GR9277 #38


Problem


\prob{38}
Tau leptons are observed to have an average halflife of in the frame in which the leptons are at rest. In an inertial frame , which is moving at a speed relative to , the leptons are observed to have an average halflife of . in another inertial reference frame , which is moving at a speed relative to and relative to , the leptons have an observed halflife of . Which of the following is a correct relationship among two of the half lives, , , and .






Special Relativity}Time Dilation Formula
The time dilation formula is given by , where time is dilated (lengthened) in all but the frame at rest (propertime ). Note that .
So, from that alone, one can deduce the following relations (without looking at the choices yet):
The latter deduction is just choice (B).


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Comments 
QuantumCat 20140925 08:49:23  Don't be like me and make the mistake of trying to correlate and . Since is at rest, it's the only one that can comfortably be dealt with (without addition of velocities or extra terms in the invariant spacetime interval). Deal only with the relationship between & and & !   ender 20091105 23:53:02  Time interval is shortest in the particle's rest frame. This means , only choices B and E remain. Choice E involves which should not be the case. Hence (B) is the answer.   Albert 20091020 13:01:25  I am not sure whether B is the right answer, despite the unlikelyhood of ETS offering a wrong answer. Or perhaps I have gotten the concept backwards :)
In frame S2 where we are moving with velocity V12 relative to S1, it becomes almost obvious to me that the time T2 should be smaller than T1, given the fact that when I speed up, time around myself speeds up, at least thats what I think because actually my own time (which is the virtual time) has slowed down. Evidently, I would witness the decay process speeding up and the time T2 getting smaller than T1, hence I feel option A should be correct. But since ETS can't be wrong, I suppose I must be!
Albert 20091103 14:44:06 
And indeed I am! Ignore my comment above, I was high on cocaine when I wrote it! :)

tommy22222 20191014 09:14:14 
Oh Albert, you should be the last person to get this relativity problem wrong even if you were high on coke! :)

  realcomfy 20081105 15:41:44  I am confused about the order that the are placed because in the official answer the first deduction corresponds EXACTLY to the answer choice A. Furthermore, the "correct" deduction has and reversed.
Also, I have on my equation sheet . Using this convention I was able to get the correct answer. Do I have this backwards, or is there a typo here?
ramparts 20090809 20:28:31 
Nope  the Lorentz factor is the reciprocial of . In the answer, they're multiplying both sides by to move it to the other side.

  Poop Loops 20081012 00:19:20  The reason why C and D don't work is because you don't know whether dt2 or dt3 is bigger, and clearly if dt2 > dt3 then C doesn't work, and if it's vice versa, D doesn't work. For example if v23 = v12, that means that S3 is also in the particle's rest frame, and only D works. On the other hand if v23 = v12, then the particle is going much faster in the S3 frame than in the S2 frame, so only C works. Clearly there is a conflict here because you aren't given enough information.
Whereas since S1 is a rest frame, it's stationary and v = 0, therefore if you move to any other frame, v can only go up, so B is the correct relationship between the two for sure.
tommy22222 20191014 09:19:20 
Even though I picked A which is correct, I wondered for a long time why C and D can\'t be correct. You explained it well!

  chrisfizzix 20081006 12:42:28  Minimal notation is a virtue on this silly timed thing:
Take to be the rest frame. Now write time dilation into the other two frames:
Now notice that all the ugly square root crap that ETS has written is just ; from that, we can see that the relations given in the choices are just
A:
B:
C:
D:
E:
Where B is clearly a correct relation.
GREmania 20081012 08:01:49 
I like that!
:)

apr2010 20100407 18:26:46 
Good one and I assume this is ETS' intention to confuse the student. Was time dilatation t = t' y or t= t' /y... I like that though, makes me think more about the correctness about what I am reading.

  callie 20080407 22:16:46  why couldn't the answer be c, if S2 were assumed to be the rest frame? (if only S2 and S3 were compared against each other) S1 cannot be an "absolute" rest frame, can it?
caffeinated 20080409 14:26:46 
I, also, am not clear on this since supposedly it doesn't matter who is moving with respect to whom, and is there really any such thing as absolute rest? Not at my desk.

rose3fa 20090831 09:23:03 
S1 can be the absolute rest frame! According to Feynman, in the twin paradox we know the twin that went away and came back is the one whose clock ran slower. Feynman says you cannot imagine that the "traveling" twin stayed still in his own reference frame and the earth moved away from it. Feynman claims that the twin who felt the movement, i.e. accelerated away and decelerated on the way back, is the one whose clock ran slower.
Now, I still have trouble understanding this concept too. But, if we accept Feynman's example, it makes sense to call S1 a rest frame with the fastest clock.
Incidentally, I only quote Feynman's example because I just finished the books, "6 Not So Easy Pieces," which deals with relativity.

apps00 20101006 10:08:20 
It's because when the formula for the time intervals' relation was derived we assumed the particle to have zero velocity in the rest frame.
Let the "2" system move with relatively to "1" system, and the particle is in "1" system. The relations between coordinates of two events in these frames are:
Now we subtract the second equation from the first one to derive relation between two time intervals:
Note, that if the particle is not moving in "1" frame, equals to zero and we derive
The (C) is not correct because is not zero when the lepton doesn't rest in the chosen frame. For this particular case and the "correct (C)" relation would be instead.

  astro_allison 20051124 21:59:38  what do we do with the information "in inertial reference frame S3, moving at speed v_23 relative to S2, the observed halflife is t_3"?
astro_allison 20051124 22:00:12 
...not supposed to be a "partial"t_3, but a "delta"t_3

yosun 20051126 01:41:37 
astro_allison: the extra information for is to confuse the testtaker who does not understand how the time dilation equation works. the time dilation equation relates the time of a rest frame (proper time) to that of a moving frame. None of the choices relate to
fyi: \Delta produces a capital delta sign , while \delta produces a lowercase delta sign . The lowercase delta is not the usual sign for a partial derivative (recall that it is used to denote variations in Calculus of Variations). The usual sign for a partial derivative is given by \partial, which produces . (Surround the LaTeX command above with dollar signs for them to take effect.)

 

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The reason why C and D don't work is because you don't know whether dt2 or dt3 is bigger, and clearly if dt2 > dt3 then C doesn't work, and if it's vice versa, D doesn't work. For example if v23 = v12, that means that S3 is also in the particle's rest frame, and only D works. On the other hand if v23 = v12, then the particle is going much faster in the S3 frame than in the S2 frame, so only C works. Clearly there is a conflict here because you aren't given enough information.
Whereas since S1 is a rest frame, it's stationary and v = 0, therefore if you move to any other frame, v can only go up, so B is the correct relationship between the two for sure.

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