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The Empire Strikes Back length contraction.

  1. Jul 23, 2010 #1
    1. The problem statement, all variables and given/known data
    The starships of the Solar Federation are marked with the symbol of the Federation, a circle, whereas starships of the Denebian Empire are marked with the Empire's symbol, an ellipse whose major axis is n times its minor axis (a=nb in the figure ).


    2. Relevant equations

    L=(1-(v/c)^2)l

    3. The attempt at a solution

    x = rest length of major axis = nb
    L = observed length of major axis when ship is travelling at speed v and must be equal to b since you want the ellipse to look like federation circle

    b = nb[SQRT(1 - v^2/c^2)] then
    to this (1/n)^2 = 1 - v^2/c^2
    solve for V....v^2 = (c^2)[1 - (1/n)^2] = (c^2)[ (n^2 - 1)/n^2]
    then arrive at v = (c/n)*SQRT[(n^2 - 1)].

    i also did this.
    L=(1-(v/c)^2)l

    b=nb[SQRT(1 - (v/c)^2)]

    then i got ((1/n)^2)^1/2)= 1-v/c ---> 1-1/n=v/c

    solving for v i got v= c(1-c/n)

    i'm not sure which one is correct and don't under stand y different answer.

    #1v = (c/n)*SQRT[(n^2 - 1)].

    #2 v= c(1-c/n)

    im lost
     
  2. jcsd
  3. Jul 23, 2010 #2
    i think i made a mistake some where not sure where tho
     
  4. Jul 23, 2010 #3

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    Gold Member

    It's kinda difficult to read your notation in pure type. This kind of problem is where LaTeX comes in real handy.

    Do you mean,

    [tex] L' = L \sqrt{1 - \frac{v^2}{c^2}} \ ?[/tex]

    Here is a link to Physic's Forum LaTex guide.
    https://www.physicsforums.com/misc/howtolatex.pdf" [Broken]

    Re-writing in LaTeX gives,
    ==========================================================

    [tex] b = nb \sqrt{1 - v^2/c^2} [/tex]

    [tex]\frac{1}{n^2} = 1 - \frac{v^2}{c^2} [/tex]

    [tex] v^2 = c^2 \left( 1 - \frac{1}{n^2} \right) = c^2 \left( \frac{n^2 - 1}{n^2} \right)[/tex]

    [tex] v = (c/n) \sqrt{n^2 - 1} [/tex]

    ============================================================

    Okay, so far I follow you. :approve:
    Rewriting,

    ============================================================

    [tex] b = nb \sqrt{1 - v^2/c^2} [/tex]

    [tex] \sqrt{ \left( \frac{1}{n^2} \right)^2} = \left( 1 - \frac{v}{c} \right) \ \longrightarrow \ \left( 1 - \frac{1}{n} \right) = \frac{v}{c} [/tex]

    ==============================================================

    Ouch! I see the problem,

    You had,

    [tex] \sqrt{ \left( \frac{1}{n^2} \right)^2} = \left( 1 - \frac{ \color{red}{v}}{ \color{red}{c}} \right) [/tex]

    But it should be:

    [tex] \sqrt{ \left( \frac{1}{n^2} \right)^2} = \left( 1 - \frac{v^{\color{red}{2}}}{c^{\color{red}{2}}} \right) [/tex]

    The above one looks good to me. :approve:
    And, followed by not so good. :frown:
     
    Last edited by a moderator: May 4, 2017
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