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On to Spacetime Diagrams, Part 2.

What we want to do here is explore the concept

of velocity as it's represented on a spacetime diagram.

But let's just remind ourselves here, what a spacetime diagram is.

We actually have four cases this time so we're going to imagine that we have

our spaceship or car travelling along the x-axis and

it's going to give off flashes at certain points in time and of course we have our

grid or lattice of clocks all along the x axis, at any given point.

So that when that flash occurs, a photograph will be taken,

and it will record the location of that clock where the photograph occurred, and

the time on that clock.

And all the clocks along the x axis, we will assume,

are synchronized using a method that we reviewed in a previous video clip.

So, got our spaceship.

We're going to imagine four trips now.

Four passes or four runs of our spaceship along the x axis and

all of them will start at t equals 0 and x equals 0.

So they'll all be starting at the origin here.

We don't have to do that of course,

we did an example where we started at some place different than the origin but for

our purposes here we'll just always start at the origin.

t = 0, x = 0, and then we've got run one, run two, run three, and run four.

So let's just review what's going on there.

So, run one starts at x = 0 then at t = 1, it's at x = 1, t = 2,

x = 2, and so on and so forth.

And that is this spacetime diagram here, or

I should say this world line right here.

Remember, a world line represents the path of an object through space and time.

And so at t equals 1, it's at position 1, x equals 1.

At t equals 2 it's at position 2,

at t equals 3 it's at position 3 and so on and so forth.

So this represents this is the world line for

run number one of our spaceship as it's traveling along the x axis.

You'll probably get sick of me saying this, reminding you of this but

this is not, the spaceship is not going up at some angle.

The spaceship was constrained to move only in one direction along the x axis.

We'll assume it's moving in the positive direction.

It could move the other direction as well.

But only one direction back and forth along the x axis,

all our examples here it's moving just to the right in the positive direction.

So that's run number one then we reset things and come back and

it starts another run at t equals 0 is it x equals 0?

And then 2, 4, 6, 8.

So, x equals 1 it's at 2.

x equals 2 it's at 4.

x equals 3 it's at 6, and x equals 4 it's at 8.

So that is run number two and

this is the world line of our spaceship in run number two.

Run number three goes, at t equals 0, again, starting at x equals 0.

Then, t equals 1/2 and then 1, 1 and 1/2, and 2.

So t equals 1, it's at 1/2 here.

At t equals 2, it's at 1.

T equals 3, it's at 1 and 1/2.

T equals 4, it's at 2.

And we will assume that we have clocks in between here at every single point that

that we need, and then the fourth run, just do something different.

We'll assume it's actually moving in the other direction now.

So it starts at zero again, but then goes at time t=1 it's at negative one, and

then negative two, negative three, four.

So at time t=1 it's at negative one, at time t=2 it's at negative two,

t=3 it's negative three and then at time t=4 it's at negative four.

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Okay, distance divided by time, right, miles per hour, kilometers per hour,

feet per second, meters per second.

And so on and so forth with that.

So how does that relate then to our diagrams here, or our world lines here?

Let's take the first one, number one here.

Run number one, just looking at it we can say, well we went one.

We'll just say we're using meters here.

So we went one meter in one second.

And then we went two meters in two seconds and

then three meters in three seconds and four meters in four seconds.

Pretty clear we're going one meter per second.

But if we want to calculate that we could say, just take the three here,

x=3 and 3 seconds.

Three divided by three is one, one meter per second.

Or two divided by two is one, one meter per second.

And because it's a straight line as we talked about before

that's a constant velocity situation.

And we're going to use that in just a minute.

Let's look at number two though, which is this one here.

Note that in one second we're going two meters.

So, the velocity at that point, or over this distance here,

would be two meters in one second, two meters per second.

Or we got four, in two seconds.

Four divided by two, two meters per second.

Or six in three seconds.

Six divided by three, again, two.

And so on and so forth.

Eight divided by four, is two.

So velocity in this case would be one meter per second,

velocity in this case is two meters per second.

Velocity in this case, again because it is a constant velocity,

we're not changing the velocity, we can just pick any given point and

read it off, but we've got one half meter in one second.

So it's one half meter per second.

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If we did 2 meters in 4 seconds, 2, distance divided by 4 seconds,

one half again.

One half meter per second.

And then the final version is -1 meter in 1 second,

-2 meters in 2 seconds, so we're going negative one meters per second.

The negative sign indicating motion to the left, okay.

We take motion to the left as negative, motion to the right as positive.

And again this is motion just along the x-axis.

These are the worldline representations of that motion along the x-axis.

We're plotting the time as the vertical axis here.

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Okay, what's the big point here because that's pretty straightforward,

maybe pretty obvious.

Well, let's relate the velocity here to the slope of these lines.

And again for those of you who haven't had math recently,

don't worry we're not going to do too much with slope, you may way back in your

memory some place it may be there, but you haven't done much with it for

a long time so we're not going to go into the details too much of it.

But here's what we are going to do,

we say okay, velocity is distance divided by time essentially.

So let's look at, do this example right here.

Let's look at number two here.

So the distance traveled say from here to here,

that's not a very straight line, so there's the distance travelled.

And here's the time.

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Okay?

So essentially, dropping things today.

Essentially, what we are doing is,

distance travel is two divided by the time one.

Again, depending on how much algebra and the like you remember,

sometimes we call this the run, okay, the run here,

and we call this the rise, sort of intuitive.

The rise, how far you've risen over that given distance.

You have a distance of two for

the run there, getting all the way over to the four.

And the rise is one.

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Run is a horizontal distance, rise is a vertical distance.

Run on our spacetime plot represents x distance,

how far it's gone along the x axis.

Rise represents distance in the sense in time.

How much time has elapsed.

So we can say that on our space time diagram,

velocity equals, remember velocity is distance divided by time.

Velocity is run divided by rise,

is run divided by rise.

So if I didn't have these numbers here and

I wanted to calculate the velocity of any given world line here or

the really the object which the world line represents it's motion.

I just pick two points and I calculate, okay, how far has it gone, run.

How long did it take to rise and that's the velocity forming.

And I say okay, well that's again not too complicated or complex or anything.

You're just stating almost the obvious here.

But let's look at this in terms of the slope of the lines.

And for our purposes the slope of the line is how,

whether it's sloping sort of downward or sloping upward, okay?

So it's the angle of the line compared to the x-axis or the time axis here.

So clearly we can see number two has a higher velocity,

because you're getting more run for a given amount of rise.

You're covering more ground for each second that you're going here.

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episodes when we talk about the special theory of relativity and

do a few more spacetime diagrams and things like that okay.

So, the lower the slope, the smaller the slope,

the less the steepness of a line really represents more velocity because

you're covering more ground in a shorter amount of time.

Whereas, if you have a steep slope you're covering a short amount of ground

in a longer period of time so the velocity is less.

And just remind you the official definition,

or say the first definition of slope you probably learned,

is that the slope of a line equals rise over run, okay?

So it's a measure of the steepness of a line, so

it's how far does it rise within a given run.

So what's its y value for a given x value?

If you have a large difference in y for a small difference in x,

you have big difference in the rise, a big rise over a relatively short run

gives you a larger number for the slope and a steeper line, okay?

But velocity, we can see here then, Is

inversely related on a spacetime diagram to the slope.

Steeper slope, like number three has a steeper slope here, but

it represents the slowest velocity of our examples.

Whereas number two has the lowest slope and it represents the faster velocity and

again, its nothing profound here, just because we're measuring the run here,

how far it's traveled x in a given amount of time.

So if it travels a long distance in a given amount of time, like a second,

it just means it's going faster.

That's all we're really saying, but again the concept here is to

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recognize that steeper slope means lower velocity, smaller slope,

lower slope as it were, less steep of a line, higher velocity.

And then, this one just means we've got a velocity going in the other direction,

so these lines have positive slope, because the rise and

run are both positive values that run that way and rise up.

This one actually has a negative slope because has a positive rise but

a negative run there.

And again we're not going to get into all the niceties of that,

but I want you to remember the concept of just being able to look at the picture for

a spacetime diagram and see a given world line.

Which will be for

our cases straight lines because we'll have constant velocity situations.

And recognize this represents a slower velocity than this one,

even though this is a steeper slope.

And then one more thing here.

We talked about these as being four different runs for a spaceship.

We could also represent them as four spaceships all

taking off at the same time from our origin.

And spaceship number one follows that path along the x axis.

Spaceship number two goes faster, spaceship number three goes slower, and

then spaceship number four actually goes off in the other direction.

And so again, if you use sort of the animation principle that at x = 0,

that's what it looks like, they're all at 0.

At x = 1 here, we took a snapshot of the whole x axis,

these would be the locations of our four ships.

This is one going that way and this is number three, number one, number two.

And then take another snapshot of the whole x axis where

are the locations of our ships at that point, the dots represent them.

And then t = 3 and t = 4 and so on and so forth.

Okay, so good to just play around with this a little bit,

spacetime diagrams, get a feel for what's going on.

Make up your own little numbers there and do some plots and things like that.

And just that writing down of things hopefully will get to stick in your brain

a little bit more.

So moving on some of the next topics we'll be covering in the video clips,

a big topic is the concept of frames of reference, and

then we'll move on to something called the Galilean transformation.

Again, this is laying the foundations for some of the key things we'll be doing in

the special theory of relativity in weeks to come.