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Lorentz Contraction with small velocity

  1. Nov 24, 2008 #1
    1. The problem statement, all variables and given/known data
    About how many femtometers shorter than its rest length is the length of a car measured in the ground frame if the car is traveling at 30 m/s in that frame? Assume for the sake of argument that the car's rest length is 5.0 m. Remember that 1 fm = 10^-15 m.


    2. Relevant equations
    L=Lr*[tex]\sqrt{1-\beta^{2}}[/tex]


    3. The attempt at a solution

    This is a pretty straight forward Lorentz contraction problem, but all attempts to actually calculate it fail because the velocity is so slow (1*10-7). How can I scale the numbers so solving 1-[tex]\beta^{2}[/tex] doesn't just give me 1?

    Thanks for the help.
     
  2. jcsd
  3. Nov 24, 2008 #2

    Chi Meson

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    It takes quite a bit of space (plus a masterful use of latex) to take you through the derivation of the "low velocity" formula. But it ends up as √(1-ß^2) ≈ 1-(0.5)ß^2.

    The derivation involves going through a binomial expansion of the original formula. Do you recall doing this in class?
     
  4. Nov 24, 2008 #3
    Thanks for the reply. We talked about the binomial approximation when we were learning about about the relationship between spacetime and proper time, but I didn't think to apply it in this case.
    Thanks for the help.
     
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