04:56.129
is three times 10 to the 8th meters.

00:04:56.129 --> 00:04:58.398
Three times 10 to the 8th meters.

00:04:58.398 --> 00:05:02.869
So in her frame of reference,
it took that photon of light

00:05:02.869 --> 00:05:05.938
two seconds to go three
times 10 to the 8th meters,

00:05:05.938 --> 00:05:09.175
or it looks like the
velocity of that photon

00:05:09.175 --> 00:05:13.146
is one and a half times 10
to the 8th meters per second

00:05:13.146 --> 00:05:15.214
in the positive x direction.

00:05:15.214 --> 00:05:17.950
And this should hopefully
makes sense from a Newtonian

00:05:17.950 --> 00:05:21.854
point of view, or a
Galilean point of view.

00:05:21.854 --> 00:05:24.824
These are called Galilean transformations

00:05:24.824 --> 00:05:27.760
because if I'm in a car

00:05:27.760 --> 00:05:30.830
and there's another car and
you see this on the highway

00:05:30.830 --> 00:05:32.999
all the time, if I'm in a
car going 60 miles per hour,

00:05:32.999 --> 00:05:35.401
there's another car
going 65 miles per hour,

00:05:35.401 --> 00:05:37.040
from my point of view,
it looks like it's only

00:05:37.040 --> 00:05:39.405
moving forward at five miles per hour.

00:05:39.405 --> 00:05:41.841
So that photon will look slower to Sally.

00:05:41.841 --> 00:05:45.845
Similarly, if we assume this
Newtonian, this Galilean world,

00:05:45.845 --> 00:05:49.691
if she had a flashlight,

00:05:49.691 --> 00:05:51.756
if she had a flashlight right over here

00:05:51.756 --> 00:05:53.786
and right at time equals
zero she turned it on,

00:05:53.786 --> 00:05:57.127
and that first photon we
were to plot it on her

00:05:57.127 --> 00:06:01.168
frame of reference, well, it
should go the speed of light

00:06:01.168 --> 00:06:04.440
in her frame of reference.

00:06:04.440 --> 00:06:07.077
So it starts here at the origin.

00:06:07.077 --> 00:06:09.772
And then after one second, in the s prime,

00:06:09.772 --> 00:06:12.672
in the s prime coordinates,
it should have gone

00:06:12.672 --> 00:06:14.741
three times 10 to the 8th meters.

00:06:14.741 --> 00:06:17.176
After two seconds, it
should've gone six times 10

00:06:17.176 --> 00:06:18.621
to the 8th meters.

00:06:18.621 --> 00:06:21.714
And so it's path on her space-time diagram

00:06:21.714 --> 00:06:25.993
should look like that.

00:06:25.993 --> 00:06:28.821
That's her photon, that
first photon that was emitted

00:06:28.821 --> 00:06:29.889
from it.

00:06:29.889 --> 00:06:32.058
So you might be noticing
something interesting.

00:06:32.058 --> 00:06:34.560
That photon from my point
of view is going faster

00:06:34.560 --> 00:06:36.062
than the speed of light.

00:06:36.062 --> 00:06:40.166
After one second, its x coordinate is 4.5

00:06:40.166 --> 00:06:41.734
times 10 to the 8th meters.

00:06:41.734 --> 00:06:44.704
It's going 4.5 times 10 to
the 8th meters per second.

00:06:44.704 --> 00:06:46.906
It's going faster than the speed of light.

00:06:46.906 --> 00:06:50.162
It's going faster than my photon,

00:06:50.910 --> 00:06:53.813
and that might make intuitive sense

00:06:53.813 --> 00:06:56.616
except it's not what we
actually observe in nature.

00:06:56.616 --> 00:06:58.031
And anytime we try to make a prediction

00:06:58.031 --> 00:07:00.061
that's not what's observed
in nature, it means that

00:07:00.061 --> 00:07:02.755
our understanding of the
universe is not complete

00:07:02.755 --> 00:07:06.636
because it turns out
that regardless of which

00:07:06.636 --> 00:07:08.913
inertial reference frame we are in,

00:07:08.913 --> 00:07:11.898
the speed of light,
regardless of the speed

00:07:11.898 --> 00:07:14.767
or the relative velocity of
the source of that light,

00:07:14.767 --> 00:07:19.172
is always going three times 10
to the 8th meters per second.

00:07:19.172 --> 00:07:22.041
So we know from
observations of the universe

00:07:22.041 --> 00:07:24.801
that Sally, when she looked at my photon,

00:07:24.801 --> 00:07:26.551
she wouldn't see it going
half the speed of light,

00:07:26.551 --> 00:07:28.753
she would see it going three
times 10 to the 8th meters

00:07:28.753 --> 00:07:29.840
per second.

00:07:29.840 --> 00:07:31.695
And we know from
observations of the universe

00:07:31.695 --> 00:07:33.725
that Sally's photon,
I would not observe it

00:07:33.725 --> 00:07:37.365
as moving at 4.5 times 10 to
the 8th meters per second,

00:07:37.365 --> 00:07:39.185
that it would actually still be moving

00:07:39.185 --> 00:07:41.915
at three times 10 to the
8th meters per second.

00:07:41.915 --> 00:07:43.770
So something has got to give.

00:07:43.770 --> 00:07:47.410
This is breaking down our
classical, our Newtonian,

00:07:47.410 --> 00:07:51.120
our Galilean views of the world.

00:07:51.120 --> 00:07:52.205
It's very exciting.

00:07:52.205 --> 00:07:55.527
We need to think of some other
way to conceptualize things,

00:07:55.527 --> 00:07:59.762
some other way to visualize
these space-time diagrams

00:07:59.762 --> 00:08:02.287
for the different frames of reference.


Starting to set up a Newtonian path–time diagram

WEBVTT
Kind: captions
Language: en

00:00:00.523 --> 00:00:03.083
- [Voiceover] Let's say
this is me right over here,

00:00:03.083 --> 00:00:05.440
and I'm drifting through space

00:00:05.440 --> 00:00:06.984
at a constant velocity

00:00:06.984 --> 00:00:08.751
relative to any other

00:00:08.751 --> 00:00:10.539
inertial frame of reference,

00:00:10.539 --> 00:00:11.625
and so I am,

00:00:11.625 --> 00:00:14.506
I am in an inertial frame
of reference myself,

00:00:14.506 --> 00:00:16.233
and in fact I'm going to define

00:00:16.233 --> 00:00:17.762
my frame of reference by me,

00:00:17.762 --> 00:00:19.142
I'm gonna say I'm at the origin

00:00:19.142 --> 00:00:20.869
of my frame of reference.

00:00:20.869 --> 00:00:22.374
So at all times,

00:00:22.374 --> 00:00:25.237
I consider myself to be stationary,

00:00:25.237 --> 00:00:26.640
and I am at the point,

00:00:26.640 --> 00:00:28.520
'x' equals zero,

00:00:28.520 --> 00:00:29.641
and we're gonna focus on

00:00:29.641 --> 00:00:30.848
just the 'x' dimension,

00:00:30.848 --> 00:00:32.142
to simplify our discussion,

00:00:32.142 --> 00:00:33.136
and I have, you know,

00:00:33.136 --> 00:00:34.274
my oxygen and everything,

00:00:34.274 --> 00:00:35.165
and food,

00:00:35.165 --> 00:00:38.725
so no need to worry about me.

00:00:38.725 --> 00:00:40.356
Now, what I've drawn here

00:00:40.356 --> 00:00:42.145
are some axes,

00:00:42.145 --> 00:00:45.209
so that I can plot the path of things

00:00:45.209 --> 00:00:47.016
as time progresses

00:00:47.016 --> 00:00:49.760
in my frame of reference.

00:00:49.760 --> 00:00:50.877
And one thing that

00:00:50.877 --> 00:00:52.136
many of ya'll might have noticed

00:00:52.136 --> 00:00:54.108
is that I have plotted time in seconds

00:00:54.108 --> 00:00:55.510
on the vertical axis,

00:00:55.510 --> 00:00:57.418
and our position, our 'x' position,

00:00:57.418 --> 00:01:00.528
in meters on the horizontal axis.

00:01:00.528 --> 00:01:01.929
And that might be a little
bit counter-intuitive

00:01:01.929 --> 00:01:03.003
for a lot of ya'll,

00:01:03.003 --> 00:01:04.241
especially with math backgrounds,

00:01:04.241 --> 00:01:06.237
in fact, it's a little
bit uncomfortable for me,

00:01:06.237 --> 00:01:09.777
we often prefer to put time
on the horizontal axis,

00:01:09.777 --> 00:01:12.803
and position on the vertical axis,

00:01:12.803 --> 00:01:14.775
so this is a little bit counter-intuitive.

00:01:14.775 --> 00:01:15.933
But this is,

00:01:15.933 --> 00:01:18.188
you know, our choice is
a little bit arbitrary,

00:01:18.188 --> 00:01:20.646
in math class, we liked to think

00:01:20.646 --> 00:01:22.211
of our independent variable

00:01:22.211 --> 00:01:23.389
on our horizontal axis,

00:01:23.389 --> 00:01:26.173
and we think of time as
somehow driving position,

00:01:26.173 --> 00:01:28.408
so that's why we tend to put time there

00:01:28.408 --> 00:01:29.765
than we'd position there,

00:01:29.765 --> 00:01:31.719
but you can flip them around.

00:01:31.719 --> 00:01:33.945
And as we'll get more into physics,

00:01:33.945 --> 00:01:35.510
we'll see, well maybe we shouldn't

00:01:35.510 --> 00:01:38.578
necessarily think of time
always driving position,

00:01:38.578 --> 00:01:42.073
or maybe position in some
ways are driving time.

00:01:42.073 --> 00:01:43.942
But we'll get into that.

00:01:43.942 --> 00:01:46.826
But let's just first get a
little bit comfortable with this.

00:01:46.826 --> 00:01:50.179
So what would be my position over time

00:01:50.179 --> 00:01:52.454
on this diagram right over here?

00:01:52.454 --> 00:01:53.463
And you might notice these numbers,

00:01:53.463 --> 00:01:54.437
one second, two second,

00:01:54.437 --> 00:01:55.363
three second,

00:01:55.363 --> 00:01:56.378
and then in meters

00:01:56.378 --> 00:01:57.497
I have three times to the eighth,

00:01:57.497 --> 00:01:58.329
six times to the eighth,

00:01:58.329 --> 00:01:59.284
nine times to the eighth,

00:01:59.284 --> 00:02:00.828
so these are massive, massive numbers,

00:02:00.828 --> 00:02:03.083
and you could guess where
they are coming from.

00:02:03.083 --> 00:02:04.730
They are coming from the notion

00:02:04.730 --> 00:02:05.948
that the speed of light

00:02:05.948 --> 00:02:08.792
is approximately 3x10 to the eighth

00:02:08.792 --> 00:02:10.763
meters per second.

00:02:10.763 --> 00:02:12.897
But we'll get to that in a second.

00:02:12.897 --> 00:02:14.258
But let's just think about

00:02:14.258 --> 00:02:17.122
my position over time,

00:02:17.122 --> 00:02:19.255
in my frame of reference here.

00:02:19.255 --> 00:02:21.390
So, time equals zero,

00:02:21.390 --> 00:02:23.380
my 'x' position in this frame of reference

00:02:23.380 --> 00:02:24.578
is zero,

00:02:24.578 --> 00:02:26.631
I consider myself to be stationary.

00:02:26.631 --> 00:02:28.054
After one second,

00:02:28.054 --> 00:02:30.024
well my position is still zero,

00:02:30.024 --> 00:02:31.650
after two seconds my position

00:02:31.650 --> 00:02:33.498
is still 'x' equals zero,

00:02:33.498 --> 00:02:34.758
after three seconds my position

00:02:34.758 --> 00:02:36.424
is still 'x' equals zero,

00:02:36.424 --> 00:02:38.517
so my, I guess you could almost

00:02:38.517 --> 00:02:39.858
consider my path

00:02:39.858 --> 00:02:41.096
on this diagram,

00:02:41.096 --> 00:02:43.737
where I'm plotting time and space,

00:02:43.737 --> 00:02:45.384
at least in the 'x' direction,

00:02:45.384 --> 00:02:47.444
is going to look like,

00:02:47.444 --> 00:02:50.204
is going to look like,

00:02:50.204 --> 00:02:52.568
whoops, I can do a better job than that.

00:02:52.568 --> 00:02:54.008
It's going to look,

00:02:54.008 --> 00:02:56.448
it's going to look like,

00:02:56.448 --> 00:02:58.683
it's going to look like that.

00:02:58.683 --> 00:03:00.709
That would be my path

00:03:00.709 --> 00:03:04.116
on my little time and space axes,

00:03:04.116 --> 00:03:06.615
or on this little diagram that I have.

00:03:06.615 --> 00:03:08.180
But let's think about

00:03:08.180 --> 00:03:10.272
something a little bit more interesting.

00:03:10.272 --> 00:03:13.828
Let's think about me having a flashlight,

00:03:13.828 --> 00:03:16.225
so this is, I have a flashlight here,

00:03:16.225 --> 00:03:18.480
and I, at time equals zero,

00:03:18.480 --> 00:03:20.085
I turn on that flashlight,

00:03:20.085 --> 00:03:21.447
exactly at time equals zero

00:03:21.447 --> 00:03:22.808
I turn on that flashlight.

00:03:22.808 --> 00:03:25.916
And let's think about
the very first photon

00:03:25.916 --> 00:03:28.720
of light that emits from my flashlight.

00:03:28.720 --> 00:03:30.447
And let's plot it's path

00:03:30.447 --> 00:03:31.971
on my little diagram here.

00:03:31.971 --> 00:03:33.352
So time equals zero,

00:03:33.352 --> 00:03:35.140
it's exactly where I am.

00:03:35.140 --> 00:03:36.217
So time equals zero,

00:03:36.217 --> 00:03:37.985
it's position is zero meters.

00:03:37.985 --> 00:03:39.285
But after one second,

00:03:39.285 --> 00:03:40.484
where is my photon of light?

00:03:40.484 --> 00:03:41.703
And I'm gonna put it,

00:03:41.703 --> 00:03:43.084
and the photon is moving

00:03:43.084 --> 00:03:45.244
in the positive 'x' direction.

00:03:45.244 --> 00:03:48.246
So it's moving in the
positive 'x' direction

00:03:48.246 --> 00:03:51.321
with a velocity of 'C' meters per second

00:03:51.321 --> 00:03:53.989
in the positive 'x' direction.

00:03:53.989 --> 00:03:55.960
And so, after one second,

00:03:55.960 --> 00:03:59.779
it will have moved 3x10
to the eighth meters.

00:03:59.779 --> 00:04:01.526
After two seconds,

00:04:01.526 --> 00:04:04.330
it will have moved 6x10
to the eighth meters.

00:04:04.330 --> 00:04:06.545
And I've intentionally scaled my axes,

00:04:06.545 --> 00:04:10.419
so that now if I were to
draw the position of light

00:04:10.419 --> 00:04:11.720
relative to time,

00:04:11.720 --> 00:04:14.117
it's going to be at a 45 degree angle.

00:04:14.117 --> 00:04:16.210
And that's just an artifact

00:04:16.210 --> 00:04:18.962
of how I've picked my axes.

00:04:18.962 --> 00:04:21.116
But let's see if I can plot this.

00:04:21.116 --> 00:04:23.757
So it's gonna look something like this.

00:04:23.757 --> 00:04:24.854
And this is hand drawn,

00:04:24.854 --> 00:04:26.358
so it's not going to be perfect.

00:04:26.358 --> 00:04:29.934
So the position of that photon over time

00:04:29.934 --> 00:04:31.214
is going to look like that.

00:04:31.214 --> 00:04:33.508
After one second,

00:04:33.508 --> 00:04:36.576
3x10 to the eighth meters,

00:04:36.576 --> 00:04:40.152
after two seconds, 6x10
to the eighth meters,

00:04:40.152 --> 00:04:42.434
after three seconds,

00:04:42.434 --> 00:04:45.300
we have 9x10

00:04:45.300 --> 00:04:47.373
to the eighth meters.

00:04:47.373 --> 00:04:50.217
Alright, well that's
reasonably interesting so far,

00:04:50.217 --> 00:04:51.882
we're getting a little bit comfortable

00:04:51.882 --> 00:04:53.833
with my little diagram here,

00:04:53.833 --> 00:04:55.074
maybe some of us are getting

00:04:55.074 --> 00:04:56.212
a little bit more comfortable

00:04:56.212 --> 00:04:57.775
with time on the vertical axis,

00:04:57.775 --> 00:05:00.701
and my position on the horizontal axis.

00:05:00.701 --> 00:05:03.262
But now let's introduce another character.

00:05:03.262 --> 00:05:05.070
Another character who could almost define,

00:05:05.070 --> 00:05:07.914
or who could define their
own frame of reference.

00:05:07.914 --> 00:05:11.511
So let's say that right
at time equals zero,

00:05:11.511 --> 00:05:14.496
a spaceship passes me by.

00:05:14.496 --> 00:05:15.654
So that's,

00:05:15.654 --> 00:05:17.727
I have to draw a bigger
spaceship than that.

00:05:17.727 --> 00:05:18.865
So, there you go,

00:05:18.865 --> 00:05:20.572
that's the spaceship,

00:05:20.572 --> 00:05:23.112
and in the spaceship I have my friend,

00:05:23.112 --> 00:05:26.402
so that's the spaceship.

00:05:26.402 --> 00:05:28.699
And the spaceship is travelling

00:05:28.699 --> 00:05:31.401
with a pretty incredible velocity,

00:05:31.401 --> 00:05:34.001
in the positive 'x' direction.

00:05:34.001 --> 00:05:35.545
So I guess if this is,

00:05:35.545 --> 00:05:37.801
this vector is the speed of light,

00:05:37.801 --> 00:05:39.506
which this seems like
an awfully short arrow,

00:05:39.506 --> 00:05:41.193
but if I'm gonna say
that's the speed of light,

00:05:41.193 --> 00:05:42.636
then let's say that this thing

00:05:42.636 --> 00:05:45.219
is moving with half the speed of light.

00:05:45.219 --> 00:05:48.023
Which is still incredibly fast.

00:05:48.023 --> 00:05:49.222
So the velocity,

00:05:49.222 --> 00:05:50.329
the magnitude of

00:05:50.329 --> 00:05:54.679
the velocity here is 0.5C.

00:05:54.679 --> 00:05:56.248
So let's plot,

00:05:56.248 --> 00:05:58.910
and my friend is stationary
in that spaceship,

00:05:58.910 --> 00:05:59.986
the spaceship is moving

00:05:59.986 --> 00:06:02.608
at this constant velocity, 0.5C

00:06:02.608 --> 00:06:04.945
in the positive 'x' direction.

00:06:04.945 --> 00:06:08.581
So, they also are in an
inertial frame of reference,

00:06:08.581 --> 00:06:10.572
because it's at a constant velocity

00:06:10.572 --> 00:06:11.507
relative to me,

00:06:11.507 --> 00:06:13.092
which is an inertial frame of reference.

00:06:13.092 --> 00:06:15.084
And so let's plot my,

00:06:15.084 --> 00:06:17.602
let's plot her position

00:06:17.602 --> 00:06:19.938
on my little diagram.

00:06:19.938 --> 00:06:21.443
So time equals zero,

00:06:21.443 --> 00:06:23.840
she's exactly where I am.

00:06:23.840 --> 00:06:26.582
She's at the origin of
my frame of reference.

00:06:26.582 --> 00:06:29.162
Her 'x' position is zero.

00:06:29.162 --> 00:06:30.544
Now after one second,

00:06:30.544 --> 00:06:33.206
she's moving half the speed of light,

00:06:33.206 --> 00:06:35.379
so after one second she

00:06:35.379 --> 00:06:39.341
would have gone 1.5x10 to
the eighth meters per second.

00:06:39.341 --> 00:06:43.059
So she would have gone about that far.

00:06:43.059 --> 00:06:45.091
After two seconds,

00:06:45.091 --> 00:06:47.712
she would have gone as far as light can go

00:06:47.712 --> 00:06:48.729
in one second,

00:06:48.729 --> 00:06:50.537
because she's going
half the speed of light,

00:06:50.537 --> 00:06:52.812
so she would go about that far.

00:06:52.812 --> 00:06:54.397
After three seconds,

00:06:54.397 --> 00:06:58.381
she will have gone 4.5x10

00:06:58.381 --> 00:07:00.249
to the eighth meters.

00:07:00.249 --> 00:07:02.221
So let me put that

00:07:02.221 --> 00:07:03.786
right over there,

00:07:03.786 --> 00:07:05.146
and you see where this is going.

00:07:05.146 --> 00:07:07.279
So if I were to plot her path,

00:07:07.279 --> 00:07:08.681
if I were to plot her path,

00:07:08.681 --> 00:07:12.178
it would look like this.

00:07:12.178 --> 00:07:15.091
It would look like this,

00:07:15.091 --> 00:07:18.078
path of my friend.

00:07:18.078 --> 00:07:20.739
That's the path of my
friend right over there.

00:07:20.739 --> 00:07:23.361
And, I want you to just
kind of think about this,

00:07:23.361 --> 00:07:24.640
think about this a little bit,

00:07:24.640 --> 00:07:26.428
think about how these different lines are,

00:07:26.428 --> 00:07:27.505
what they represent,

00:07:27.505 --> 00:07:28.764
once again get familiar with

00:07:28.764 --> 00:07:30.228
the 't' on the vertical axis.

00:07:30.228 --> 00:07:31.873
But now in the next video,

00:07:31.873 --> 00:07:33.072
we're gonna use this,

00:07:33.072 --> 00:07:34.373
what I just drew,

00:07:34.373 --> 00:07:35.835
to draw her frame of reference,

00:07:35.835 --> 00:07:37.135
to draw a coordinate system,

00:07:37.135 --> 00:07:38.943
or I guess you could say

00:07:38.943 --> 00:07:41.077
a space time diagram for
her frame of reference,

00:07:41.077 --> 00:07:42.479
on top of this one.

00:07:42.479 --> 00:07:44.449
And at least in the context of this video

00:07:44.449 --> 00:07:45.568
and the next few videos,

00:07:45.568 --> 00:07:47.192
we're going to assume a Newtonian world,

00:07:47.192 --> 00:07:48.270
we're not gonna get to Einstein

00:07:48.270 --> 00:07:49.406
in special relativity,

00:07:49.406 --> 00:07:50.605
we're just going to assume,

00:07:50.605 --> 00:07:51.418
well,

00:07:51.418 --> 00:07:54.082
we'll get to that in the next video.


Visualizing multiple Newtonian path–time diagrams

WEBVTT
Kind: captions
Language: en

00:00:00.213 --> 00:00:01.072
- [Voiceover] In the last video,

00:00:01.072 --> 00:00:04.021
we started to construct
a space-time diagram

00:00:04.021 --> 00:00:05.994
for my frame of reference

00:00:05.994 --> 00:00:08.061
and I'm just drifting through space

00:00:08.061 --> 00:00:10.569
and I'm assuming that I'm in
an intertial frame of reference

00:00:10.569 --> 00:00:12.496
which means I'm moving
at a constant velocity

00:00:12.496 --> 00:00:15.027
relative to all other frames of reference

00:00:15.027 --> 00:00:17.348
and we set up a situation where

00:00:17.348 --> 00:00:19.995
I emitted a photon right at time zero,

00:00:19.995 --> 00:00:21.714
so after one second, it would have moved

00:00:21.714 --> 00:00:23.409
3 times 10 to the 8th meters.

00:00:23.409 --> 00:00:24.570
After two seconds, it would have moved

00:00:24.570 --> 00:00:26.543
6 times 10 to the 8th meters,

00:00:26.543 --> 00:00:27.913
and then we had a,

00:00:27.913 --> 00:00:30.305
we added a little bit of flavor

00:00:30.305 --> 00:00:31.930
to our little scenario

00:00:31.930 --> 00:00:34.647
where right at time zero,
a friend passes me up

00:00:34.647 --> 00:00:36.713
in her spaceship and she is traveling

00:00:36.713 --> 00:00:39.732
relative to me in the positive-x direction

00:00:39.732 --> 00:00:41.728
at half the speed of light,

00:00:41.728 --> 00:00:44.352
and so we plotted her
path right at time zero.

00:00:44.352 --> 00:00:45.606
Her spaceship is right there.

00:00:45.606 --> 00:00:46.906
I could draw the spaceship.

00:00:46.906 --> 00:00:48.207
It's at the origin.

00:00:48.207 --> 00:00:49.901
Then after one,

00:00:49.901 --> 00:00:51.341
after one second,

00:00:51.341 --> 00:00:53.937
she would have traveled

00:00:53.937 --> 00:00:56.168
1 1/2 times 10 to the 8th meters.

00:00:56.168 --> 00:00:57.654
After two seconds, she would have traveled

00:00:57.654 --> 00:00:59.465
3 times 10 to the 8th meters,

00:00:59.465 --> 00:01:00.881
so this blue line in the last video

00:01:00.881 --> 00:01:02.414
was her path.

00:01:02.414 --> 00:01:03.993
But now I want to make
it even more interesting.

00:01:03.993 --> 00:01:05.873
Let's assume that we have
actually a whole train

00:01:05.873 --> 00:01:09.496
of spaceships, all traveling
in the positive-x direction

00:01:09.496 --> 00:01:11.887
at the same velocity as her spaceship.

00:01:11.887 --> 00:01:13.884
So this is her spaceship right over here,

00:01:13.884 --> 00:01:16.067
at time equals zero,

00:01:16.067 --> 00:01:18.389
she is exactly where I am,

00:01:18.389 --> 00:01:21.105
but let's say that 3 times
10 to the 8th meters in front

00:01:21.105 --> 00:01:23.218
of her, there's another spaceship,

00:01:23.218 --> 00:01:26.051
traveling at exactly

00:01:26.051 --> 00:01:29.952
1 1/2 times 10 to the
8th meters per second,

00:01:29.952 --> 00:01:34.085
so they're traveling at the
same relative velocity to me,

00:01:34.085 --> 00:01:35.849
but if you think about it, from
each of their point of view,

00:01:35.849 --> 00:01:37.591
they would seem to be stationary,

00:01:37.591 --> 00:01:39.077
because the distance between them,

00:01:39.077 --> 00:01:41.213
in this x-direction is
going to stay the same.

00:01:41.213 --> 00:01:43.372
So this person, at time
zero, is going to be

00:01:43.372 --> 00:01:46.437
3 times 10 to the 8th meters
in the positive-x direction

00:01:46.437 --> 00:01:47.343
from me.

00:01:47.343 --> 00:01:49.990
Now, if I were to wait two seconds,

00:01:49.990 --> 00:01:51.197
they are going to be,

00:01:51.197 --> 00:01:54.192
so if I were to wait two seconds,

00:01:54.192 --> 00:01:55.214
they are going to be,

00:01:55.214 --> 00:01:59.161
they are going to be 6
times 10 to the 8th meters

00:01:59.161 --> 00:02:01.413
away from me, because they're going half

00:02:01.413 --> 00:02:03.271
the speed of light,

00:02:03.271 --> 00:02:05.059
and so I could draw their path

00:02:05.059 --> 00:02:07.845
and I'm going to do it
in a slightly more muted,

00:02:07.845 --> 00:02:10.562
thinner color, so that we,

00:02:10.562 --> 00:02:12.558
what I'm essentially
going to be doing here

00:02:12.558 --> 00:02:15.484
is I'm setting up gridlines for

00:02:15.484 --> 00:02:19.431
my friend's alternate frame of reference.

00:02:19.431 --> 00:02:22.241
I'm going to put that on
top of my frame of reference

00:02:22.241 --> 00:02:23.727
and to be clear,

00:02:23.727 --> 00:02:26.025
I'm not assuming special relativity.

00:02:26.025 --> 00:02:27.813
I'm assuming a Newtonian world,

00:02:27.813 --> 00:02:29.183
classical mechanics.

00:02:29.183 --> 00:02:31.157
Just to get familiar with these ideas

00:02:31.157 --> 00:02:34.825
and to see where the Newtonian
world is going to break down

00:02:34.825 --> 00:02:36.288
but let's say that's not the only ship,

00:02:36.288 --> 00:02:37.356
let's say there was another ship,

00:02:37.356 --> 00:02:40.491
that at time equals zero is
6 times 10 to the 8th meters

00:02:40.491 --> 00:02:42.697
away from me in the positive x-direction.

00:02:42.697 --> 00:02:44.415
Well, where are they going to be?

00:02:44.415 --> 00:02:47.782
Where are they going to be after

00:02:47.782 --> 00:02:49.105
two seconds?

00:02:49.105 --> 00:02:50.104
Well, after,

00:02:50.104 --> 00:02:51.566
let's see, after one second,

00:02:51.566 --> 00:02:54.469
they're going to be 1 1/2
times 10 to the 8th meters

00:02:54.469 --> 00:02:55.723
further.

00:02:55.723 --> 00:03:00.499
Then after two seconds,
they're going to be

00:03:00.499 --> 00:03:03.616
3 times 10 to the 8th meters further,

00:03:03.616 --> 00:03:05.891
and then after three seconds,

00:03:05.891 --> 00:03:08.724
after three seconds,
so they're going to be

00:03:08.724 --> 00:03:12.718
1 1/2 times 10 to the 8th
meters further than that,

00:03:12.718 --> 00:03:15.481
so their path is going to look like,

00:03:15.481 --> 00:03:17.710
so let me draw this.

00:03:17.710 --> 00:03:21.146
I'm really just doing these
to show what stationary

00:03:21.146 --> 00:03:23.584
objects in my friend's frame of reference,

00:03:23.584 --> 00:03:25.442
how they actually look to me,

00:03:25.442 --> 00:03:27.647
and these should be parallel lines.

00:03:27.647 --> 00:03:29.157
These should be parallel lines,

00:03:29.157 --> 00:03:31.780
so that's a pretty good attempt,

00:03:31.780 --> 00:03:32.407
and so I'll,

00:03:32.407 --> 00:03:35.054
let me, and so let me actually just set up

00:03:35.054 --> 00:03:37.028
that alternate frame of reference,

00:03:37.028 --> 00:03:40.209
so this is going to be not only my x-axis,

00:03:40.209 --> 00:03:42.693
I'm also going to call
this my x-prime axis

00:03:42.693 --> 00:03:44.644
and I'll do it in a
second of thinking about

00:03:44.644 --> 00:03:46.455
how we can read this.

00:03:46.455 --> 00:03:49.589
So, I can color this
in with the blue color.

00:03:49.589 --> 00:03:51.308
Actually, let me do it
in that light blue color

00:03:51.308 --> 00:03:53.955
that we're going to be
using for my friend's

00:03:53.955 --> 00:03:54.907
frame of reference,

00:03:54.907 --> 00:03:56.323
so this is also,

00:03:56.323 --> 00:04:00.224
this is also going to be
our x-prime axis overlaid,

00:04:00.224 --> 00:04:03.428
and this one, let me call
this the t-prime axis,

00:04:03.428 --> 00:04:04.891
so my friend's frame of reference,

00:04:04.891 --> 00:04:06.702
I'm going to call that the
prime frame of reference,

00:04:06.702 --> 00:04:07.607
or if I called,

00:04:07.607 --> 00:04:10.881
if I called my frame of reference
the S frame of reference,

00:04:10.881 --> 00:04:13.551
I can call my friend's frame of reference

00:04:13.551 --> 00:04:18.311
the S-prime frame of reference,

00:04:18.311 --> 00:04:21.074
and so, let me draw
horizontal lines to show the

00:04:21.074 --> 00:04:23.327
passage of each second,

00:04:23.327 --> 00:04:26.995
so, that's one second,

00:04:26.995 --> 00:04:29.271
two seconds,

00:04:29.271 --> 00:04:31.314
three seconds,

00:04:31.314 --> 00:04:33.543
so on and so forth.

00:04:33.543 --> 00:04:36.352
So let's make sure we're
comfortable with reading

00:04:36.352 --> 00:04:38.373
what's going on here.

00:04:38.373 --> 00:04:40.137
So, a point,

00:04:40.137 --> 00:04:41.693
let's say this point on our diagram.

00:04:41.693 --> 00:04:42.436
Let me just see.

00:04:42.436 --> 00:04:45.361
This point on our diagram,

00:04:45.361 --> 00:04:48.705
from my frame of reference,
in the S frame of reference,

00:04:48.705 --> 00:04:51.514
this is the point time is,

00:04:51.514 --> 00:04:52.745
or space,

00:04:52.745 --> 00:04:54.045
or my distance,

00:04:54.045 --> 00:04:57.342
my distance from the origin
in the positive-x direction

00:04:57.342 --> 00:04:59.107
is going to be,

00:04:59.107 --> 00:05:00.570
I just drop a,

00:05:00.570 --> 00:05:02.961
I just go parallel to my vertical axis,

00:05:02.961 --> 00:05:04.308
so I'm just going to do that,

00:05:04.308 --> 00:05:08.604
and I get 4.5 times 10 to the 8th meters,

00:05:08.604 --> 00:05:11.715
so 4.5 times 10 to the 8th meters,

00:05:11.715 --> 00:05:13.364
and what is our,

00:05:13.364 --> 00:05:15.128
so I can write the units there if I want,

00:05:15.128 --> 00:05:17.636
and our time is at one second,

00:05:17.636 --> 00:05:18.286
one second.

00:05:18.286 --> 00:05:20.190
So this is our coordinates from,

00:05:20.190 --> 00:05:22.883
this is the coordinates
from my frame of reference,

00:05:22.883 --> 00:05:24.346
from the S frame of reference,

00:05:24.346 --> 00:05:25.670
I guess S for Sal,

00:05:25.670 --> 00:05:27.202
and so, what would be the frame?

00:05:27.202 --> 00:05:28.549
What would be those same coordinates

00:05:28.549 --> 00:05:30.313
from my friend's frame of reference?

00:05:30.313 --> 00:05:33.587
Let's say her name is Sally, or S-prime.

00:05:33.587 --> 00:05:35.515
Well, from her frame of reference,

00:05:35.515 --> 00:05:37.581
our time is still one second.

00:05:37.581 --> 00:05:39.044
We see the horizontal line there.

00:05:39.044 --> 00:05:42.016
We're just going parallel
to our horizontal axis

00:05:42.016 --> 00:05:43.084
or our x-axis

00:05:43.084 --> 00:05:46.172
so the time is still one second,

00:05:46.172 --> 00:05:47.263
but her,

00:05:47.263 --> 00:05:49.632
the distance that the,

00:05:49.632 --> 00:05:53.765
or the x-coordinate is 3
times 10 to the 8th meters,

00:05:53.765 --> 00:05:57.131
3 times 10 to the 8th meters,

00:05:57.131 --> 00:05:59.453
and I really want you to
sit and think about this

00:05:59.453 --> 00:06:01.752
and even take out a ruler
and draw this on your

00:06:01.752 --> 00:06:04.167
own paper and feel good about this.

00:06:04.167 --> 00:06:05.978
Now, why does this make sense?

00:06:05.978 --> 00:06:09.182
Well, this, you could
view this point as where

00:06:09.182 --> 00:06:10.598
that second spaceship was

00:06:10.598 --> 00:06:14.453
after a second, and from
Sally's point of view,

00:06:14.453 --> 00:06:16.241
from the S-prime frame of reference,

00:06:16.241 --> 00:06:17.564
at time equals zero,

00:06:17.564 --> 00:06:20.379
that person, that ship was
3 times 10 to the 8th meters

00:06:20.379 --> 00:06:23.351
in front of her, and after a
second, that spaceship is still

00:06:23.351 --> 00:06:25.603
3 times 10 to the 8th
meters in front of her.

00:06:25.603 --> 00:06:26.764
After two seconds, it's still

00:06:26.764 --> 00:06:29.806
3 times 10 to the 8th
meters in front of her,

00:06:29.806 --> 00:06:32.522
and so from her point of view
and her frame of reference,

00:06:32.522 --> 00:06:34.542
it is stationary,

00:06:34.542 --> 00:06:36.539
so the way to read this for any point,

00:06:36.539 --> 00:06:38.234
if you want my frame of reference,

00:06:38.234 --> 00:06:40.092
you, the time is easy,

00:06:40.092 --> 00:06:42.158
you just go horizontally to either one,

00:06:42.158 --> 00:06:45.246
so on her axis, we've essentially
just taken the coordinate

00:06:45.246 --> 00:06:47.545
axis and sheared it and skewed it.

00:06:47.545 --> 00:06:48.961
This is still one second,

00:06:48.961 --> 00:06:51.097
this is two seconds,
this is three seconds,

00:06:51.097 --> 00:06:53.768
but it turned all of these kind of a grid

00:06:53.768 --> 00:06:57.459
of squares if I were to draw
a grid for my coordinates

00:06:57.459 --> 00:06:58.969
into a bunch of,

00:06:58.969 --> 00:07:02.243
into a bunch of parallelograms

00:07:02.243 --> 00:07:04.054
which we see right over here,

00:07:04.054 --> 00:07:06.840
and so for any given point in,

00:07:06.840 --> 00:07:09.255
remember we're thinking of Newtonian space

00:07:09.255 --> 00:07:11.089
and time,

00:07:11.089 --> 00:07:13.504
to read it, we go horizontally for time,

00:07:13.504 --> 00:07:14.990
in either case,

00:07:14.990 --> 00:07:17.149
but for me, you would go straight down

00:07:17.149 --> 00:07:19.239
because my time axis is vertical.

00:07:19.239 --> 00:07:21.120
You go straight down and
you read where it intersects

00:07:21.120 --> 00:07:23.442
the x-axis to figure out our position

00:07:23.442 --> 00:07:24.835
but from her frame of reference

00:07:24.835 --> 00:07:28.294
we move parallel to her t-axis,

00:07:28.294 --> 00:07:30.036
or her t-prime axis, I should say,

00:07:30.036 --> 00:07:32.334
to get what her x,

00:07:32.334 --> 00:07:35.144
I guess we could say what
her x-prime coordinates

00:07:35.144 --> 00:07:38.464
are going to be, so this
is x-prime, t-prime.

00:07:38.464 --> 00:07:41.552
This is x and this is t.

00:07:41.552 --> 00:07:43.619
Now, what's clear, at any given point,

00:07:43.619 --> 00:07:45.244
at any given point,

00:07:45.244 --> 00:07:49.424
t is going to be equal to t-prime,

00:07:49.424 --> 00:07:52.419
or we could say t-prime is equal to t,

00:07:52.419 --> 00:07:53.324
either way,

00:07:53.324 --> 00:07:56.830
but what's the relationship
between x-prime and x?

00:07:56.830 --> 00:07:58.851
Well, let's think about that for a second,

00:07:58.851 --> 00:08:03.285
so when x is 4.5 times
10 to the 8th meters,

00:08:03.285 --> 00:08:07.047
x-prime is 3 times 10 to the 8th meters,

00:08:07.047 --> 00:08:08.626
and so it looks like,

00:08:08.626 --> 00:08:12.712
it looks like x-prime
is going to be equal to

00:08:12.712 --> 00:08:16.311
x minus, well what's it going to be minus?

00:08:16.311 --> 00:08:17.867
It's going to be minus

00:08:17.867 --> 00:08:22.867
1.5 times 10 to the 8th meters

00:08:24.182 --> 00:08:28.641
but that difference is going
to grow as time goes on,

00:08:28.641 --> 00:08:31.729
meters per second times,

00:08:31.729 --> 00:08:34.956
times the amount of
time that has passed by,

00:08:34.956 --> 00:08:36.071
and you should see that

00:08:36.071 --> 00:08:37.951
because after one second,

00:08:37.951 --> 00:08:42.061
after one second, the difference
is 1.5 times 10 to the 8th

00:08:42.061 --> 00:08:44.592
meters, after two seconds,

00:08:44.592 --> 00:08:46.403
after two seconds, so
let's look at this point

00:08:46.403 --> 00:08:47.703
right over here,

00:08:47.703 --> 00:08:51.372
in my frame of reference,
that person is sitting at

00:08:51.372 --> 00:08:55.691
6 times 10 to the 8th meters
in the positive-x direction,

00:08:55.691 --> 00:08:57.455
but in Sally's frame of reference,

00:08:57.455 --> 00:08:59.104
in the S-prime frame of reference,

00:08:59.104 --> 00:09:03.051
they're still at 3 times
10 to the 8th meters

00:09:03.051 --> 00:09:04.816
in the positive-x direction,

00:09:04.816 --> 00:09:06.929
so we'd multiply two seconds

00:09:06.929 --> 00:09:09.436
times 1.5 times 10 to the 8th

00:09:09.436 --> 00:09:12.664
so we have 3 times 10 to the 8th meters,

00:09:12.664 --> 00:09:14.777
so we take 6 times 10 to the 8th

00:09:14.777 --> 00:09:15.798
which would be that.

00:09:15.798 --> 00:09:18.097
We subtract out 3 times 10 to the 8th

00:09:18.097 --> 00:09:20.581
and we get the coordinates in S-prime,

00:09:20.581 --> 00:09:22.346
so this might get a little bit confusing

00:09:22.346 --> 00:09:25.550
and the key here is really
try to draw this yourself

00:09:25.550 --> 00:09:27.083
and really try to plot points

00:09:27.083 --> 00:09:29.335
and think about how they
are different coordinates

00:09:29.335 --> 00:09:32.980
in the different frames of reference,

00:09:32.980 --> 00:09:34.513
or different time-space
coordinates in the different

00:09:34.513 --> 00:09:36.481
frames of reference.


Example: Correlation coefficient intuition

WEBVTT
Kind: captions
Language: en

00:00:00.614 --> 00:00:02.885
- I took some screen captures from the

00:00:02.885 --> 00:00:07.464
Khan Academy exercise on
correlation coefficient intuition.

00:00:07.834 --> 00:00:09.600
They've given us some
correlation coefficients

00:00:09.600 --> 00:00:11.428
and we have to match them to the various

00:00:11.428 --> 00:00:12.670
scatterplots on that exercise.

00:00:12.670 --> 00:00:13.901
There's a little interface
where we can drag

00:00:13.901 --> 00:00:15.763
these around in a table to match them

00:00:15.763 --> 00:00:17.667
to the different scatterplots.

00:00:17.667 --> 00:00:19.572
The point isn't to figure out how exactly

00:00:19.572 --> 00:00:22.405
to calculate these, we'll
do that in the future,

00:00:22.405 --> 00:00:25.279
but really to get an intuition
of we are trying to measure.

00:00:25.750 --> 00:00:28.422
The main idea is that
correlation coefficients

00:00:28.422 --> 00:00:31.626
are trying to measure
how well a linear model

00:00:31.626 --> 00:00:35.965
can describe the relationship
between two variables.

00:00:36.435 --> 00:00:37.827
For example,

00:00:40.942 --> 00:00:43.348
let me do some coordinate axes here.

00:00:44.166 --> 00:00:45.724
Let's say that's one variable.

00:00:45.724 --> 00:00:47.675
Say that's my y variable

00:00:47.675 --> 00:00:51.230
and let's say that is my x variable.

00:00:51.230 --> 00:00:54.019
Let's say when x is low, y is low.

00:00:54.019 --> 00:00:55.831
When x is a little higher,
y is a little higher.

00:00:55.831 --> 00:00:58.294
When x is a little bit
higher, y is higher.

00:00:58.294 --> 00:01:00.845
When x is really high, y is even higher.

00:01:01.593 --> 00:01:05.333
A linear model would
describe it very, very well.

00:01:05.333 --> 00:01:07.304
It's quite easy to draw a line

00:01:08.434 --> 00:01:10.635
that essentially goes
through those points.

00:01:10.635 --> 00:01:14.489
So something like this
would have an r of 1,

00:01:14.489 --> 00:01:15.560
r is equal to one.

00:01:15.560 --> 00:01:17.629
A linear model perfectly describes it

00:01:17.629 --> 00:01:19.742
and it's a positive correlation.

00:01:19.742 --> 00:01:22.273
When one increases, when
one variable gets larger,

00:01:22.273 --> 00:01:24.114
then the other variable is larger.

00:01:24.694 --> 00:01:26.258
When one variable is
smaller then other variable

00:01:26.258 --> 00:01:28.167
is smaller and vice versa.

00:01:28.167 --> 00:01:30.572
Now what would an r of
negative one look like?

00:01:30.572 --> 00:01:32.195
Well, that would once again be a situation

00:01:32.195 --> 00:01:34.006
where a linear model works really well

00:01:34.006 --> 00:01:35.424
but when one variable moves up,

00:01:35.424 --> 00:01:37.510
the other one moves down and vice versa.

00:01:37.910 --> 00:01:41.742
Let me draw my coordinates,

00:01:41.742 --> 00:01:43.525
my coordinate axes again.

00:01:45.133 --> 00:01:48.222
I'm gonna try to draw a dataset where the

00:01:48.222 --> 00:01:49.821
r would be negative one.

00:01:50.453 --> 00:01:53.495
Maybe when y is high, x is very low.

00:01:53.495 --> 00:01:57.093
When y becomes lower, x become higher.

00:01:57.093 --> 00:01:58.788
When y becomes a good bit lower,

00:01:58.788 --> 00:02:00.901
x becomes a good bit higher.

00:02:00.901 --> 00:02:04.691
Once again, when y decreases,

00:02:04.691 --> 00:02:07.987
x increases or as x
increases, y decreases.

00:02:07.987 --> 00:02:10.216
They're moving in opposite directions

00:02:10.216 --> 00:02:13.162
but you can fit a line
very easily to this.

00:02:13.701 --> 00:02:17.596
The line would look something like this.

00:02:17.904 --> 00:02:21.365
This would have an r of negative one,

00:02:21.365 --> 00:02:24.221
and r of zero, r is equal to zero,

00:02:24.221 --> 00:02:26.752
would be a dataset which a line

00:02:26.752 --> 00:02:29.144
doesn't really fit very well at all.

00:02:29.144 --> 00:02:30.557
I'll do that one really small,

00:02:30.567 --> 00:02:32.395
since I don't have much space here.

00:02:32.395 --> 00:02:35.410
An r of zero might look

00:02:36.460 --> 00:02:38.157
something like this.

00:02:38.157 --> 00:02:39.505
Maybe I'll have a data point here,

00:02:39.505 --> 00:02:41.734
maybe have a data point here,

00:02:42.990 --> 00:02:44.336
maybe I have one there.

00:02:44.336 --> 00:02:46.556
There, there.

00:02:47.867 --> 00:02:49.660
And it wouldn't necessarily
be this well organized

00:02:49.660 --> 00:02:51.178
but this gives you a sense of things.

00:02:52.188 --> 00:02:54.491
How would you actually
try to fit a line here?

00:02:54.491 --> 00:02:57.370
You could equally justify
a line that looks like that

00:02:57.370 --> 00:02:59.228
or a line that looks like that,

00:02:59.228 --> 00:03:01.498
or a line that looks like that.

00:03:04.324 --> 00:03:06.336
A linear model really does not describe

00:03:06.336 --> 00:03:08.982
the relationship between
the two variables that well,

00:03:08.982 --> 00:03:10.377
right over here.

00:03:10.377 --> 00:03:12.048
So with that, is a primer.

00:03:12.048 --> 00:03:14.721
Let's see if we can
tackle these scatterplots.

00:03:14.721 --> 00:03:15.815
The way I'm gonna do it is I'm just gonna

00:03:15.815 --> 00:03:18.556
try to eyeball what a linear
model might look like.

00:03:18.556 --> 00:03:21.017
There's different methods of
trying to fit a linear model

00:03:21.017 --> 00:03:22.969
to a dataset, an imperfect dataset.

00:03:22.969 --> 00:03:24.201
I drew very perfect ones, at least for the

00:03:24.201 --> 00:03:26.130
r equals negative one and r equals one

00:03:26.130 --> 00:03:28.695
but these are what the real
world actually looks like.

00:03:28.695 --> 00:03:31.846
Very few times will things
perfectly sit on a line.

00:03:31.846 --> 00:03:35.282
For scatterplot A, if I
were to try to fit a line,

00:03:35.282 --> 00:03:39.185
it would look something like that.

00:03:39.185 --> 00:03:41.090
If I were to try to
minimize distances from

00:03:41.090 --> 00:03:44.379
the points to the line,
I do see a general trend

00:03:45.169 --> 00:03:46.529
if we look at these data points over here,

00:03:46.529 --> 00:03:49.629
when y is high, x is low.

00:03:50.989 --> 00:03:54.270
When x is larger, y is smaller.

00:03:54.600 --> 00:03:56.296
Looks like r is going
to be less than zero,

00:03:56.296 --> 00:03:58.179
and a reasonable bit less than zero.

00:03:58.179 --> 00:04:00.268
It's going to approach this thing here.

00:04:00.268 --> 00:04:02.056
If we look at our choices,

00:04:02.056 --> 00:04:04.171
it wouldn't be r equals 0.65.

00:04:04.171 --> 00:04:06.982
These are positive so I wouldn't
use that one or that one.

00:04:06.982 --> 00:04:08.979
And this one is almost no correlation.

00:04:08.979 --> 00:04:13.531
R equals negative 0.02, this
is pretty close to zero.

00:04:13.531 --> 00:04:17.451
I feel good with r is
equal to negative 0.72.

00:04:20.195 --> 00:04:22.173
I wanna be clear, if I didn't
have these choices here,

00:04:22.173 --> 00:04:23.370
I wouldn't just be able to say,

00:04:23.370 --> 00:04:24.660
just looking at these data points without

00:04:24.660 --> 00:04:25.937
being able to do a calculation,

00:04:25.937 --> 00:04:28.237
that r is equals to negative 0.72.

00:04:28.237 --> 00:04:30.350
I'm just basing it on
the intuition that it is

00:04:30.350 --> 00:04:33.439
a negative correlation,
it seems pretty strong.

00:04:33.439 --> 00:04:34.663
The pattern kind of jumps out at you,

00:04:34.663 --> 00:04:37.669
that when y is large, x is small.

00:04:37.669 --> 00:04:39.643
When x is large, y is small.

00:04:39.643 --> 00:04:41.407
So I like something that's approaching

00:04:41.407 --> 00:04:42.475
r equals negative one.

00:04:42.475 --> 00:04:44.299
I've used this one up already.

00:04:44.789 --> 00:04:47.415
Now scatterplot B, if I were
to just try to eyeball it,

00:04:47.415 --> 00:04:48.737
once again this is gonna be imperfect.

00:04:48.737 --> 00:04:50.432
But the trend, if I were
to try to fit a line,

00:04:50.432 --> 00:04:52.032
it looks something like that.

00:04:52.032 --> 00:04:54.562
It looks like a line
fits in reasonably well.

00:04:54.562 --> 00:04:56.645
There's some points that
would still be hard to fit.

00:04:56.645 --> 00:04:58.692
They're still pretty far from the line.

00:04:58.692 --> 00:05:00.726
It looks like it's a positive correlation.

00:05:02.076 --> 00:05:05.373
When y is small, x is
relatively small and vice versa.

00:05:05.373 --> 00:05:09.608
As x grows, y grows and
when y grows, x grows.

00:05:09.608 --> 00:05:11.573
This ones going to be positive

00:05:11.573 --> 00:05:13.219
and it looks like it would
be reasonably positive.

00:05:13.219 --> 00:05:14.335
I have two choices here.

00:05:14.335 --> 00:05:16.536
I don't know which of
these it's going to be.

00:05:16.536 --> 00:05:19.928
It's either going to be r is equal to 0.65

00:05:19.928 --> 00:05:24.009
or r is equal to 0.84.

00:05:24.339 --> 00:05:26.568
I also got scatterplot C,
this ones all over the place.

00:05:26.568 --> 00:05:28.952
It kinda looks like what we did over here.

00:05:29.608 --> 00:05:30.685
What does a line look like?

00:05:30.685 --> 00:05:31.767
You could almost imagine anything.

00:05:31.767 --> 00:05:34.033
Does it look like that?

00:05:34.033 --> 00:05:36.201
Does a line look like that?

00:05:37.522 --> 00:05:39.170
There's not a direction
that you could say,

00:05:39.170 --> 00:05:42.071
"Well, as x increases, maybe
y increases or decreases."

00:05:42.071 --> 00:05:43.627
There's no rhyme or reason here,

00:05:43.627 --> 00:05:46.251
so this looks very non-correlated.

00:05:46.251 --> 00:05:48.688
So this one is pretty close to zero.

00:05:48.688 --> 00:05:49.755
I feel pretty good that this is the

00:05:49.755 --> 00:05:53.680
r is equal to negative .02.

00:05:53.680 --> 00:05:55.489
In fact, if we tried
probably the best line

00:05:55.489 --> 00:05:57.161
that could be fit, would be one with

00:05:57.161 --> 00:05:59.365
a slight negative slope.

00:05:59.365 --> 00:06:01.216
It might look something like this.

00:06:02.985 --> 00:06:04.471
And notice, even when
we try to fit a line,

00:06:04.471 --> 00:06:06.584
there's all sorts of points
that are way off the line.

00:06:06.584 --> 00:06:08.860
So the linear model did
not fit it that well.

00:06:08.860 --> 00:06:12.593
R is equal to negative 0.02,

00:06:13.224 --> 00:06:14.757
So we'll use that one.

00:06:14.757 --> 00:06:16.220
Now we have scatterplot D.

00:06:16.220 --> 00:06:18.008
That's gonna use one of the
other positive correlations

00:06:18.008 --> 00:06:21.351
and it does look like there
is a positive correlation.

00:06:21.351 --> 00:06:23.146
When y is low, x is low.

00:06:23.966 --> 00:06:27.107
When x is high, y is high and vice versa.

00:06:27.107 --> 00:06:28.987
We could try to fit something that looks

00:06:28.987 --> 00:06:31.281
something like that.

00:06:31.926 --> 00:06:33.817
But it's still not as good as that one.

00:06:33.817 --> 00:06:36.253
You can see the points
that we're trying to fit,

00:06:36.253 --> 00:06:39.178
there's several points that
are still pretty far away

00:06:39.178 --> 00:06:41.518
from our model.

00:06:42.197 --> 00:06:44.029
The model is not fitting it that well,

00:06:44.029 --> 00:06:46.698
so I would say scatterplot
B is a better fit.

00:06:46.698 --> 00:06:49.136
A linear model works
better for scatterplot B

00:06:49.136 --> 00:06:50.577
than it works for scatterplot D.

00:06:50.577 --> 00:06:53.827
I would give the higher r to scatterplot B

00:06:53.827 --> 00:06:58.076
and the lower r, r equals
0.65, to scatterplot D.

00:06:58.076 --> 00:07:01.256
R is equal to 0.65.

00:07:01.256 --> 00:07:03.092
Once again that's because
with a linear model

00:07:03.092 --> 00:07:06.382
it looks like there's a
trend but there's several

00:07:06.382 --> 00:07:09.510
more data points are way off the line

00:07:09.510 --> 00:07:12.534
in scatterplot D than in
the case of scatterplot B.

00:07:12.534 --> 00:07:14.206
There's a few that are
still way off the line

00:07:14.206 --> 00:07:17.776
but these are even more
off of the line in D.


Ordering triangle sides and angles

WEBVTT
Kind: captions
Language: en

00:00:00.340 --> 00:00:02.210
- [Voiceover] We're asked
to order the side lengths

00:00:02.210 --> 00:00:04.822
of the triangle from shortest to longest.

00:00:04.822 --> 00:00:06.041
And we have the three sides here,

00:00:06.041 --> 00:00:07.202
and we could use this little tool

00:00:07.202 --> 00:00:09.462
to order them in some way.

00:00:09.462 --> 00:00:10.680
If we look at the triangle,

00:00:10.680 --> 00:00:14.384
we've been given the interior
angles of the triangle,

00:00:14.384 --> 00:00:17.228
and they haven't told us
the actual side lengths.

00:00:17.228 --> 00:00:19.062
So, how are we supposed
to actually order them

00:00:19.062 --> 00:00:20.722
from shortest to longest?

00:00:20.722 --> 00:00:23.961
Well, the realization
that you need to make here

00:00:23.961 --> 00:00:28.096
is that the order of the lengths

00:00:28.096 --> 00:00:29.757
of the sides of a triangle

00:00:29.757 --> 00:00:33.414
are related to the order
of the measures of angles

00:00:33.414 --> 00:00:35.372
that open up onto those sides.

00:00:35.372 --> 00:00:36.812
What do I mean by that?

00:00:36.812 --> 00:00:39.702
Well, let's think about these
three angles right over here.

00:00:39.702 --> 00:00:42.802
57 degrees, that is the
smallest of these three,

00:00:42.802 --> 00:00:45.588
and so the side that
this angle opens up to,

00:00:45.588 --> 00:00:47.260
or you can think of it
as the opposite side,

00:00:47.260 --> 00:00:50.208
is going to be the shortest
side of the triangle.

00:00:50.208 --> 00:00:52.542
So, b is going to be the shortest side.

00:00:52.542 --> 00:00:55.199
So, the next largest angle is 58 degrees,

00:00:55.199 --> 00:00:57.743
and so a is going to be the middle side.

00:00:57.743 --> 00:01:00.644
It's not going to be the
longest nor the shortest.

00:01:00.644 --> 00:01:01.720
So, a.

00:01:01.720 --> 00:01:05.261
Then 65 degrees, that
opens up onto side c,

00:01:05.261 --> 00:01:07.722
or the opposite side of that angle is c.

00:01:07.722 --> 00:01:09.753
So, c is going to be the longest side.

00:01:09.753 --> 00:01:12.342
To get an intuition for why that is,

00:01:12.342 --> 00:01:16.080
imagine a world where the 65 degree angle,

00:01:16.080 --> 00:01:17.415
if we were to make it bigger.

00:01:17.415 --> 00:01:19.343
If we were to make the
65 degree angle bigger,

00:01:19.343 --> 00:01:21.369
maybe by moving this point out

00:01:21.369 --> 00:01:23.272
and that point out, what would happen?

00:01:23.272 --> 00:01:25.549
Well, side c would get bigger,

00:01:25.549 --> 00:01:28.415
and because the angles of a triangle

00:01:28.415 --> 00:01:29.914
have to add up to 180 degrees,

00:01:29.914 --> 00:01:31.433
if this one's getting bigger,

00:01:31.433 --> 00:01:34.035
these will have to get smaller.

00:01:34.035 --> 00:01:37.030
Likewise, if I were to take angle...

00:01:37.030 --> 00:01:39.328
let's say, if I were to
take this 58 degree angle,

00:01:39.328 --> 00:01:41.511
and if I were to make it smaller,

00:01:41.511 --> 00:01:42.423
what's going to happen?

00:01:42.423 --> 00:01:44.722
Well, side a is going to get smaller.

00:01:44.722 --> 00:01:45.824
So, the general principle,

00:01:45.824 --> 00:01:47.762
I'm not giving you any formal proof here,

00:01:47.762 --> 00:01:49.180
but the intuition is,

00:01:49.180 --> 00:01:52.430
is that the order of
the angles will tell you

00:01:52.430 --> 00:01:54.148
what the order of the
sides are going to be.

00:01:54.148 --> 00:01:55.669
So, the smallest side is going

00:01:55.669 --> 00:01:57.422
to be opposite the smallest angle.

00:01:57.422 --> 00:02:01.287
The largest side is going to
be opposite the largest angle.

00:02:01.287 --> 00:02:03.388
We can check our answer,
make sure we got it right.

00:02:03.388 --> 00:02:06.082
Now, let's do one that
goes the other way around.

00:02:06.082 --> 00:02:08.868
Here, we want to order
the angles of the triangle

00:02:08.868 --> 00:02:09.902
from smallest to largest,

00:02:09.902 --> 00:02:11.179
and we're given the sides.

00:02:11.179 --> 00:02:12.607
Well, same, exact idea.

00:02:12.607 --> 00:02:14.418
The smallest angle is going to be

00:02:14.418 --> 00:02:16.101
opposite the smallest side

00:02:16.101 --> 00:02:17.377
or the shortest side.

00:02:17.377 --> 00:02:20.083
Well, the shortest side is
this side of length 7.2.

00:02:20.083 --> 00:02:23.438
The angle that opens
up onto it is angle a.

00:02:23.438 --> 00:02:26.816
So, that's going to be the smallest angle.

00:02:26.816 --> 00:02:31.541
Then, the next smallest side
is the side of length 7.3,

00:02:31.541 --> 00:02:33.453
and the angle that opens up to it

00:02:33.453 --> 00:02:36.496
is angle b right over here.

00:02:36.793 --> 00:02:37.896
So, angle b.

00:02:37.896 --> 00:02:40.670
Then angle c opens up
onto the largest side.

00:02:40.670 --> 00:02:42.725
So, it's going to be the largest angle.

00:02:42.725 --> 00:02:43.386
So, we're done.

00:02:43.386 --> 00:02:45.267
We've ordered the angles of the triangle

00:02:45.267 --> 00:02:47.201
from smallest to largest.