6 - 29 - Combining Velocities (25-16, Low-Def)

8/10/2019 6 - 29 - Combining Velocities (25-16, Low-Def)

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In this video lecture, we want to returnto the topic of combining velocities,adding or subtracting velocities.You may remember hopefully in week two wedid this from the perspective of sort ofcommon sense and also the GalileanTransformation.And the essential idea there was thatvelocities add, in other words, if youhave, we did an example of, say, Bob inhis spaceship, he had an escape pod.If he sets of the escape pod such thatit, it moves away from him with velocity,here we're calling it u sub r for therocket frame.We'll get back to that in a minute, butjust some velocity u, and then if he'smoving with velocity v with respect toAlice.So, again, the escape pod is moving withrespect to him, with velocity u, thenhe's moving with respect with respect toAlice velocity v.As far as Alice is concerned, the

velocity of the escape pod that she seesin her frame of reference is v plus, plusu.And we did, had other examples, ofcourse.If you're throwing a ball or something,you could throw with a certain velocity,and then you, you get on a car or bicycleor something.And you're moving in a certain directionwith an additional velocity, and then youthrow the ball say again the velocityadd.

Or if it's a car and it's going onedirection and you throw it the otherdirection then the velocity subtract, andso on and so forth.But very, just very simply, the velocityis either add or subtract.Now we want to analyze that in terms ofEinstein's theory of relativity.And before we get into those details,let's remind ourselves about the Lorenztransformation.So, I've written the two basic equationshere where the R's stand for in this case

the rocket frame.So, that's what we're going to call itthis time.We could call it Bob's frame of course,but we'll say it's, it's the rocketframe, moving with some velocity v to theright, compared to Alice.And we'll say she's in the lab framethat's the ls here.So, for a given event with x and t


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coordinates a space coordinate x, and atime coordinate t, some place along thex-axis here.In terms of, as Bob essentially looks atthat event in his coordinates, and thenwe can get Alice's coordinates for thatevent using Lorentz transformation.So, I want to do one quick note, herewe've got a little bit of algebra, as I'msure you've, you've seen,and may, you'reattention may be drawn over here.So, let's just take a look at this,because this is going to be helpful in aminute, when we, when we look at our,situation of combining velocities, inother words we are going to do that.The escape pod example again, but thistime from a relativistic perspective.So, note that if we have two events inthe lab frame at positions x2 and x1.Okay, so that's within Alice's frame.She just says, okay, this happens atposition x2 and then something elsehappens at x1, or just, the difference

between two positions is x2, over here,minus x1.And so that gives you the distancebetween x2 and x1, in the layout frame.Using Lorentz transformation equations,we can, for x2 sub l, we can plug inthis, with the 2s here.Okay, so we got gamma x2 sub r, plus v,times t sub 2 sub r, in other words inthe rocket frame there.So that's for the x2 sub l, we just tookthe equation there, put that in for x2l,and then similarly for minus x1 sub l.

we plugged in the same equation.We're not doing the t equation here,we're just working with the x equationand plug that in over here with the x1and t1.And then we just multiply it through bygamma here.So, gamma times this gives us this, andthen we know that over here we have minusgamma times x1 sub r, so we brought thatover here, just to sort of get the x'stogether.And then over we here we have gamma v 2

sub r, the rocket frame.And then over here minus gamma vt1 sub r.So we put the time coordinates togetherthere.And, and then just simplifying somealgebra here we just said for these 2terms here lets just bring.We got a gamma in both of them.So we're just bringing the gamma overhere.


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And call this gamma, times x2 sub r minusx1 sub r.And similarly over here I've got a gammathere, actually a gamma v here, and agamma v here, so pull those both out andjust left with t2 sub r minus t1 sub r.And then I note, well that's our deltanotation that we've used before.In other words if I've got here x2 sub lminus x1 sub l, we'll call that delta xl,the change in x sub l in the lab frame.And here, I've got the same thing, butit's delta x sub r, the change in x inthe rocket frame.This the x2 coordinate in the rocketframe minus the x1 coordinate in therocket frame.And then in terms I've got the same, orsimilar, thing, but for time.T2 in the rocket frame minus t1 in therocket frame.So, that's just delta tr, the differencein time between t1 and t2 in the rocketframe.

Again, one and two representing twodifferent events that we can specify inrocket coordinates, Bob's coordinates, orin Alice's lab coordinates.And, then we just clean up a little bitmore.We've got a gamma here and a gamma here.So we'll bring that gamma out.I can see if, if you caught that slightmistake.We don't want a gamma in here.We've got too many gammas in there.So it becomes, now as we pulled this

gamma out, and this gamma out to put itin front.Now these delta x sub r, plus v timesdelta t sub r.And then, so.Actually, that's the final equation forthat that we want.So we noted that, delta x sub l.Really, if you just put delta x sub lhere, and a delta x sub r there.And delta t sub r there.Its' the Lorentz transformation equation.When you have a difference in the

coordinates.Again, difference in the rocket framecoordinates for x, x2 minus x1 in therocket frame, delta t sub r, t2 in therocket frame minus t1 in the rocketframe, the difference in time in therocket frame.And that's, that combination, there, inthe Lorentz transformation format, willgive us delta x of l, the difference, in


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Alice's lab coordinates between those twothings.Similarly you can do the same thing forthe t equation.We're not going to do it.I'll leave that to you as a littleexercise to practice on if you like.You've got delta t sub l equals gammadelta t sub r plus v over c squared deltax sub r.Now you might be asking, you know, why,that's sort of nice, but why are weinterested in this.Well let's come back to our escape podsituation, and remind ourselves whatvelocity is all about.Velocity is simply change in distance,the distance covered divided by an lapsedtime.And we can write that as so it's thedistance covered, I'm going to saydistance divided by the time it takes tocover that distance.In other words we can write this of

course as delta, delta x divided by deltat, the distance covered, the change in x.How far have you gone, in this certainamount of time?Delta t.X2 minus x1, divided by t2 minus, minust1.And so clearly for, for Bob here, interms of the escape pod, we could write usub r.Okay?This is the velocity that Bob ismeasuring for the escape pod is simply

going to be delta x sub r divided bydelta t sub r.And again those are in his coordinates.And as always, Bob has his lattice ofclocks, his measuring system he's usingAnd they're all synchronized.Alice has her lattice of synchronizedclocks that, that she's using, and theLorentz transformation allows us toswitch back and forth between those twosystems.And so now we're saying, let's look atthe velocity of the escape pod in Bob's

frame of reference.And clearly, using his measuring system,he's at rest as far as he's concerned,very simple to just say okay, how far didthis escape.And he shoots off the escape pod,measures how far did it travel in acertain amount of time and that's thevelocity he's going to get for his escapepod.


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Now what we'd like to find though is whatis the velocity that Alice sees?In her frame of reference, for thatescape pod going off.And as you might guess, since we're doingthis, it's not going to simply be u sub rplus v or v plus u sub r, whichever wayyou want to do that in the simplerGalilean transformation example we didbefore.So what we can say though, right off thebat, is just from the basic definition.That the velocity of the escape pod, inAlice's frame reference is going to be usub l which is definition.It's going to be the distance covered inher frame of reference, divided by thetime elapsed in her frame of reference.And now hopefully you can see why we justdid this little example over here.Because the Gal, the Lorentztransformation allows us to have a deltaxl, here, and a delta tl.And so now what we have to do is erase

the board a little bit to give us someroom, but take these equations here andput them into the equation.Alice's equation for the velocity of theescape pod u sub l, in the lab frame ofreference, her frame of reference, andsee what we get for it, okay?So we'll leave that up just a secondhere.now let's just, have this, I think wedon't need too much room actually,fortunately for this.So let's rewrite this we'll say okay

Alice says u sub l is delta x sub l overdelta t sub l.Lorentz Transformation, okay?And we know u sub r up here or minus lusub r equals delta x sub r over delta tsub r and that's given to us, Bobmeasures that for us.Well say hey I measured my state partgoing off and use delta x sub r, delta tsub r and gives us some value for that.So we know what u sub r is, now wewant to find out what that value as faras Alice sees it.

That's going to be just discovered in herframe.And, of the escape pod, and divided bythe, the elapsed time.So, using our results from over here, andlet's give us a little more room here,we'll say this equals.So, u sub l equals, here's delta x sub r,so we write gamma times, delta x sub rplus v, delta t sub r, and this by the


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way is one of the examples where we'refinding the lens transformations veryuseful.Why we spent the time to actually deriveit, see where it comes from so it's notjust a, a magic formula or anything likethat.so that's delta x sub l.In terms of delta x sub r and delta t subl is gamma from this equation right heretimes delta t sub r plus v over c squareddelta x sub r.'Kay?One nice thing right away we see.The gammas cancel.So we don't have to worry about gammahere.And then you look and they say well,what's going on?Well, we'd like to get u sub r in heresome place.Because we know what u sub r.Bob has told us what the velocity is, isin terms of his measurement.

Can we get something in here that's thatinvolves u sub r?And in fact, we can if we do this, let'sfactor out a delta t sub r from each ofthese terms here.Okay, and rewrite this and see what weget.So, remember the gammas cancel, so wedon't have to worry about those any more.So, I've got delta t sub r times, wellthis term then becomes delta x sub r overdelta t sub r, right?This term here delta x sub r can be

written in this form, delta t sub r timesthis, the denominator here, up here thatcancels that's just delta x sub r, in alittle more complicated format there.Plus the second term we pull the delta tsub r out here, so that's just going tobe v times 1, or v.Oh a great v, there, I'll make it alittle better.Looks a little better.Okay?So, that's the numerator, the top partthere.

Again, all I did was pull out the delta tsub r from this term, and the delta t subr from this term, and rewrote it likethis.Alright?Just a little bit of algebra, there.Now, let's do the same thing, actually,for the, the bottom denominator, thebottom part.So, we're going to pull out a delta t sub


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r.And, this first one just becomes 1, then,because delta t sub r times 1 is delta tsub R.And then I've got a, the v over csquared.That's not going to change.And I've pulled out a delta t so far fromthis term, so just like we did up herethis becomes delta t sub r.Oops, sorry.I meant to say delta x sub r over delta tsub r.So, I just used a little algebraic trickthere to pull out delta t sub r.And one thing that we notice right awayis these delta t sub rs are going tocancel, so we get rid of those.But why do we do this?Well, look what we've got inside here.I've got a delta x sub r over delta t subr.Another one down here actually, delta xsub r over delta t sub r.

And that is u sub r, by definition.So, I can replace this and this with justthe value u sub r, that Bob was going totell us and now we'll erase this overhere so we can write it nicely.So, this comes up here, and what have wegot?We've got, we'll just rewrite ul here.Okay remember this is the velocity thatAlice sees and, of the pod, escape podand her reference frame.So, I've got u sub l equals on the top.This is just a u sub r by definition,

delta x sub r over delta t sub r, plus v.And on the bottom, I've got 1 plus v overc squared times u sub r.And that's my answer, okay?Now I have found if, if I know how fastbob is moving with respect to Alice, andBob shoots off his escape pod.And he tells me how fast its going in hisframe of reference where again he isstationary, he's measuring it recedingaway from him, going away from him.And he's, it gives me that velocity here,then I can find out what Alice is going

to measure of the velocity of the escapepod as it's going to be velocity that Bobmeasures.Plus the velocity between, referenceframe velocity here, the difference ofvelocity between the two referenceframes.And then on the bottom 1 plus v over csquared v sub r.sometimes we write this like this just


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so, you know, it's slightly differentform.Obviously [UNKNOWN] do v plus 2 sub r useof r plus v if you want.Into that, and then the bottom sometimeswe just write it as one plus u sub r, vover c squared or I could do v sub r csquared.It doesn't really matter there.Okay, let's look at what this is tellingus.First of all, so that's our final answer.It was our derivation, not too difficult,hopefully.And remember where it came from.We just said, by definition, velocity ischange in position divided by the theelapsed time.And we just use the Lorentztransformation equation, to go from Bob'sframe of reference to the version inAlice's frame of reference.And again this is what we get here.So, this comes over here.

Well, it sort of looks like the Galileantransformation, if you think about it.because look at this, on the bottom, onthe top, the numerator, we have u sub rplus v.That's just what we did normally in, ineveryday common sense experience.Galilean experience you could call it Isuppose.You just add the velocity.You say Alice would say okay, It's justgoing at u sub r for Bob and then I'mjust going to add the velocity to it.

And that will give me the velocity that Isee, except we've got this denominatorfactor down here.And know what is down here, we've got a 1plus a v over c squared factor or urvover c squared.Note that this factor here, is going tobe very small unless both v and ur arepretty close to the speed of light.Under normal circumstances, here I've got1 plus 0.0000000.You know, a whole bunch of 0's out there15 or so.

and therefore in real life, even thoughthis relative, relativistic effect reallyis true, would happen if we could measurethat precisely, in real life we don'tnotice it.This, this becomes u sub r plus v over 1when v, okay, the relative loss betweenthe two frames of reference.And, u sub r, velocity of the escape pod,or the velocity of the object in the


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other frame of reference, is small,compared to the c.So, we see that this reduces to theGalilean, are normal Galileantransformation, addition of velocities,combining of velocities that way.When u sub r and v are, arenon-relativistic as we, as we might say,in other words, not near the speed oflight.Now, let's look at a couple other thingstoo, though.So, so that makes as a good news.If we didn't get that, something wouldhave been wrong because we know in oureveryday experience, the Galileantransmission, addition of velocitiesworks.Lets also look at the case, what if u subr equals 0.It's always good as we've done before tojust try some, some simple numbers inhere and see what happens.So if u sub r equals 0, notice if the

velocities of the escape pod is 0 to Bob.Which means he hasn't shot it off fromhis ship.Okay, we would except then, it's justgoing to be travelling along with theship.Alice should see the velocity of theescape just to be v.Because, that's what the velocity of theship is, and if the escape pod is juststuck to the ship, it's velocity has justgot to be v as well, in terms of whatAlice observes.

So, lets just see if our equation works.U sub r equals zero.So let's plug that in here.So we'll say that leads to u sub l equalszero plus v over 1 plus, well if u sub ris zero this whole thing over herebecomes zero.Zero times anything is zero of course, soit's just 1 plus zero, I get v divided by1 that's just v.And that's exactly what we'd expect toget.So, that if the velocity of the pod, or

whatever the object is, is zero, then thevelocity as Alice sees it of the objectis just the velocity of the difference ofthe frame of reference, really, therelative velocity between them.So that's, that's one thing to check.Let's do another one, here, okay, so thatchecks out.What if v is zero?Okay, if v is zero then, think about


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what's happening here, that means Bob andAlice could, they're standing next toeach other.Bob shoots off the escape pod.In other words Bob is not moving withrespect to Alice.They're side by side, stationary withrespect to each other.If Bob shoots off the escape pod, Alice,and he sees it at velocity u sub r, wellthen of course, Alice should see it atvelocity 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 plus 0over 1 plus,this time v.Velocity between the frames of referenceis 0.They're essentially the same frame ofreference.So, this whole thing becomes 0 again,again you got 1 plus 0 and this justequals u sub r, alright so that checksout as well.So that's, that's good news.

now though let's do another case here,where what happens if, let's see this.Let's say, let's let v here, let's get upto relativistic speed and see whathappens.Let's say v is 0.9c, okay, 9 10th thespeed of light.And Bob have, has a super fancy escapepod, as well as a super fancy ship.And he can shoot of his escape pod as hemeasures it at 0.7c, 0.7 times the speedof light.So, that means it's receding away from

him at 7 tenths the speed of light.Again, he, he's, as far as he'sstationary in his frame of reference.He sees the escape pod going out.So, the question is what does Alice seein terms of the velocity of the escapepod.So, let's try out our formula here.So we got u sub l, the velocity in thelab frame that Alice is going to see.So u sub R is .7 c.Plus v is 0.9c, for example.On the bottom we've got 1 plus, u sub r

0.7c times v.Or the other way around if you want to doit that way, but we'll do the bottom onehere, times v is 0.9c, all over, csquared, okay?Let me just put that there.so what do we have here?We'll do one more step squeeze it inhere.So, 0.7c plus 0.9c, 1.6c, mm.


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Greater than the velocity of, let mewrite that so it's a little bitter, 1.6c.Definitely greater than the speed oflight there.What did we have on the bottom?We got 1 plus 0.7c times 0.9c.So, that's 0.7 times 0.9, it's going tobe 0.63, 9 times 7 is 63.We got the decimals in there, it'sgoing to be 0.63 and I've got a c squaredthere.So, I get 0.63 c squared all over csquared, from there.C squared of course cancel.Those cancel, and look what I'm left withover here.This equals, 1 equals 1.6 c.All over 1.63, the c squareds are gonethere, so this becomes, you'll have tocalculate for yourself.But it is roughly it looks like 0.9something c here 0.9, you know somethingelse there, times c.

The important point is not so much theexact figure there, is the fact that itis less than the speed of light.And in fact, the way this equation worksis, no matter what speed you put in for ur and v as long as they're, are less thanthe speed of light.Then the result will always be less thanthe speed of light.In other words, Galileo and our commonsense would say, hey, If I've got 0.9 Cand 0.7 C for the escape pod shootingoff, shouldn't Alice see the escape pod

be at 1.6 times the speed of light?And intuitively we think well that's,that's the way it should work.In actual fact, it doesn't work that way.That, because of the LorentzTransformation, how that works, we putthis together, found out the velocity inthe lab frame.We get this equation here, this equationwill always give us, no matter what thesetwo values are again, as long as they areless than the speed of light, for objectsthat we're dealing with.

Then, the result will always be less thanthe speed of light.It adds in sort of a weird way, that way.But it also shows us, there's somethingabout that ultimate speed limit, thatwe'll talk more about actually in a, in avideo lecture coming up.We've mentioned a few times before thatfor whatever reason, which is sort ofbuilt in the theory, here, you cannot go


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faster.You cannot actually get a material objectup to the speed of, of light.It seems to be a natural speed limit inthe universe.Okay, so that is our basic combining ofvelocity equation here.Again we got it from LorentzTransformation, and just basic definitionof velocity, and we see the Galileantransmission is embedded in it in a sensebecause if you are.the state pi velocity and the relativevelocity between the frames are low withrespect to speed of light.And we just get our normal u sub r plus vthis add velocities, but when get speedsup close to the speed of light, they donnot add we expect from to.So, there's that little bit later anothervideo lecture we'll look at the case wellwhat happens if escape pad actually goesoff the side.At a perpendicular angle either straight

up or, or off to the side here.So, we'll deal with that as well.

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6 - 29 - Combining Velocities (25-16, Low-Def)