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Invariance of length in classical mechanics?

  1. May 10, 2006 #1
    Recently a question asked by sparsh stimulated my gray matter regarding the invariance of lengths in galilean relativistic classical mechanics. First I state the question.

    A man X is sitting at the rear end of a long compartment of a train running at constant horizontal velocity with respect to a ground observer Z. X tosses a coin to a person Y sitting at the front end of the compartment. The trajectory of the ball, as seen by Y and Z, will have : 1) Equal vertical and horizontal ranges OR 2) equal vertical ranges but different horizontal ranges OR 3) Different vertical and horizontal ranges.

    Obviously the problem can be solved with a simple galilean transformation. However, consider that the length travelled by the coin as seen by Y is simply the distance between X and Y in the train's reference frame, that is the compartment's length. But when seen Z the point where X released the coin is much further from the point where Y catches it than the length of the compartment.

    This seems to indicate that the two inertial observers will not be able to agree on a length measurement. But I was under the impression that in classical mechanics length was strictly invariant and stayed so unless you replaced galilean transformations with relativistic Lorentz ones, when the ability to rotate lengths in an extra dimension meant that spacetime intervals rather than spatial lengths were invariant.

    What am I missing here. Can you help me out of this dilemma? Thank you.

    Molu
     
  2. jcsd
  3. May 10, 2006 #2

    Curious3141

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    Static lengths are invariant. Let's say the train is moving at speed v in the positive x-direction wrt the stationary observer Z. When the stationary observer makes the final observation of the landing of the coin, he would've observed the start (trailing end) of the train compartment having moved vt (where v is the speed of the train and t is the time taken for the coin to land). The leading end of the train compartment is at L + vt (where L is the length of the compartment from the POV of those inside it). Hence the stationary observer would also observe a train length of (L + vt) - vt = L. The static length remains invariant.

    Of course, the stationary observer would observe the coin to have travelled a horiz. length of L + vt before it lands. This would disagree with the observation of those onboard the train (who would only have observed a horiz. range of L). There's no reason why these measurements should agree. Since the vertical component of travel is unaffected, the stat. observer would see the coin taking a much "flatter" and longer parabolic arc. Indeed, the stationary observer *has* to observe a longer range because from his POV, the coin has a horiz. component of velocity that's higher by v than the horiz. velocity of the coin observed from within the train, but the same initial vertical velocity and the same vertical acceleration (-g).
     
    Last edited: May 10, 2006
  4. May 11, 2006 #3
    So what types of length have to be invariant and what types do not in classcial mechanics? How would you define static length?
     
  5. May 11, 2006 #4

    nrqed

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    a length measurement (I personally would not even call this a static length measurement, which is not standard terminology...we are talking about length measurements, period) corresponds to measuring the distance between two points *at a fixed time in your frame* .
    Take any two objects (or two points) and measure the distance between them (using local observers if you want) at a fixed time in your frame. Now I will measure the distance between these two objects at the same time in my frame.

    In Galilean physics, our results will agree. And they will agree with anyone else moving at any velocity with respect to us. (and of course, we all agree on the concept of simultaneity).

    If someone throws a basebal to someone else in a train, of course the distance travelled by the baseball will be different as measured in different frames (both in SR and in Galilean physics) but that is not a length measurement.

    (and of course the baseball will travel at different velocities in different frames, in both theories. But in Galilean physiscs, the time calculated using distance travelled over speed will give teh same result in all frames)
     
  6. May 11, 2006 #5

    Curious3141

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    Agreed, "static length" is my own way of seeing it, it is not standard terminology. I meant the same thing by it - it has to be a fixed "snapshot" in time of distance between two points, not a length "dragged out" over a time interval. Sorry, this seems intuitively obvious to me, but it's tough to put into words. :smile:
     
  7. May 11, 2006 #6

    nrqed

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    Hey, Curious.
    Don't apologize!! I thought that your expression was very good. It conveyed the idea very well I just pointed out that it was not standard terminology just to make this clear to the OP so that he/she would not expect to find it in the literature.
    You have a very nice way to put it...not a length ''dragged out in time''. That's a nice way to put it. I simply expressed it in more rigorous terms (measurements of two positions simulatenous in a given frame) because this can be generalized to special relativity where even ''static lengths'' are not absolute.

    No need to apologize!

    Best regards

    Patrick
     
  8. May 11, 2006 #7

    Curious3141

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    Thanks! :smile:
     
  9. May 12, 2006 #8
    That's clear now, length have to measured keeping time constant to qualify for invariance. Thanks for the help. Incidentally, who is this OP you keep referring to?
     
  10. May 12, 2006 #9

    Hootenanny

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    Yeah, you got it. Both curious and nrqed explained it better than I ever could! I think I'll keep this thread for reference!:biggrin:

    OP stands for "Original Poster" :smile:

    ~H
     
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