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Particle Motion (Astrophysics)

  1. Jan 31, 2017 #1
    1. The problem statement, all variables and given/known data
    This is new for me, so forgive me my clumsiness. I am working on the following problem:
    A particle p is moving with a velocity v1 = c (speed of light) towards an object q, which is moving in the same direction with the speed v2, where v1>v2. Now, v2 is a function of the distance r between p and q. I need to find the time when p "catches up" with q.

    2. Relevant equations
    v2(r)= v0hr, where h - the Hubble constant, v0 - initial velocity, and r - distance between p and q

    3. The attempt at a solution
    Particle p will "catch up" with q when the distance between them is 0, so we have:
    r - s =0 and r = s (1)
    v2(r)= v0hr
    Differential equation:
    ds/dt = v0hr

    Separating variables:
    ds/r= v0h dt
    Integrating:
    log r = v0ht + c
    r = ev0ht+c
    r = ecev0ht
    ec=R
    r=Rev0ht

    Plugging into (1)
    Rev0ht = v1t

    I am not sure how to proceed from here or if any of it makes sense.
     
  2. jcsd
  3. Feb 4, 2017 #2

    Andrew Mason

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    Science Advisor
    Homework Helper

    Welcome to PF Gregorski!

    There are two reference frames. The observer frame, relative to which q is moving at speed v2, and q's rest frame. It is not clear from the question as to the reference frame in which r or time is being measured.

    According to Special Relativity, p must have 0 rest mass and must be moving at speed c relative to all inertial reference frames. The observer and q would measure the time for p to reach q differently.

    AM
     
  4. Feb 4, 2017 #3
    Andrew,
    Thank you for your input; you're absolutely right there are two frames. I managed to do the last step by applying Lambert W function.
    Best,
    Greg
     
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