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In this video lecture, we want to return to the topic of combining velocities.

Adding or subtracting velocities you may remember, hopefully.

In week two, we did this from the perspective of sort of common sense and

also the Galilean transformation.

The essential idea there was that velocities add.

In other words, if you have, we did an example of say, Bob and his spaceship.

He had an escape pod.

If he sets off the escape pod such that it moves away from him with velocity, here

we're calling it uR for the Rocket frame and we'll get back to that in a minute.

But with some velocity u and

then if he's moving with velocity v with respect to Alice.

So again, the escape pod is moving with respect to him with velocity u,

then he's moving with respect to Alice velocity v.

As far as Alice is concerned, the velocity of the escape pod that she

sees in her frame of reference is v plus u and we had other examples of course.

If you're throwing a ball or something, you can throw it with a certain velocity

and then you get in a car or bicycle or something and you're moving in

a certain direction with an additional velocity and then you throw the ball say.

Again, the velocities add.

Or if it's a car and it's going one direction and you throw it the other

direction, then the velocities subtract and so on and so forth.

Just very simply.

The velocity is either add or subtract.

Now, we want to analyze that in terms of Einstein's theory of relativity and

before we get in to those details.

Let's remind ourselves about Lorentz transformation.

So I've written the two basic equations here where the R's stand for,

in this case, the Rocket frames.

That's what we're going to call it this time.

We could call it Bob's frame, of course.

But we'll say,

it's the Rocket frame moving with some velocity v to the right compared to Alice.

And we'll say, she's in the Lab frame.

Thus, the Ls here.

So for a given event with x and t coordinates,

a space coordinate x and a time coordinate t.

Some place along the x-axis here in terms as Bob,

essentially looks at that event in his coordinates,

then we can get Alice's coordinates of that event using Lorentz transformation.

So, want to do one quick note here,

we've got a little bit of algebra as I'm sure you have seen.

Your attention may be drawn over here.

So let's just take a look at this, because this is going to

be helpful in a minute when we look at our situation of combining velocities.

In other words, we're going to do the escape pod example again, but

this time from a relativistic perspective.

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So, that's in Alice's frame.

She just says, okay, this happens at position x2 and

then something else happens in x1 or

just the difference between two positions is x2 over here minus x1.

And so, that gives you the distance between x2 and x1 in the Lab frame.

Using Lorentz transformation equations, we can for (x2)L,

we can plugin this with the 2s here.

So we've got gamma (x2)R plus v times (t2)R.

In other words, in the Rocket frame there.

So, that's for the (x2)L.

We just took equation there.

Put that in for (x2)L and then suddenly, for minus (x1)L.

We plugged in the same equation, we're not doing the t equation.

We're just working with the x equation and plug that in over here with the x1 and t1.

And then we just multiply it through by gamma,

here's what gamma times this gives us this.

And then we know over here, we have minus gamma times (x1)R.

So we brought that over here, just to sort of get the x's together.

And then over here, we have gamma v2 sub R, the Rocket frame.

And then over here, minus gamma v(t1)R.

So, we put the time coordinates together there and

then just simplifying some algebra here with just said for these two terms here.

Let us just bring got a gamma in both of them.

So we'll just bring the gamma over here and

call it gamma times (x2)R minus (x1) R.

And, similarly, I've got a gamma there.

Actually, a gamma v here and a gamma v here.

So pull those both out and you're left with (t2) R minus (t1)R.

And then I note well, that's our delta notation that we've used before.

In other words, if I've got here, (x2)L-(x1)L, we'll call that delta xL.

The change in xL in the Lab frame.

And here I've got the same thing, but it's delta xR,

the change of x in the Rocket frame.

This is the x2 coordinate in the Rocket frame minus the the x1 coordinate in

the Rocket frame.

And then in this term, I've got the same or similar thing, but for time.

T2 in the Rocket frame minus t1 in the Rocket frame.

So that's just delta tR, the difference in time between t1 and

t2 in the Rocket frame.

Again, one and two representing two different events that we can specify

in Rocket coordinates, ball's coordinates or in Alice's Lab coordinates and

then we just clean it up a little bit more.

We've got a gamma here and a gamma here, so bring that gamma out.

I can see if you caught that, a slight mistake here.

We don't want a gamma in here.

We've got too many gammas in there.

So as we pull this gamma out and this gamma out and

put it in front, that leaves delta xR plus v rimes delta tR.

Actually, that's the final equation for that that we want.

So we noted that delta xL really, if you put delta xL here and a delta xR here and

delta tR there, it's the Lorentz transformation equation when you have

a difference in the coordinates.

Again, difference in the Rocket frame coordinates for x,

x2 minus x1 in the Rocket frame.

Delta tR, t2 in the Rocket frame minus t1 in the rocket frame.

The difference in time in the Rocket frame.

And that combination there in the Lorentz transformation format will give us delta

xL, the difference in Alice's lab coordinates between those two things.

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Similarly, you can do the same thing for the t equation.

We're not going to do it, leave that to you as a little exercise you can practice

on it late on if you would like.

You have delta tL equals gamma delta tR plus v over c delta xR.

And you might be asking why or that's sort of nice, but [INAUDIBLE] interest on this.

Well, let's come back to our escape pod situation and

remind ourselves what velocity is all about.

Velocity is simply change in distance,

the distance covered divided by an elapsed time.

And we can write that as it's distance covered and we'll say,

Distance/Time it takes to cover that distance.

In other words, we can write this, of course, as delta x divided by delta t.

The distance covered, the change in x.

How far have you gone in a certain mount of time?

Delta t.

X2 minus x1 divided by t2 minus t1.

And so clearly, for Bob here in terms of the escape pod, we could write uR.

This is the velocity that Bob is measuring for

the escape pod is simply going to be delta xR/delta tR.

Again, those are in his coordinates.

As always, Bob has his lattice of clocks.

His measuring system he's using and they're all synchronized,

Alice has her lattice of synchronized clocks that she's using.

And the Lorentz transformation allows us to switch back and

forth between those two systems.

And so now we're saying Let's look at the velocity of the escape pod in Bob's frame

of reference.

And clearly, using his measuring system he is at rest as far as he's concerned.

Very simple, just say okay, he shoots off this escape pod.

Measures how far did it travel in a certain amount of time,

and that's the velocity he's going to get for his escape pod.

Now, what we'd like to find though is what was the velocity that Alice see's

in her frame of reference for that escape pod going off?

And as you might guess since we're doing this,

it's not going to simply be uR plus v or v plus uR, whichever way you want

to do that in the simpler Galilean Ttransformation example we did before.

So what we can say though, right off the bat is just from the basic definition.

That the velocity of the escape pod in Alice's frame of reference is going to be

uL, that's just definition and it's going to be the distance covered

in her frame of reference divided by the time elapsed in her frame of reference.

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And now hopefully, you can see why we just did this little example over here, because

the Lorentz transformation allows us to have a delta xL here and a delta tL.

And so now what we have to do is erase the board to give us some room, but

take these equations here and put them into Alice's equation for

the velocity of the escape pod uL in the lab frame of reference and

see what we get for it.

So, we'll leave that up just a second here and I'll just have this.

I think we don't need too much room actually, fortunately for this.

So let's rewrite this, we'll say, okay.

Alice says, uL is delta xL over delta tL,

Laurent's transformation and we know uR up here.

We're going to remind ourselves that

uR equals delta xR over delta tR.

And that's given to us, Bob measures that for us.

Bob says, hey, I measured my escape pod going off and used delta xR, delta tR and

gives us some value for that.

So, we know what uR is.

Now, we want to find out what that value is as far as Alice sees it.

That's going to be distance covered in her frame and

the escape pod divided by the elapsed time.

So, using our results from over here and let's give us a little more room here.

We'll say this equals, so uL equals, here's delta xR.

So we write gamma(delta

xR+ v delta tR).

And this, by the way is one of the examples where we'll find the Lorentz

transformations very useful.

Why we spent the time to actually derive it, to see where it comes from.

So, it's not just a magic formula or anything like that.

So that's delta xL In terms of delta xR, delta tR.

And delta tL is gamma from this equation right here,

times delta tR + v over c squared delta xR).

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One nice thing right away, we see the gammas cancel.

So you don't have to worry about gamma here and then you look at this and say,

well, what's going on?

We'd like to make it a little simpler.

Well, we'd like to get uR in here some place, because we know what uR is.

Bob has told us what the velocity is in terms of his measurement.

Can we get something in here that involves uR?

In fact, we can if we do this.

Let's factor out a delta tR from each of these terms here and

rewrite this, and see what we get.

So remember, the gammas cancel, so we don't have to worry about those anymore.

So I've got delta tR times, well,

this term then becomes delta xR over delta tR.

This term here, delta xR can be written in this form, delta tR times this.

Denominator here, up here that cancels, that's just delta xR,

a lot more complicated format there.

Plus the second term we pulled the delta tR out here.

So, that's just going to be v times 1 or v.

Not a great v there, a little better, that's a little better.

So, that's the numerator the top part there.

Again, all I did was pull out the delta tR from this term and

delta tR from this term and rewrote it like this.

Just a little bit of algebra there.

Now let's do the same thing actually for the bottom denominator, the bottom part.

So we are going to pull out a delta tR and this first group just

becomes one then, because delta tR times 1 is delta tR.

And then I've got v over t squared here,

that's not going to change and I pulled out a delta tR from this term.

So just like we did up here,

this becomes delta xR over delta tR.

So, I just used a little algebraic trick to pull out delta tR.

And one thing we noted right away is these delta tR are going to cancel, so

we get rid of those, but why did we do this?

Well, look what we've got inside here.

I've got a delta xR over delta tR.

Another one down here actually, delta xR over delta tR and

that is uR by definition.

So, I can replace this and this with just the value uR that Bob is going to tell us.

And now we'll erase this over here, so we can write it nicely.

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So, this comes up here and what have we got?

We've got and will just rewrite uL here.

Remember, this is the velocity that Alice sees in other pod,

escape pod and he referencing.

So I've got uL equals on the top, this is just a uR.

By definition, delta xR over delta tR plus v.

And on the bottom, I've got one

plus v over c squared time uR.

That's my answer.

Now I have found if I know how fast Bob is moving with respect to Alice and

Bob shoots off his escape pod and he tells me how fast it's going in his frame of

reference where again, he is stationary.

He's measuring it receding away from him, going away from him and

he gives me that velocity here,

then I can find out what Alice is going to measure the velocity of the escape pod as.

It's going to be the velocity that Bob measures plus the reference frame velocity

here, the difference of velocity between the two reference frames.

And then on the bottom, 1 plus v over c squared uR.

Sometimes we write this like this, just slightly different form.

Obviously, if we're going to do v plus uR.

Ur plus v, if you want.

And then on the bottom, sometimes we

just write it as 1 + uR v over c squared.

Or it could be v uR over c squared, it doesn't really matter there.

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Well, it sort of looks like the Galilean transformation if you think about it.

Because, look at this.

On the top, on the numerator, we have uR + v.

That's just what we did normally in everyday, common sense experience,

Galilean Experience you could call, I suppose.

You just add the velocities.

You say, Alice would say okay, it's just going at u sub R for Bob.

Then I'm just going to add the velocity to it and

that would give me the velocity that I see.

Except, we've got this denominator factor down here.

And know what is down here,

we've got a 1 + v over c squared factor, or uRv over c squared.

Note that this factor here is going to be very small

unless both v and uR are close to the speed of light.

Under normal circumstances, here I've got 1 plus 0.0000000,

a whole bunch of zeros out there, fifteen or so.

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And therefore in real life, even though this relativistic effect really is true,

would happen if we can measure that precisely,

in real life we don't notice it.

This just becomes uR + v over 1 when v, the relative velocity

between the two frames of reference, and uR, velocity of the escape pod,

velocity of the object in the other frame of reference, is small compared to the c.

So we see that this reduces to our normal Galilean transformation,

addition of velocities, combining of velocities that way, when uR and

v are non-relativistic as we might say, in other words, not near the speed of light.

Now let's look at a couple of other things too, though.

So, that makes that good news.

If we didn't get that,

something would have been wrong, because we know in our everyday experience,

the Galilean Transformation, addition of velocities works.

Let's also look at the case, what if u sub R = 0?

It's always good, as we've done before,

to just try some simple numbers in here and see what happens.

So if u sub R = 0, notice that the velocity of the escape pod is 0 to Bob,

which means he hasn't shot it off from his ship.

We would expect then, it's just going to be traveling along with the ship.

Alice should see the velocity of the escape pod just to be v because that's

what the velocity of Bob's ship is.

And if the escape pod is just stuck to the ship,

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it's velocity is got to be v as well, in terms of what Alice observed.

So, let's see if our equation works, u sub R = 0.

So let's plug that in here.

So we'll say that leads to u sub L = 0 + v over 1 +,

well if u sub R = 0, this whole thing over here becomes 0.

0 times anything is 0 of course, so it's just 1 + 0.

I get v divided by 1, that's just v.

And that's exactly what we'd expect to get.

All right?

So that if the velocity of the pod or whatever the object is,

is zero then the velocity as Alice sees it of the object is just the velocity of

the difference in the frames of reference really,

the relative velocity between them.

So that's one thing to check.

Let's do another one here.

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Okay, if v is 0, then think about what's happening here.

That means Bob and

Alice could, they're standing next to each other, Bob shoots off the escape pod.

In other words, Bob is not moving with respect to Alice.

They're side by side, stationary with respect to each other.

If Bob shoots off the escape pod, Alice, and he sees it at velocity u sub R.

Well then, of course, Alice should see the velocity u sub R as well.

So let's see if that's what we get.

If v is 0, on the top I get u sub R + 0 over 1 +, and

this time v, velocity between the frames of reference is 0.

They're essentially the same frame of reference.

So this whole things becomes 0 again.

And again, you've got 1 + 0, and this just equals u sub R.

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Now let's do another case here where what happens if, let's say this.

Let's say, let's let v here,

let's get up to relativistic speeds and see what happens.

Let's say v is 0.9c, nine-tenths the speed of light.

And Bob has a super fancy escape pod, as well as a super fancy ship.

And he can shoot off his escape pod as he measures it at 0.7c,

0.7 times the speed of light.

So that means it's receding away from him at seven-tenths the speed of light.

Again, as far as he's concerned, he's stationary in his frame of reference.

He sees the escape pod going out.

So the question is, what does Alice see in terms of the velocity of the escape pod?

So let's try out our formula here.

So we've got u sub L, velocity in the lab frame that Alice is going to see.

So u sub R is 0.7c + v is 0.9c, for example.

On the bottom, we've got 1 + u sub R, 0.7c, times v, or

the other way around if you want to do it that way, but

we'll do the bottom one here, times v is 0.9c, all over c squared.

Okay?

We'll just put that there.

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So what do we have here?

We'll do one more step, squeeze it in here.

So, 0.7c + 0.9c, 1.6c, greater than the velocity of,

let me rewrite that so it looks a little better.

1.6 c, definitely greater than the speed of light there.

What do we have on the bottom?

We've got 1 + 0.7c times 0.9c.

So that's 0.7 times 0.9 is going to be 0.63, 9 times 7 is 63,

you get the decimals in there it's going to be 0.63.

And I've got a c squared there.

So I got 0.63c squared all over c squared, there.

c squareds, of course, cancel.

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And intuitively we think well, that's the way it should work.

In actual fact, it doesn't work that way.

That because of the Lorentz transformation,

how well that works, we put this together,

found out the velocity in the lab frame, we got this equation here.

This equation will always give us, no matter what these two values are,

again as long as they are less than the speed of light for objects that

we are dealing with, then there's, will always be less than the speed of light.

It adds in sort of a weird way that way.

But it also shows us there's something about that ultimate speed limit

that we'll talk more about actually in a video lecture coming up.

We've mentioned a few times before that for whatever reason,

which is sort of built in the theory here, you cannot go faster,

you cannot actually get a material object up to the speed of light.

Seems to be a natural speed limit in the universe.

Okay, so that is our basic combining of velocity equation here.

Again we got it from Lorentz transformation and

just basic definition of the velocity, and

we see the Galilean transformation is embedded in it in a sense.

Because if uR, the escape pod velocity, and the relative velocity

between the frames are low with respect to the speed of light,

then we just get our normal uR plus v, just add velocities.

But, when we get speeds up close to the speed of light,

they do not add as we expect them to.

So there's that a little bit later here in another video lecture, we'll look at

the case where, what happens if the escape pod actually goes off to the side?

In the perpendicular angle.

Either straight up or off to the side here.

So, we'll deal with that as well.