0:04

Now let's move on to our second diagram, second major diagram.

This is actually the first part of it, Diagram 2.1,

under the topic of the relativity of simultaneity.

So this time we're going to do a paintball experiment.

And we're going to have Alice moving and Bob observing.

So Alice is going to be in her spaceship this time.

In previous examples we've done, we often have had Bob moving, so

it's time to get Alice involved here.

So she can move this time in her spaceship.

So here's the situation.

Here's her spaceship, similar to what we had before in the previous video clip, but

just two clocks at the end of it.

Some sort of apparatus in the exact middle such that it can shoot

one thing one way and another thing the other way.

She is moving, the whole spaceship is moving with velocity v.

So this whole thing is moving to the right here.

0:55

Bob, we've replaced his spaceship, but really,

we could imagine his spaceship is still here, stationary.

But we've replaced it with a representation of his frame of reference,

in other words, his lattice of clocks.

So whether if the spaceship was there or not, remember this is where we sort of get

the payoff for this emphasis we had before about events and observers.

They do it with a lattice of clocks.

So at any given point we have the photo principle, the photo flash principle,

the camera principle, that you take a picture at that point.

You look at the clock at that point so you know where it happens.

Bob knows, therefore, where something happens along his lattice of clocks,

and he knows the exact time.

All his clocks are synchronized,

and we have them all reading 0 at this point, okay?

So Bob has his lattice of clocks.

He's stationary, he's observing what's going on with Alice.

And this is a paintball experiment, so we're not too relativistic effects yet.

We'll just keep things slow.

And we say that the paintballs are shot with velocity v equals u.

So Alice shoots one paintball that way and one paintball that way.

And they're shot off right when Bob observes his clocks at time T equals zero,

or TB equals zero.

And we'll imagine there's a little flash of something,

that when the paintballs are shot off, so, get a little flash of

light that lingers in space there for a little bit after this happened.

Okay, so paintballs go off, one goes that way, one goes that way.

And now we need to analyze, in our next section here,

exactly what goes on as Alice is moving that way and

the paintballs are heading towards her two clocks here.

So I suppose we can call this Diagram 2.2 is our next version here.

2:52

We're keeping track of that.

And again, this is on the handout, so you can sort of see it step by step.

Here, now we're going to move Alice's spaceship, its velocity, so

we'll just move it over one clock, as it were.

In the actual diagram in the handout, I think we move it over two clocks.

It really doesn't matter how far we move it,

we just want to be able to fit everything in here.

So okay, so now paintballs have been shot.

Alice is moving.

The paintballs are in transit toward the two clocks.

And so, oops, we don't want to erase.

This stays there, this is a permanent feature now,

that little flash of smoke or something there that indicates

that that's the location where the paintballs were shot.

And so, We draw this part now.

So we've moved Alice over.

She started off right here.

So let's move her, with the right color here.

So now her spaceship goes here.

Make sure I get that, so that's three, one, two, three.

So she's now at four here, like that.

Okay, so her clock is here now.

And clock is here now.

The apparatus that shot the paintballs has now moved over one clock, as it were.

So it's still here, sort of something like that, sitting there.

And the actual paintballs, remember, this is the position they were shot at.

So if we're observing from Bob's perspective,

this paintball is going to be a little bit traveling that way.

And this paintball over here is going to be also traveling

4:30

that way, some place in between.

Once the paintballs leave that apparatus, this end,

of course, of Alice's spaceship is traveling away from them.

Remember, it's traveling with velocity v, in that direction.

This end is traveling towards this paintball.

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So we might expect that this paintball will hit first.

And that paintball has a far longer way to go to catch up to that clock,

might hit second.

But we have to remember, for regular objects like paintballs, or

if you're throwing a ball, the velocities add.

So even though this paintball was shot at velocity

u with respect to the paintball gun here, it also, the paintball gun itself was

moving with velocity v when it was shot with velocity u.

And therefore the velocity of this paintball,

we'll write it in red, is going to be u + v.

5:26

Meanwhile, for this paintball, the paintball apparatus,

the gun was moving that direction with velocity v.

It was shooting it this way with velocity u.

And so its velocity is going to be u- v for

this paintball.

So this paintball will actually be traveling more slowly, this paintball will

be traveling more quickly, and the end result is, it will cancel out.

And in our next diagram we'll draw them.

They'll hit at exactly the same time there.

So let's, just like this, and

so what, you can see it in your own diagram on the handout.

We're just going to, in this case,

we'll move over one more space in terms of the clocks.

Bob's clocks here, and sees something like this.

So this is what it looks like in transit, we call this Diagram 2.2,

the paintballs are in transit.

Again, the key thing here is, this paintball, even though

this clock is moving away from it, it's moving faster than this paintball,

where this clock is moving towards it with velocity v.

So they end up, in our next diagram, remember,

this is the initial launching point.

They end up, in the next diagram.

6:45

Let's see, so where were we here?

I think we're now here.

We start off one, two, three, four, so

we're right here in terms of Alice's spaceship now.

6:58

There's one clock and two clocks more or less lined up there.

Let's draw this clock a little bit better here.

There we go.

The gun apparatus in the center there.

And we got one paintball splattering on that clock and

one paintball splattering on that clock simultaneously.

So actually one thing I forgot to do, and actually you just probably saw it

on your diagram, as we went to that middle situation where the paintballs were in

transit, we should have changed the times on the clock here for Bob.

So we'll just do that, you can see it on your own, Diagram there.

7:45

So put this, and so this will be,

we'll say, in the intermediate position.

When they were in transit, that might have been TB1.

Now when they've hit, we'll call that TB2.

So all Bob's clocks will read TB2.

We won't write all of them in there, but you can.

They all read the same thing because, of course, Bob's clocks are synchronized,

as far as he's concerned.

And what about Alice?

She also sees, we'll call that, we haven't given her times on her clocks before this,

just because it actually gets a little tricky here later on.

And we're just emphasizing that Bob is the one who's doing the observing.

But certainly she also would see that

the paintballs hit her clocks at the same time.

Again, the key point here is, the reason they hit at the same time

is that even though this clock is moving away, the motion of her ship

on the paintball gun adds that velocity to the paintball velocity, and

going this direction subtracts it off that velocity, the paintball velocity.

So everything hits at the same time there.

8:56

So that is an example of a paintball experiment.

And again, you might say, what's the point?

Well, the point is, in a minute, we're going to do this with light pulses.

And we get a very different result when we do with light pulses.

Because, as we've emphasized before,

9:17

light pulses do not add in terms of velocity like we expect paintballs or

other common objects to do, especially at speeds much lower than the speed of light.

If we were doing this experiment with paintballs and

things were travelling near the speed of light, this analysis wouldn't be correct.

But we're just going to assume, regular situation,

speeds aren't very great, we're dealing with paintballs, of course, and

therefore the paintballs will hit at the same time here.

9:45

Okay, so that's Diagram 2, our paintball experiment.

And really I should, just to be precise about this,

the diagram we have on the page now, or on the board now, is Diagram 2.3.

So we did the diagram first when they're right here and the paintballs were

shot off and then halfway through when the paintballs were in transit.

And then Diagram 2.3 is when they actually hit.

And you see something, then, very similar to this on your handout.

Again, to emphasize, all Bob's clocks are synchronized here.

So he sees the paintball start off at time TB equals zero.

And then at TB1, they see them in transit.

And then at TB2, all of his clocks would register TB2, as we see here.

Okay, so that's the paintball experiment.

Now we want to do exact same thing but with light pulses, and

see what the difference is between them.