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Spacetime Interval Problem

  1. Dec 12, 2011 #1
    1. The problem statement, all variables and given/known data
    Two friends with super-synchronized clocks. One goes on merry-go-round the other stands outside at the same point. The merry-go-round has a constant v of 45 m/s and takes 300s to go around. Find dt, ds, and dτ.


    2. Relevant equations
    v=dx/dt
    v=2πR/p using p=period
    ds^2=dt^2-dx^2
    dτab=(1-v^2)^1/2 x dtab
    dt≥ds≥dτ


    3. The attempt at a solution
    Put into SR Units first:
    45 m/s (1 s/3x10^8m)=1.5 x 10^-7
    300 s (1 x 10^9 ns/1 s)=3 x 10^11 ns

    v=dx/dt
    1.5 x 10^-7=(3.0 x 10^11 ns)/dt
    dx= (1.5 x 10^-7)(3.0 x 10^11 ns)
    dx=45000 ns

    ds=√(3.0 x 10^11)^2 - (45000 ns)^2
    ds=√9.0 x 10^22
    ds= 3.0 x 10^11

    v=2πR/p
    1.5 x 10^-7= 6.28R/(3.0 x 10^11 ns)
    R=7165 ns

    dtab=2πR/v
    =2π(7165 ns)/(1.5 x 10^-7)
    =3.0 x 10^11


    dτab=(1+(-v^2))^1/2 x dtab
    = (1-1/2(v^2)) x dtab
    = (1-1/2(2.25 x 10^-14)) x dtab
    because v ≪ 1:
    dτab=dtab

    My answer fits the equation dt≥ds≥dτ but it seems wrong that they're all the same. Never had any examples before where this has happened. Was hoping someone can just check to see if I made any mistakes.

    Thanks
     
  2. jcsd
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