Astronomy to Relativity and Gravitation Electricals.
Electric tutorial so-called
and it's fairly three format. I never I've never convinced I
have a a really good plan for what to do in this hour, but
unimaginative. But I think going through a couple of the
exercises in the exercise collection is probably a decent
idea, if not necessarily terribly exciting one.
Does anyone have any meeting suggestions for a better for
better plans?
Other particular areas you are particularly hoping I'll cover?
Yeah, yeah, I'll certainly mention those. Yeah.
In fact, let's just keep going on that.
So as I
mentioned in the
I mentioned a couple of times
and also on the on the padlet, the
aims objectives are several pedagogical fashions out of
date. But I find that the distinction useful.
The distinction being that the aims are the point and the
objectives are the accessible things.
So you, you, you, you, you've seen this document, it's no you
haven't seen this compendium version. But you have the the
aims, objectives at the beginning of each chapter and
and and I will if necessary tweak those next semester once I
I start thinking about this a little bit more. But they have
settled down over the last couple of years to to to what
they are.
The aims, as I mentioned, are the high level
point of the course, but they tend to be expressed in terms of
understand, appreciate, blah blah blah rather than things
that you can actually put in an exam and so the objectives
which roughly match. But I'm not the same thing, as the aims are
the things that I think are fair game for the examiner.
There's rather a long list of these. If you if you put them
all together in one document that you do end up with quite a
few of them. I'm not going to go through them step by step,
but the reason that they're
there's lots of them. It's simply because
I I find it useful to,
you know, concretize. What do I think is the accessible thing in
this in this chapter?
So, so that's why there's several of them.
There's also quite a few of them to give me a little bit of scope
when I'm making the exam. So these are all individually,
there's too small to see, but you've got them in in in the, in
the, in the notes. These are all individually quite small things.
Very few of them are. Remember such and such.
Basically I don't have a good memory for for, for, for, XYZI.
Don't feel it's it's fair game to get you to memorise things,
which I, I I wouldn't. And similarly derivations people
sometimes people would like to put. It's an easy exam question,
you recite the derivation of of such and such. It's not a very
exciting exam question. It's not actually very useful thing to be
able to do,
so there aren't. There are a couple of things where I see you
should be able to
remember this blah blah. I think one of them is for example,
to quote the expression for the scalar product between 24
vectors.
I think you should have to do that because of heaven's sake.
If you can't do that, you know that's pretty bad, but that but
that should that should just come naturally in a sense,
because you have used it several times in doing the exercises. So
take those seriously. Things that are clearly not covered by
the objectives are.
I will regard it out of scope for the exam.
Things which are clearly, you know covered by that I very much
regarded in school for the exam
Italy and it's borderline. Do feel free to ask for
clarification, OK, but that that's the I I look at that when
I'm when I'm making the exam.
Any questions,
less of them, I will. Yes. Yeah
it's just duplicating what's in each chapter but I
in previous years I have sometimes used slightly edited
the list when making turn it into a a full document just
because I I've, I've I've gone through them and thought that's
not very clearly expressed or something. So I think yes, I
will. I, I will put up the I can bend your version of of that and
probably next around the time when you're starting to worry
about you
anything else on that particular topic.
OK. So does that, does that answer you the question you had
to some extent,
Yep.
OK
and you were also in the lecture. That folder there is
the list of all the exercises.
And again this is we will, I will be going through to
touching on this
you the rest of the hour
just to repeat myself again, the A2 tutorial handbook that's
mentioned in in the Moodle includes a section on includes
exercises for the vote, the four sub courses within a 2, the
section of relativity. They're good exercises. They're not
particularly keys to the way I've ended up doing these
lectures. So when I talk about the exercises, it's these ones.
I mean,
OK and in
it's a long story, but in previous years these exercises
were included in the
the lecture notes document chapters that you you downloaded
and they were taken out because folks thought there were too
many pages in those files and and and OK, but I think I
probably should put them back in because it's a bit confusing
having them separate. But you will find that document in the
Election Notes folder.
There are quite a few of them, as you will see if you examine
it
and
for example.
But you didn't have example. OK, it's slightly too small to see,
but at the end of each of them you'll see which objectives they
actually I I feel they
touch on
and some of them marked as slightly more difficult or
slightly less difficult than than the main and some are
marked as particularly useful.
So this
the minus is I I think that all all the quick questions they ask
in the in in the lecture are D minus are particularly easy
ones. The one at the top there is a hard one and so on. So
that's what that notation means in case you're thinking of what
wants to select
and
neither the the the the the total exercise vary quite a lot
they from from very simple very simple things just like the
quick questions in the lectures to things which are quite
thoughtful and we'll we'll we'll have you puzzling over them for
an afternoon. So there's quite a lot of variation in those. So
those really aren't. I mean, that's a good question. Those
are are are a poor guide to what's in the in a test or an
exam
because some of them are
yeah. The the the fairly variable and and and and and.
Next next semester I will post a a documentary includes notes on
the on on solutions of them.
I don't do that in the first semester because reasons, but I
I will post an answer. This and some of some of those notes are
like a couple of lines. Some of them are are extensive
discussion of of of the of the point.
So as usual the best guide to what's in the class test or
exams or the like is previously previous papers as usual and
those tend to have when I'm marking them. I tend to write
notes about what people did and what people the the various
things that odd things people and put when answering questions
and to those those notes and fairly chatty for past papers
for the relative reputation stuff. And those are much longer
than would be model model answers.
They tend to be as they chatty or anything else. So. So that's
I'm talking too much.
I thought of a couple of these exercises which it would be
useful to go through
slowly now. Can I get rid of this
thing on the right? Probably not,
you know, a bit bigger,
I think. I thought of
Access 4.7.
OK,
this is quite a nice exercise. Let's see if we can make that a
bit bigger still.
Is excess 4.7
caught Mcrae's?
And this is the real thing? This isn't just a I mean, I mean that
thing is a real bit of of observational particle physics.
If you like
quick raise from the top of the atmosphere, collide with it and
it just shows of elementary particles, there's this happens.
So, companies
coming from the cosmos.
Mysterious source.
We showed them elementary particles including
a variety of exotic particles and all ones that reached the
ground are muons, which are sort of heavy electron,
which are unstable and decay with a half life of of 2.2
microseconds.
And they're the troubling relativistic speeds because the
Cosby rates at the atmosphere at very high speeds at at ultra
relativistic speeds. So the muons are moving at similarly at
relativistic speeds.
And the majority of the of the muons that are produced behind
the atmosphere reach the ground
to produce interesting. And you and you can you can test that
you can observe that by observing the the Muon flux at
altitude on top of the mountain and the ground level and it it
all matches up. So, so, so this is happening.
So supposing that these muons are created at a nominal height
of 15 kilometres currently, how long it takes them to reach the
ground in the earth frame giving the answer in seconds and then
half lives.
So this is setting up our our
leading you through as as solving this.
So what I will do therefore is. I think it's
and this one which is the one which is on the E360 rather
unfortunately would be better than the other one. Could you
could talk on the on my left knee that adequately here.
OK. Then
what can do is we can talk about
no.
Can we see that
you only just
So where are we? We have, we have our
an altitude of 15,
there's metres
we have.
Uh,
well, we want to know how long it takes to reach the ground. So
what speed are these? Are these moving at
anyone
speed of light? Yes.
So you could work out exactly how fast the moving given the
gamma factor of or 40. But it's going to be so close to the
speed of light it makes no difference.
So you would just use the speed of light
I at their speed. So.
So there's no need for excess precision and a complicated
calculation to try and turn gamma equals 40 into A into a
very precise value.
Because your teams tend to the
8 metres per second and I'm using physical units here rather
than natural natural units. Because the the question is
being framed in terms of of physical units. And the answer
I'm looking for is in terms of seconds and half lives and so
on. So there's no need to put any other complications there.
They're travelling a distance of 15,015 thousand metres in at at
a speed of of that. So the the time is equal to.
This has been a time
H over
V is going to be 15,
1000
metres
over the times 10 to the eight
which is second and I'm being careful to put in the units and
I'm looking at that and going yes, I have that, that those
units cancel and give seconds. So I I do have that the right
way up.
And it's all the one of the reasons why you always put in
units for a numerical calculation is because that
means you can check that the units end up the right way up.
Because that's an easy, cheap check that you haven't made
some, you know, silly mistake and got things upside down.
I would then 15 / 3 is 510 to 3 -. 10 to the eight is 5 * 10 to
the minus
31885. It's 5 * 10 to the minus
5
seconds.
Unless anyone is going to disagree with that,
which is
50
microseconds.
And the half life is 2.2 microseconds, so that you know I
divide that one by the other, giving 22 point, 22.7
T half
and
so I've done. There's no relativity here,
I've just done this is the Speed Times thing.
I'm being reasonably careful about this. I I think if this
were an assignment I'd I would make a stomach for a copy of
this before submitting it, but I'm being fairly systematic
about going about going through this. Good, so A is done and we
haven't had to think very hard, which is always very nice.
Calculate how long it takes the muons to reach earth in the
muons frame, giving the answer again in.
Seconds and half likes.
Now why does it see in the muons stream?
Yeah,
Temperance transformation
mostly right.
You don't have to use the range transformation because it's just
a matter of time dilation. So you can just use the time
dilation formula.
So, and I think that that's a a good point to emphasise, you
don't always have to use the right transformation
equation, and when you're doing these sort of calculations
you'll end up using a mixture of the transformation, time
dilation, length, contraction distance and speed times time
in using the right the right thing at the right time. So in
this case the question is asking that's time of 50 microseconds.
What? So between the Muon hitting the top of the
atmosphere and the Muon hitting the ground takes 50 microseconds
in the Earth frame.
What is the separation between those two events in the new on
stream? And you could set it up with the answer that the answer
is not wrong,
but you'd be making it hard for yourself because then you'd have
to think OK, what are the events, but you know and write
things down and then go turn the handle and get the answer. You
can just leap straight to the answer by going time relate.
How do you work out the the time in the beyond stream?
Why? Why over gamma?
That that's right. And and in a sense the the slightly cheating
answer is say because we expect it to be smaller
because you can remember L naughty L = L naughty over gamma
or lol or gamma times laughter, whichever. Whichever one it is.
I I can't remember which way up it is, but I know the lengths
contract
that moving, moving, moving, moving length, moving rod gets
shorter, eventually shorter, and that time dilates.
So that in this case I expect to um
that the answer in the burn frame to be to be less so I know
I'm gonna I'm gonna have to divide by gamma. So that's
that's a bit like cheating but it's it's quite legitimate
because you are thinking through that that that you know what
physical answer you should get which is a lot smaller and so
you know where to multiply or divide. So I think for for this,
for Part B
and
this time,
and
let's rate, I'm going to jump back a bit and say that's that
That's time
T prime equals to T over gamma equal to.
Did prime equal to over gamma
equals.
With 50
microseconds
over, we were told the camera was 40,
which is equal to
1.25
microseconds.
A question there
and frame travelling at the speed of light. No the frame.
The frame isn't travelling at the speed of light but we're
this this is as I'm making a central approximation so so the
muestrame isn't at the spiral light. When we were working
doing the calculation, the numerical calculation that
worked out what the the time elapsed time was we thought we
might as well take it to be special late because the
difference from that is going to be ignorable. If we're talking
about two significant figures in the in in the nominal height
then it doesn't matter. So I think that that that's being
applying a bit of physical common sense in important
number. Should we pick if we did it
this exact way and went from gamma equals 40 to V equals?
2.9978 whatever. Then we would be injecting spurious accuracy.
So, so, so, so, so, so yes, the mainstream isn't moving at the
speed of light, but we can take it to be for medical purposes.
OK. Sorry, do you? Yes,
due to time dilation, whenever I'm writing carefully, there's
never enough time to write carefully. But due to time
relation, less time would pass in the nuance frame. Thus time T
prime equals T over gamma.
So although I said
that we know the number of good has got to get smaller,
therefore we divide, I don't write that down.
What I write down is an explanation which which
indicates that I have a clue
that I understand what what what's happening here and you
know, write it the right way around. OK, so I could give a
more elaborate explanation in terms of I could, I could, I
could write pages on that I'm sure, but I've written down
enough to make it clear to the person marking this what
that I understand what's happened.
OK, because the the, the, the secret with all exams and it
took me, I I never worked this out. When I was an
undergraduate, I never really realised that the people marking
my exams were the people teaching
and the humans teaching them. And there was a human sitting
there in the dark of the night, slogging through my, my, my exam
papers. Because if I had, I realised that the point the way
you do an exam is you're talking to someone,
you're talking to him and saying look, I'm clever,
bear that in mind. And so this due to time dilation less time
would be would pass in the millions frame. That's saying I
have a clue.
OK, so bear that in mind.
You don't have right? You don't have to write an epic, but you
have to write something. So. So just writing down t = 2 over
gamma T frame equal to over gamma would not get marked as
far as I'm concerned.
Just quoting the answer and jumping straight there wouldn't
get marked. You have to have something in there which says I
have a clue.
OK.
That by the way is my nature handwriting
part C and we've done it in ah yeah I need a lost a mark
I I failed to read the question and I failed to to to to write
down that that was what the number I have is 0.57
0.57 half lives.
That was a free market nearly missed. Read the question.
OK, so we've done I I just checked and we have done Part B.
Now
which part C
can this be regarded as a test of special relativity?
What's the answer to that?
I mean, yes, yeah, yeah, but why? Why is it special?
That's it. Yeah. So so the the mere fact that the, the the
that the the naive calculation in part a is saying it takes the
muons 22 half lives to get to earth
means that the number of muons that you get to get to Earth
would be 1 / 2 to the 22
of the number that were at the end of the atmosphere. So almost
none.
So the the mere fact that, as the question said, most of the
muons arrive at Earth is telling you that the explanation that
you would that you would naively jump to from the calculation
part A can't be right.
And the explanation that if there's less than one half life
in the Muon frame, it takes less than one half life in the muons
frame to get down to earth makes it perfectly reasonable that
most of the muons make it to Earth.
So the question then is how do I write that down in one sentence?
I don't have to write a book here, I can just write down
something like.
How do I freeze it
if
the muons
actually
it took 22
half
lives to reach
F
capital E
and
almost
none
would arrive
observationally,
most
2
so
much less
time than
1
half life
would
pass.
I am just getting really messy. Sorry about that,
Beyoncé.
And you're not here to watch me, right? Slowly. As you. Slowly.
Neatly. But
right now that's not quite enough,
because I haven't said I haven't mentioned the word special
relativity in that sentence yet.
How do I inject that?
Only special relativity
provides
this explanation,
right? What I've written down there is,
but that's gone off the bottom.
If the muons actually took 20 to half life to reach the Earth,
almost none would arrive. Observationally, I am linking it
to the question.
Most do so much less time than 1/2 life must would pass in the
muons frame. Only Sr provides this explanation and I should
probably therefore this confirms this corroborates Sr.
SO
I I think if I were about to submit that as as assessment I
would go back over that and thinking of I can reword that
bit a bit a little bit better. But the key points are I have
mentioned I I haven't just. I haven't stopped at the first
sentence there. I linked it to the question I I looked at what
I was actually asked to see
and I I I I made some sort of link to that. I said
observationally, most do that. That's the thing that is that is
the key thing.
And is this a taste of special relativity
such sentence? Yes it is basically. I could freeze that
better, but but, but but the, the, the, the. The point is to
read the question and make sure you actually do answer it.
If you haven't written a lot, I would
tater that little bit in in wording terms to to, to to to to
work an assessment. There's less time, of course, and exams, but
the the, the, the goal of the scene,
OK,
you have it.
No?
In the frame of the muons, the Earth is moving towards them at
approximately the speed of light.
Calculate the distance between the altitude of which the under
created and the surface of the earth in the Muon frame and thus
how long it takes for the mules to traverse it. Give you answers
again in seconds and half lives. So what's happening here? What
am I asking you to do
that students?
That's right.
Length contraction. Yeah, length contraction is the key thing.
So the in the burns frame the distance from the top of the
atmosphere to the bottom is much less than 15,000 metres. By how
much less? By that divided by a factor of of gamma.
So I have to indicate that I understand this. So
in the bum stream,
the atmosphere
is length.
Yeah,
by our factor of gamma.
Thus
each prime equals H over
gamma equals 15
those metres
over
40, which is a less pretty number. What number do they cope
with? That should be the 105 metres
and
in the UMM stream.
OK
Do I stop there? No, I do not. The question didn't ask me how
he took asked me to do that, but it also said and thus how long
it takes the means to traverse it.
What speed is the Earth moving at in the beyond stream?
Should be like, yes. So the time it takes for the Earth to travel
305 metres in the Muon stream.
Yeah,
travels with each prime
at
speech C in time
T Prime equals distance is speed.
It's each frame over C which is 3 and 75
metres
times 10 to the eight metres per second
because because that turned the number of seconds. That's good.
I've got something done stupid to put that upside down which is
equal to
and that is equal to 0 point 1.25 microseconds
which is equal to 0 point
57T half. And I have remembered that I'm being asked for it to
put this in seconds and in microseconds and and in in
seconds and in numbers of half lives,
have I done everything? How long it takes for years, right? Yes,
in second-half. Yes, I've done that. So I have. I've done
answered the quest
good. No last bit
E
comment on the equality or inequality of your answers to
Part B and and D
No comment is a keyword that there There are numerous
keywords in exams.
One of the very important ones is write down. If you're asked
to write down something, it means you know this. You don't
need any explanation. It means that it's trivial to to just put
on paper, but you'll be asked to write down. So write down is a
keyword. It's just I'm not expecting you to to explain
this. You don't have to explain this
comment is another keyword for exams. It means say something
intelligent about.
So what is being?
What is it you're supposed to notice about the answers to part
B&D?
The equal Yeah.
So
the the time it takes for the Muon to get from the top of the
atmosphere to the bottom in the Muon frame. So the amount of
time it's got to decay
and the amount of time it takes for the Earth to get from
1500 metres away to to
to the the the thickness of the atmosphere and the muons frame
is such that it takes only only that that time to get all the
way through it are the same.
If you just write down the other seam, I'll write down the seam
and
answer this. No, that's good. Correct tick.
Not, not all the marks, because I haven't said what's
interesting about that
and and and there's a couple of things I could write down there.
A sensible thing would be,
um to note that
and calculations in different frames must agree
the they have different explanations. In one case the
the time it the the the the the muellers clocks are being slowed
down because of time relation. In the other case, the
atmosphere is being compressed because of length contraction.
Different explanations which which come to the same
the same result.
Uh,
Calculations
India.
Different frames
of degree.
And I think if if I were writing an assignment, I would, you
know, obsessed a little bit about how to frame how to freeze
that. In writing that in an exam, I'd make sure I say
something like that, but not. I wouldn't obsess about it.
OK.
You know, you can see that in the.
In the question I said comment on the equality or inequality of
your answers to part B&D.
So I say hey, I I have been deliberately vague about whether
I whether those answers should be the same. Of course they
should be the same. But see, one way or another you
go different answers for those two for what would be and D
Perhaps just your arithmetics poor. OK,
so you so you think of that. Are you think? Should I have got the
same? Should they be the same?
Are they interestingly different? Or have I just made a
mistake
and you think about that, and you might be that you conclude
these should be the same.
I made a mistake.
That's fine. You write down these are different but they
should be the same because blah blah gives us an excellent
explanation
because that shows me you have a clue
and that's a general thing. In exams if things have gone all
out of port and you you, you, you realise you've written down
rubbish. You, you, you're the answer, you've written. Your
writing down is is wrong. Can't be right,
then see through. Don't just have a panic attack and score at
the last five pages of of your exam exam question. See, this
can't be right because
because if you if you're if the person making it sees that
you're thinking about this and that you you know what the
answer should be, roughly, and that you give a a reasonable
explanation for why it can't be right,
then OK, you lose some marks for whatever, whatever mistake it
was. But you'll get some marks. They'll find some marks for you.
For you being switched on enough to know this is this is clearly
wrong
for this intelligent reason.
So remember that
again talk to your mother
so that that's why I, you know the the order inequality is
there just in case
you said you retain the fact of the question but the the comment
that you had to it can be various.
So that's quite a a I think that was that was A
and I think assignment one or something a few years ago. So
that's that's a sort of Simon type question rather than exam
type question. So. So they're not very good guide to exam
things but I think it's a nice question to pick here because it
covers quite a lot of things And interestingly this is from
Chapter 4 so it's before the Lorentz transformation. So we
haven't had to use the range transformation at all in this
one. We just had to think about what's happening here just to
speed times time length contraction time relation and
and so on.
So the more elaborate version of that we we we could ask but you
don't have to use as I I think I said at one point that all
relatively exam questions are the same, but just you know here
are some numbers use the rent transformation to them from one
frame to another. That's not actually true as you can see,
but it's just that the range transformation is a nice
straightforward way of
doing this sort of calculate
and that I meant that to take about 20 minutes, 25 minutes
rather than
I was chatting a lot at the begin.
Other questions about that
OK,
Is that reassuring or terrifying or or,
you know,
are you happy?
Actually rather 4 miles there.
OK
and
OK, let's try another one. Um,
is it the other quite nice one?
The traditional traffic lights for example,
so 615
OK,
you're driving towards some traffic lights
showing red
and and no one will wavelength there.
How fast do you have to be driving through a Doppler?
Shifted enough that they appear to be green
normal wavelength,
and the question gives hints on how to how to approach this.
OK, So what this is. This is a question about the Doppler
shift. Obviously
A
SO 614.
So here, what with with most of the of the
questions about the Doppler shift,
the question the the The challenge with them is always
basically the same.
Try not to get confused with which frame you're talking about
and don't lose track of maintains,
because you'll always end up having to think in which we're
under my pointing here and it. Therefore, this is one of the
cases where it's very important to be very clear about what is
what moving what direction in which frame.
So
frame,
yes.
Yeah,
Fear of Lights.
S Prime is the
with the car.
OK, I'll write it down
straightforwardly,
and let's say we have and in this case we. I think we want a
diagram.
So this is,
yes, Prime
and we're moving towards
some traffic lights,
yeah.
And here I'm
this is, I mean Koski diagram. It's just as an ex Y diagram
but I'm writing I'm I'm writing down that this is moving at
speed V
towards the right.
OK,
just because I'll mess up the minus signs otherwise.
And
in the frame of the traffic lights,
the
has
if you can see F equals, we said 600 metres.
No it doesn't. It doesn't.
It has frequency Lambda equals 600,
90 metres. One year I actually
wrote and distributed and people sat the exam in which I wrote
and I talked about frequency of 600 nanometers and no one
noticed. I didn't notice. The internal checkers didn't notice.
None of the students noticed because they were. They somehow
magically fixed it in their heads. But it was slightly
embarrassing, but
easier than you think to do
anyway. So frequency, wavelength, Wavelength is
different frequency as you learned in first year and our
wavelength is 600 metres.
And
what do we have to the the, the
in?
It's primed
observed at
Lambda prime which we don't know.
Ohh no, we do not equal 600
500 nanometers. Ohh, numbers
500 nanometers. OK, So what do we have to play with here? What?
What do we know about the Doppler shift? What we can do is
we can jump back to.
We'll scan through our notes here and and and and recall
we have an expression
Lake
UH
from Notes F primed.
You know F
equals F primed
gamma
one plus
Cos Theta V Cos Theta.
OK, now did that help us?
It does because it has
F&F prime unit which can be proxy for V and for Lambda and
Lambda prime. We're going to think about that,
and it has
the speed V which is what we're trying to find out here,
so we don't. So this is a case where we're doing the Doppler
effect backwards, trying to find V given the change in frequency,
and it's got this angle Theta frame.
But what is Theta framed? A question.
Yes. If this were an exam question I would give that I
would give you that formula. Yeah. Because I I I wouldn't
expect you to That's one of the. I mean I think yeah I wouldn't
expect to memorise that partly because I don't think memorising
is terribly useful but also because it would it would be it
would be just so easy to get a frame with the wrong place and
and and scupper the whole question. So I think if if we're
an an exam question
I would either give it in the question or since last year the
question is the the question papers have been redesigned to.
There's a crib at the at the beginning, so there might be a
crib at the beginning of the of the of the paper which is just a
list of of equations with no context, no explanation of what
other symbols mean. Is just that that that that list there simply
so that you you can spot which one of the is the one with all
the primes in the right place.
So that would be quite memory. Well, running time almost.
But then what do you see? The frame here,
and this is the other bit that that that all of the Doppler
shift questions are all always getting you know our our
testing, what direction are we talking about here
and here the
the, the the the directions are in the
XY plane
of the of the street or of the car or whatever. And the
question is, in what direction in this plane is this light
moving?
OK,
So what direction is the light moving?
What? What?
That's right? So Theta prime is 0
new pie.
Yes. And that's and that's and that's the thing that what all
these doctors your questions are are are testing but can you tell
the difference 0 and π
and in this case the lights moving towards you so the light
coming from the traffic light
to the driver is moving
in the negative X direction
so Theta prime so so and and that's true in in both frames
but we'll be but this requires in the in the car drivers frame
what directions are moving in is moving in the direction opposite
to the X prime X prime axis. So the light is moving in direction
theatre frame.
Yeah,
Light moving
at Peter Prime
equals π,
so
F = F prime gamma 1 -, V
OK,
so the
and
this would be good progress here.
We can rewrite that as F / F prime equals gamma 1 minus
V
and uh
the the the the the question hints at. Right. Row equals F
primed over F, which is also Lambda over
Lambda prime either way up. So I'm I'm rushing slightly could
at the end here but this I think this would not be a a terrific
exam question because it requires you going back at you
you're thinking a little bit more carefully and going back I
think I've got that upside down and and so on so that there's
there's too many hostages to fortune for like this to be a
good exam question basically a very nice assessment question or
excess question. So I'm going to
skip a little bit
and and and and and. Note that at this point we've essentially
done all our work.
We've got
F / F prime,
which we know. We know that from from the the the data given in
the question we've got something on the right hand side which is
depends on on V So this is one over rho squared equals 1 -, v
^2 / 1 -, v ^2.
And then it's just a bit of algebra
to turn.
I mean you know not trivial but not really hard to to to to
invert that and turn that from role as a function of V into V
as a function of row.
We rearrange it and then you could you you discover what V is
as a function of the role which you know is a six fifths blah
blah. Turn the handle and you get a speed which is I think
eleven 6080 / 81 disputed later, which is very fast.
So
I'll stop talking in just one moment. I'll point out that yes,
that's not a terrific exam question, but it's quite a good
exercise and quite assignment, and that it involves first
writing down what you know from the note and then thinking
carefully about what the what the symbols mean and not get and
and and watching your mind sings
so good they have only two more lectures to go and then at the
end of the
a few.