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In a previous video clip, we looked at combining velocities really adding

velocities is what we were doing when we had motion in the x direction.

So remember, we used Lorentz transformation, and

you got an equation like this, and what I've done here is added these x Subscripts

to the velocity U sub R remember is the velocity in the rocket frame or

example was Bob here going with some velocity in

Alice's you're save [INAUDIBLE] in a picture

going with some velocity v with respect to Alice's frame of reference.

And showing off an escape pod in his direction of

motion with velocity u sub R compared to him.

And we discovered then the velocity that Alice would observe

the escape pod to go followed this equation how.

We did a few examples with that just to check out how it might work.

And in fact, we saw that even though the two velocities here,

the velocity of Bob with respect to Alice of the space ship and

then plus the velocity of the escape pod.

If they were both added up so

they were greater than c just by adding them together.

This equation would be such that the result would never be greater

than c in terms of what Alice observed in terms of the velocity of the pod.

What we want to do now is do another case where what happens above shoots it up

maybe straight up or just perpendicularly off to the side here and

figure out what would Alice see in that situation, and interestingly enough this

will bring us back to the light clock that we are talking about recently.

So, here's the situation, Bob shoots it up at some velocity u sub R, and

we'll use a subscript y emphases just in the y direction now.

You can say it's the z direction, but

we'll choose the y direction for that direction.

And of course, definition of velocity, delta y sub R over delta t sub R.

This is for Bob now, it says, how far does he go in the y direction?

And with the elapsed time for how far it's going in that direction.

That's going to be the definition of the velocity.

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over delta TL just by definition of the velocity, and

this why I as reminder added the Y and Z components, so

that's transformation remote they don't transform it all they're the same and

so delta Y sub L is going to be the same as delta Y sub R.

So this is going to be delta y sub R over delta t sub L.

But that does change of course by our equation we used before.

Remember, we could use deltas in here to indicate change of quantity.

So we'll write that as delta y sub R on the top.

And then delta t sub L, remember, using the equation here, is going to be gamma

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Times delta t sub r plus v over c squared delta x sub r.

Okay again, these are the quantities that Bob is measuring, and we're

assuming everything is constant velocity here for those of you who had some physics

you say well what if it's accelerate, well we're not dealing with that case here.

And therefore we can define velocity as,

not the change of direction, but the lapse direction if you want.

Direction covered divided by the elapsed time.

So, delta y sub R over this quantity here.

Again we'd like to simplify,

we'd like to be able to get some of the u sub R rocket frame y components in here.

And so let's pull out this delta t sub R, sort of like we did before, and

see what happens here.

So, we have delta y sub R on the top still,

and got to gamma here.

Gammas don't cancel this time.

And we're going to pull out a delta t sub R from that expression.

So we've got delta t sub R.

Times 1 plus v over c squared delta x sub r over delta t sub r.

Okay so delta t sub r times 1 is delta t sub r.

Delta t sub r plus this thing gives us,

because the delta t sub r's cancel there of course.

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okay well so but in our example here, note what happens up for

Bob, not what happens, but what is happening for Bob here.

There is no velocity of the pod in the x direction.

Remember, Bob is stationary.

As far as he's concerned, he's shooting the pod straight up,

as far as he's concerned, straight off to the side.

And therefore there is no velocity in the X direction for this example.

So, in this example this equals 0.

No velocity of the pod in the X direction, as far as Bob is concerned.

And so, this that's nice, because this whole term then here is 0 for us.

So, this thing here turns into zero for this case.

And look at what we're left with here, delta y sub R divided by delta t sub R,

and also divided by gamma.

Delta y sub R divided by delta t sub R,

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in the y direction divided by gamma okay.

Now we're going to come back to that in a minute, but

let's do one other thing here too just to make sure here, because for

bob it's going just straight up Right but for Alice Bob is moving that way.

So Alice may and probably will and

actually will see will observe that the pod is going to go up at an angle,

because you do have the velocity of motion here of the space ship involved.

And so let's see if we can get that in here as well, so

we'll squeeze it in right here I just did that.

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Right? As far as Bob is concerned,

no velocity in the x direction, so I get zero plus v

all over one plus, well again u of our x is zero to Bob in this example.

Therefore, this whole thing is zero, so I just get one plus zero, and

in fact we did this example, when we're talking about

the transformation equations here for the velocity, this just gives us, v.

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So, there are true components as we are talking about velocity here for Alice.

One component is just the velocity V and the X direction,

because that's just the motion of the spaceship here, and the other compound of

the velocity in the Y direction for Alice is U sub Y R over gamma, so it is whatever

velocity Bob sees here shooting up then that's going to be divided by gamma.

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Note one very interesting thing here, Gamma is the denominator, so

as the velocity of light gets,

not the velocity of light the velocity of the pod shooting up here gets faster and

faster closer to the speed of light remember what happens to gamma.

Gamma gets close to infinity, and

therefore this whole thing no matter how big this velocity is

this is like something divided by a very, very, very big number this goes to zero.

So this approaches zero as v

approaches c or gamma another way to say it gamma approaches infinity.

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So that's very interesting, because what that's saying is no matter how fast Bob

shoots up his escape pod, if his relative velocity is close to the speed of light,

Alice sees almost no upward velocity at all.

This is Alice's upward velocity in the y direction that she's observing.

If gamma is very large, if Bob's velocity is very close to the speed of light,

this whole thing goes to zero.

And all she sees is this sideways motion from the x direction v.

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And the reason, I mentioned the light clock a minute ago.

Because remember,

we did an example with the light clock where of course, the light clock.

So, here's Bob's light clock moving with velocity v in that direction.

And of course we saw from Alice's perspective,

the light beam goes up here and then a little down there.

So, light clock has to get a little further over there, and

we talked about as v gets closer and closer to the speed of light remember that

green in here again, that this light beam can never catch up to that upward mirror.

Because if this gets close to c,

then this light beam has to travel too far to the upper mirror.

By the time it essentially gets there, the mirror would be farther away.

And in fact as it gets very, very close to c this triangle gets very, very elongated,

and in fact this light beam never gets up to that top mirror and it's frozen.

The light clock as Alice would observe it,

if Bob could go by at the speed of light, which he can't but if he gets very,

very close to it It's almost like the light clock is frozen.

It ticks very, very slowly.

It's just a time dilation effect.

Gamma's very large and, therefore, Bob's clock ticks very slow.

If you could actually reach the speed of light, it's frozen.

And that's what this Mathematics here is telling us another way that we argued for

qualitatively earlier, I never put the green in here,

that if this is the velocity of light, if we're going c here, and it can get up here

at c, but it never gets up to that top mirror, that's what this is telling us.

It's saying if, we can go with the velocity of light, If Bob could shoot, or

if Bob could have a space ship with velocity v of light here then this would

be infinite, and Alice's observed velocity for the escape pod going up no

matter what the velocity of the escape pod was would be zero in the upward direction.

And should just see the C factor going across there.

So tells us another way.

It's good to have these consistency checks here, because you could say,

just to make the point in one other way here this light clock is sort of like this

scape pod if this scape pod instead of being a scape pod were just a beam of

light going up in that direction, make it squiggly for beam of light here.

A beam of light going up in that direction c, that's exactly what the light clock

is for Bob's perspective, is going up and down like that.

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She just sees Bob go this way, and the light clock is frozen in that instance.

So an interesting confirmation of some of the other things we've done.

This is one of those things that takes a while just to mull it over,

think about it, work through it.

Convince yourself that it actually makes sense,

put in some numbers, try things out, sort of get a feel for what is going on here.