About Taylor & Wheeler 3-1 practice problem


Main Question or Discussion Point

I'm an amateur italian physics addict, so please sorry for my english, eventually.
I'm reading carefully the Taylor & Wheeler 2nd edition, and I'm trying to do
all the problems before jumping to the next chapter.

Practice problem 3-1, page 78: let's talk about point b. Here we have two floats
which are towed at constant speed at 1/3 m/s through a lake.
I'm assuming the swimmer is swimming with constant 1 m/s velocity respect to
water, as pointed out few lines above for the "stationary" pool ( all in all the lake
is a sort of big pool, so I guess the 1 m/s still holds ).

That said, we have to compute the total swimming time from A -> B and B -> A,
keeping into account the floats are actually moving.

I'm able, as you will see, with my reasoning to reach the correct answer, but the intermdiate
results have not exactly the same meaning as the answers at the end of this holy book.

First of all: as you've noticed, I'm italian, so may be I've misunderstood the word "towed".
I'm actually assuming this means within the lake there are no flows or water current ( otherwise
the problem would have been probably stated a river and not a lake ). So to me, "towed" means
someone/thing is actually towing with constant speed the 2 floats, keeping them at a fixed
30m distance apart.

this is important, since actually there's not any constant water flow which could be added to
swimming motion.

that said, we have 2 situations: 1) the swimmer swims in the same direction as the floats are moving.

2) after having reached the B float, the swimmer come back to float A.

I've used the same approach Feynman adopted for the hat falling into a river ( hope you know
about this problem, but I'm quite sure yes ):

1) the floats and the swimmer start together, heading the same direction. So, actually, we can
"forget" about the floats moving, imagining them at rest. This is achievable thinking about the
swimmer as moving with 1 - 1/3 = 2/3 m/s relative speed related to the flows.
So this answers also to the first question of b: the relative speed of the swimmer respect to
the floats is 2/3 m/s.
This also means we could sit in the "floats frame" seeing the swimmer heading point B with 2/3
m/s speed. Since the floats are at rest for us, we can answer to the second question about
distance: since the floats are kept at 30m apart during their motions, the swimmer has still
to cover this distance.
that said, the time needed by the swimmer to go from A -> B is actually:

t1 = x / ( 2/3 ) = 90 / 2 = 45s

If we compare this time with the pool time ( point a) ) for going from A -> B, i.e. 30s we
can see it's slower. This makes sense, since the relative speed between the swimmer and the
floats is now just 2/3, instead of 1 as in the pool case, so the swimmer takes more time
to cover the distance.

2) Now the swimmer is coming back to B from A. Again, sitting at rest within the floats frame,
we now modify the relative speed like this: 1 + 1/3 = 4/3 m/s.
We have to think about this as for relative collisions: if you are colliding with something which
heading towards you with its own speed, you have to SUM the speeds. This is just the case.
So actually, since the distance is always 30m in this case as well, we have:

t2 = x / ( 4/3) = 90 / 4 = 22.5s

So in this case the swimmer takes less time than the 30s pool case, since it's moving
faster than 1m/s.

The total trip time is actually: t1 + t2 = 67.5s

If you look at Taylor & Wheeler result you can see it's correct. But actually the 2 times are swapped, leading to me in troubles.
As you can see, the answer is talking about a "current", so the 22.5 is actually the time
when the swimmer swims with the current, while 45 when it's swimming against it.

This is unfortunately the opposite of my results, and I cannot understand them properly.

Of course the Taylor & Wheeler result makes sense if there would a river with a water current:
of course, looking from the shore, the swimmer would take less time when she swims with the
current than against, but this is true if points A & B are kept fixed with the shore, and not
moving as in our case.
There's is not current at all in our case, just 2 frames moving with 1/3 and 1 m/s against the water.

So, probably, I've misunderstood the word "towed". So the first question is: have I missed
the problem completely?

2) forgetting eventually about my misunderstanding about the "towed" word, may you confirm
my reasoning still holds? ( i.e. imagining the water is at rest and the floats are simply "moved"
at constant speed respect to the water?

Thanks for you help in advance



EDIT: I'm pasting here the part of the exercice we are talking about:

"Check how very different the story is for the swimmer plowing along at 1 m/s with respect to the water.

a) How long does it take her to swim down the length of a 30mt pool and back again?

b) How long does it take her to swim from float A to float B and back again when the
two float are still 30mt apart, but now are being towed through a lake at 1/3 m/s?

Discussion: When the swimmer is swimming in the same direction in which floats are
being towed, what is her speed relative to the floats?
And how great is the distance she has to travel expressed in the frame of reference
of the floats? So, how long does it take to travel that leg of her trip?
Then consider the same three questions for the return trip".

And their correct answer is: "a) 60 sec b) 45 seconds against the current, 22.5 seconds
with the current, 67.5 seconds round trip."
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Answers and Replies

I'm not sure if no one is answering because:

1) the question is too odd it doesn't worth answering

2) this is actually a "Galilean relative motion" problem, and not properly a special relativity one.
( this is not fully true: the problem 3-1 is actually a problem in which we have to compare
the light behaviour between frame of reference Vs a swimmer with a relative "floats" system )

3) this is a problem taken from a book, and hence should be posted in a different "frame".

In case the problem is the 3rd one sorry for that. I've relalized too late probably this is the wrong place.
I've tried to move it, but I cannot find any command for that. If anyone would suggest me how to move
it, I will be happy to achieve it.
Of course the Taylor & Wheeler result makes sense if there would a river with a water current
In the frame in which the two floats are at rest, there *is* a water current. The floats are moving relative to the water. In the frame in which the water is at rest, this relative motion appears as the floats moving. In the frame in which the floats are at rest, the relative motion appears as the water moving, i.e., a current.

So I think it's just a matter of which frame you want to view the results from.

So, probably, I've misunderstood the word "towed". So the first question is: have I missed the problem completely?
I don't think so; I think you've correctly interpreted what "towed" means, it means the floats are moving relative to the water. I just think you didn't fully consider the implications of that; see above.

2) ... may you confirm my reasoning still holds? ( i.e. imagining the water is at rest and the floats are simply "moved" at constant speed respect to the water?
Yes, this reasoning is valid, but you have to be clear about which frame you are using. Your reasoning is done in the frame in which the water is at rest; but it looks like Taylor and Wheeler's reasoning was done in the frame in which the floats are at rest (and in which the water is therefore moving). Both answers are valid descriptions of the same situation, just in different frames.
You have been very clear, thank you very much. You are right, of course. I'm just using the "wrong" frame of reference, where wrong means different from T & W are choosing.
This is a good point: I will try to develop completely the two views ( floats at rest and water at rest ), and I will
eventually let you know new doubts coming from that.

Anyhow, since we are talking about SR here, the main point of this exercice, I guess, is to see how the light
actually doesn't add to any speed, no matter the frame we are working with, while the swimmer actually
does, getting to different times according to with or against motion.

So, replacing the swimmer with a light ray, actually it would take the same time for going and coming back,
since doesn't add. More or less, like the swimmer in the pool...

Is this analysis correct, from your POV, please?

So, replacing the swimmer with a light ray, actually it would take the same time for going and coming back,
since doesn't add. More or less, like the swimmer in the pool...
It's true that a light ray, unlike the swimmer, will take the same time going and coming back; but that's not supposed to be obvious. Indeed, when Michelson and Morley did their experiment in the 1880's, they *expected* light to take different times to travel depending on the motion of the apparatus, just as the swimmer takes different times to travel depending on the motion of the floats. The fact that this did *not* happen for light was quite unexpected.

It's been a while since I looked at T&W, but don't they follow up with a problem that addresses exactly this question?
Yes you are right, I didn't meant this was obvious, don't let me be misunderstood. The point the light doesn't add up
has been guessed by Einstein, and this lead to all the "counterintuitive" results we are now "used" in SR, like
the time not been absolute as Newton guessed and of course this lead to non-simultaneity between events in different frames.
The problem you are talking about is exactly the 3-1 I've posted. the purpose of it was to compare a light trip
Vs the swimmer, which lead exactly to our conclusions.

I've just figure out both cases, anyhow:

1) thinking the floats to be at rest, and sitting to the start float A, we can see the water current working
against the swimmer until it reaches the float B. This actually leads to the same as mine result for t1,
i.e. 45, since the speeds are actually subtracting. The same for the second "back" swim, in which now the
current is with the swimmer, making the speed add up and leading to a "fast" trip of 22.5s.

So this results are actually perfectly aligned with my approach also for "partial" trips, leading to the same
reality, as you were depicting ( this was actually my main doubt, you helped really a lot in solving ).

Nevertheless, I've seen I've used ( without thinking about it ) a slight "mixed" approach, since I've
user the water rest frame to fix the 1/3 and 1 m/s speed related to the water, but then I've ideally
jumped onto the floats A, as it was at rest, just imagining the swimmer now would swim related to me
with 2/3 for the first halftrip, and 4/3 for the second.
So I totally missed the water current, "simply" thinking to relative motion between the floats and the swimmer.
I've to admit this is a sort of lucky strike, but I've done it automatically.
Is this completely correct from your POV?
I know it should be, but some voice inside is actually telling me I could not sit down to the floats
A without taking into account the current, as I did.

Am I missing something, or am I too rigorous, may be?

thanks for your precious support.


EDIT: after sleeping, I've realized there's nothing wrong in sitting on the floats A and imagining what
I would see there: if I'd measure the progressive distances and time between me and the swimmer,
I would really measure a "real" speed of 2/3 during the first trip and 4/3 on it's back one.
Of course, if I'd look to the water below me, I would notice there would be a water current
heading towards me. So, I could eventually say: "yes, I'm measuring 2/3 and there's a water
flow against me and against the swimmer ( on the first trip ), so it' actually performing to
to a greater speed, to overcome the water current". But this is another part of it.

The "reality2 is I would measure those speeds between me and the swimmer when I'd
sit ontop the float A.
So my mixed approach should be actually correct also from a "rigorous" POV.
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