1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Relativistic Trajectory

  1. Dec 16, 2009 #1
    1. The problem statement, all variables and given/known data
    Given that the trajectory in frame S is:
    [tex]x=\frac{c^2}{a}(\sqrt{1+\frac{a^2t^2}{c^2}}-1)[/tex]

    Show that in S' that is moving in the +x direction at speed u:
    [tex]x'=\frac{c^2}{a}\sqrt{1+\frac{a^2(t'-A)^2}{c^2}}-B[/tex]

    Find the constants A and B.
    2. Relevant equations
    Lorentz transformation:
    [tex]x'=\gamma (x-ut)[/tex]
    [tex]t'=\gamma (t-\frac{ux}{c^2})[/tex]



    3. The attempt at a solution
    [tex]x'=\gamma (x-ut)[/tex]

    plug in x:
    [tex]x'=\gamma \left ( \frac{c^2}{a}(\sqrt{1+\frac{a^2t^2}{c^2}}-1)-ut \right )[/tex]

    Since we want x' in terms of t'... i used:
    [tex]t=\gamma (t'+\frac{ux'}{c^2} )[/tex]

    [tex]x'=\gamma \left [ \frac{c^2}{a}(\sqrt{1+\frac{a^2\gamma^2 (t'+\frac{ux'}{c^2})^2}{c^2}}-1)-u\gamma (t'+\frac{ux'}{c^2}) \right ][/tex]

    I can group some terms and play around with it:

    [tex]x'= \frac{c^2}{a}\sqrt{\gamma^2+\frac{a^2\gamma^4 (t'+\frac{ux'}{c^2})^2}{c^2}}-\frac{\gamma c^2}{a}-u\gamma^2 (t'+\frac{ux'}{c^2}) [/tex]

    [tex]x'= \frac{c^2}{a}\sqrt{\gamma^2+\frac{a^2 (\gamma^2t'+\frac{\gamma^2ux'}{c^2})^2}{c^2}}-\left ( \frac{\gamma c^2}{a}+u\gamma^2 (t'+\frac{ux'}{c^2}) \right ) [/tex]

    [tex]x'= \frac{c^2}{a}\sqrt{\gamma^2+\frac{a^2 (\gamma^2t'+\frac{\gamma^2ux'}{c^2})^2}{c^2}}-\left ( \frac{\gamma c^2}{a}+u\gamma^2 t'+\frac{u\gamma^2ux'}{c^2} \right ) [/tex]

    [tex]x'-\frac{\gamma^2u^2x'}{c^2}= \frac{c^2}{a}\sqrt{\gamma^2+\frac{a^2 (\gamma^2t'+\frac{\gamma^2ux'}{c^2})^2}{c^2}}-\left ( \frac{\gamma c^2}{a}+u\gamma^2 t' \right ) [/tex]

    [tex]x'\left (1-\frac{\gamma^2u^2}{c^2} \right )= \frac{c^2}{a}\sqrt{\gamma^2+\frac{a^2 (\gamma^2t'+\frac{\gamma^2ux'}{c^2})^2}{c^2}}-\left ( \frac{\gamma c^2}{a}+u\gamma^2 t' \right ) [/tex]

    As you can see, it becomes very messy quickly. Since this is featured in a past paper, I think there should be some trick to do it quickly.
     
    Last edited: Dec 16, 2009
  2. jcsd
  3. Dec 17, 2009 #2
    what is 'a' (i.e. the small a) , is it the accn of the body
     
  4. Dec 17, 2009 #3
    what is 'a' (i.e. the small a) , is it the accn of the body

    takin it as a const its already pretty complicated
     
  5. Dec 17, 2009 #4
    "a" is just a constant.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook