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Length Contraction

  1. May 25, 2008 #1
    A stick of 1m in length travels at v = 1/2 c along its axis away from the observer.

    Question 1:
    Show that the observer perceives the length of the stick to be shorter without theory of relativity. Calculate the length as perceived by him if he calculates it by the difference in length between both ends which have been photographed at the same time.

    Question 2:
    Use theory of relativity to solve the first question.

    Regarding 1:
    The light emitted by the far end of the stick has to be emitted a little bit sooner so that it reaches the observer at the same time as light from the front end of the stick. Since light travels at c towards the observer and the stick travels at 1/2 c away, it's necessary to know how fast 1/2 c (the difference between the speed of light and the speed of the stick) needs to cover 1m. The time it takes for this determines how much later the front end has to emit a photon so that it reaches the observer at the same time as a photon emitted by the far end.

    Is that correct so far?

    Now I'm not sure how to calculate the perceived length. I guess I have to find out where the photon of the far end has to be emitted relative to the front end, but how would I do that?

    Regarding 2:
    I'm not sure about the approach. I know how to use Lorentz transformation to get the new length, but would I include the effect calculated for question 1 as well or not?
  2. jcsd
  3. May 25, 2008 #2
    for question 2, you can use lorentz position transformations to find this but it is more difficult than to use the length contraction relation (if you have studied it). Ultimately, though, they will reduce to the same thing, and you will find:
    [tex] l = l_{o}\sqrt{1-\frac{v^{2}}{c^{2}}}[/tex]
    If you use the lorentz transofmrations, be sure to remember that some relations must cancel ([tex]\Delta ? = 0[/tex] -- i leave the question mark for you to fill in) based on the way you must make measurements.
    Last edited: May 25, 2008
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