Relativity and Length: A Simple Homework Question

[kostasdb]

Homework Statement

A 20 m long stick is flying with 90% of the speed of light along the x-axis relative to S. From our vantage point (S), it flies behind and parallel to a building that is 10 m long in the x-direction. Which of the following statements is not true?

1 The stick appears to have a length of only 8.72 m in S.

2 At some point in time, the stick is observed (according to careful measurements in S) to be behind the building in its entire length.

3 The stick does not have a "true" (rest)length, it is completely arbitrary which coordinate system one uses to measure its length, and every coordinate system will give a different but equally valid answer.

4 From the point of view of S', the building is way too short to obscure the stick at any time.

The Attempt at a Solution

All 4 statements seem true to me The stick does seem to be 8.72 m long according to S (L=L’/y=20m/2.294=8.72m). The building according to S’ would be 4.4m according to it(using the same formula as before) and too small to conceal the stick. All frames of reference are equally valid and technically, from S frame of reference the 8.72m stick can be behind a 10 m building. I must be missing something small here...

 

[Simon Bridge]

For #3 - what does the inclusion of the word "rest" in parentheses indicate?
Is there a difference between a coordinate system and reference frame?
"...every coordinate system will give a different but equally valid answer..." to what question?

The problem itself is badly phrased - it does not state which reference frames the measurements are taken in.
The word "appears" technically does not belong in #1.

 

[kostasdb]

Rest length would seem to correspond to the S' since that in is the frame of reference in which it seems like other reference frames are moving while it is at "rest". So in that case i guess it would have a true rest length corresponding to S' in this case. Am i on the right track here?

 

[Simon Bridge]

The meaning of the words "rest length" is the length measured in a frame where the object is at rest.
Can there be more than one "rest length"?

 

[phinds]

Rest length would seem to correspond to the S' since that in is the frame of reference in which it seems like other reference frames are moving while it is at "rest". So in that case i guess it would have a true rest length corresponding to S' in this case. Am i on the right track here?

ALL reference frames see other reference frames as moving, not just S'. Any reference frame X' that is not moving relative to reference frame X IS reference frame X by definition.

 

[kostasdb]

This is where my confusion comes in. Focusing on #3, saying that there is a true length, whether at rest or not, would say one frame of reference is more "true" than another frame of reference which seems to go against the whole point that all reference points are equally valid. As far as the (rest) part comes in, i would guess there is only one "rest length" since there seems to be only one reference point that the stick doesn't seem to be moving and that is with the same velocity that the stick is traveling.

 

[Simon Bridge]

This is where my confusion comes in. Focusing on #3, saying that there is a true length, whether at rest or not, would say one frame of reference is more "true" than another...

The problem is sloppily written - yes. Q3 does not make a clear definition about what it means by "true". This means you have to use context to figure it out.

In this case, #3 explains, parenthetically, that "true" means "rest".
But then muddies the waters by talking about "validity".

This sort of confusion is why the correct word for a measurement taken in the rest frame is "proper".

... frame of reference which seems to go against the whole point that all reference points are equally valid.

What is valid or invalid depends on what you want the measurement for.
The correct term to describe the concept you are describing is "absolute" ... there is no absolute reference frame, no absolute speed.

As far as the (rest) part comes in, i would guess there is only one "rest length" since there seems to be only one reference point that the stick doesn't seem to be moving and that is with the same velocity that the stick is traveling.

That is correct ... there is only one rest frame.
Thus the stick only has one valid rest length.

But like I said - I'd also take issue with the word "appears" used here.
There is no "appears". It "is" or it "is not".
The stick is 8.72m long in S and it is 20m long in S'

You could also treat the problem as a process of elimination - of all the statements there, provided we accept that one of them must be false, which is the best candidate? Which is the most vague or tricky to interpret?

 

[kostasdb]

Ok ok, I'm pretty sure i have enough to answer the question. Thank you so much for your help! It is definitely appreciated.