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Relativistic relative velocity

  1. Jan 9, 2015 #1
    1. The problem statement, all variables and given/known data
    In a given inertial frame two particles are shot out simultaneously from a given point with equal speeds u at an angle of 60 degrees with respect to each other. Using the concept of 4-velocity or otherwise, show that the relative speed of the particles is given by

    ##u_R = u(1-3u^2/4c^2)/(1-u^2/2c^2)##

    I have tried this a number of ways but always end up getting

    ##u_R = u(1-3u^2/4c^2)##

    So I guess my question is which answer is correct?

    2. Relevant equations

    ##u'_x = (ux-v)/(1-v*ux/c^2)## and ##u'_y = uy/\gamma(1-v*ux/c^2)##

    3. The attempt at a solution

    I set S' as the stationary state of particle A, moving at a velocity u in the x direction. This meant that particle A was at rest in this frame. Therefore the velocity of B in the S' frame is the relative velocity of the two particles.

    For particle B I obtained:

    ##u'_x = -u/2## and ##u'_y = usin60/\gamma##

    Then using those values calculated the relative velocity to be

    ##u_R = u(1-3u^2/4c^2)##


    I apologise for the layout of the equations because it is my first time posting.
     

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    Last edited: Jan 9, 2015
  2. jcsd
  3. Jan 9, 2015 #2

    PeroK

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    You got off on the wrong foot. You need to work out the velocity of particle B in the initial frame, where the angle is valid.

    Also, should the answer you're given have ##2c^2# in the denominator?
     
  4. Jan 9, 2015 #3

    TSny

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    Hello, and welcome to PF!

    I believe there is a misprint in this expression. Should the expression in parentheses in the numerator be raised to the 1/2 power?

    I do not get these results using your relevant equations. In particular, what happened to the denominators of ##u'_x## and ##u'_y##?
     
  5. Jan 9, 2015 #4
    For particle B I set the velocity in the x-direction to be ##ucos60## and in the y-direction to be ##usin60##. I then just substituted those values into the equations for u'_x and u'_y. I obtained those values for u'_x and u'_y because both u_x and u_y are zero since the we are considering the rest frame of particle A.
     
  6. Jan 9, 2015 #5
    Is the angle still not valid if we consider the frame where particle A is stationary? Surely the relative velocity of particle B in that frame it the overall relative velocity?
     
  7. Jan 9, 2015 #6

    PeroK

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    No. You need to use the velocity transformation formula, which is more than just ##\gamma## factor.
     
  8. Jan 9, 2015 #7

    TSny

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    ux and uy are the x and y components of the velocity of B in the original unprimed frame.
     
  9. Jan 9, 2015 #8
    Thanks for all your help. I managed to solve it.
     
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