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Relativity question, photon and proton

  1. Dec 8, 2011 #1
    Relativity question, photon and proton....

    1. The problem statement, all variables and given/known data
    Proton of gamma factor 10^12 and a photon set off from one side of our galaxy to the other take the distance to be 9.3x10^20m.
    What is the time interval that the photon precedes the proton?


    2. Relevant equations
    \gamma = sqrt(c/2epsilon). (epsilon being c-v)


    3. The attempt at a solution
    Worked out epsilon to be 1.5x10^(-18)


    Not sure where to go from there anyhelp/hints would be most appreciated.
     
    Last edited: Dec 8, 2011
  2. jcsd
  3. Dec 8, 2011 #2

    vela

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    Re: Relativity question, photon and proton....

    You should try to form ratios that are dimensionless. In this case, you could write
    [tex]v = c - \Delta v = c\left(1 - \frac{\delta v}{c}\right) = c(1-\varepsilon)[/tex]where [itex]\varepsilon = \Delta v/c[/itex]. Why? Because it avoids any complication due to units, for one thing. Regardless of which set of units you use, [itex]\varepsilon[/itex] defined this way will always turn out to have the same value. Moreover, an expression is often more naturally expressed in terms of these unitless ratios.

    You can show that
    [tex]\gamma = \frac{1}{\sqrt{1-(v/c)^2}} = \frac{1}{\sqrt{1-(1-\varepsilon)^2}} \cong \frac{1}{\sqrt{2\varepsilon}}[/tex]

    Now write down expressions for the time each takes to travel the given distance.
     
    Last edited: Dec 8, 2011
  4. Dec 8, 2011 #3
    Re: Relativity question, photon and proton....

    [itex] \Delta t = \frac{(1- \epsilon )x-x}{c (1- \epsilon) }[/itex]


    after putting [itex] t_1 = \frac{x}{c} [/itex] and [itex] t_2 = \frac{x}{c(1- \epsilon )} [/itex], then to find [itex] \Delta t = t_1 - t_2 [/itex]

    but I cant seem to get it , did I do this correctly?
     
  5. Dec 8, 2011 #4

    vela

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    Re: Relativity question, photon and proton....

    Because ε is so small, you can approximate t2 well with a series expansion to first order.
     
  6. Dec 8, 2011 #5
    Re: Relativity question, photon and proton....

    so [itex] t_2= \frac{x}{c} \cdot (1- \epsilon )^{-1} = \frac{x}{c} \cdot (1) [/itex]

    or, because that completely gets rid of [itex] \epsilon [/itex] :-

    [itex] t_2= \frac{x}{c} \cdot (1- \epsilon )^{-1} = \frac{x}{c} \cdot (1+ \epsilon ) [/itex]

    That after finding [itex] \Delta t [/itex] gave me and answer of [itex] -1.55 \cdot 10^{-12} s[/itex]

    only problem is you have to find the time as being in the reference frame of the proton later and need [itex] t_2 [/itex].
     
    Last edited: Dec 8, 2011
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