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High-Energy Approximation

  1. Jan 16, 2017 #1
    1. The problem statement, all variables and given/known data
    Show that, for an extremely relativistic particle, the particle speed u differs from the speed of light c by
    $$ c - u = (\frac {c} {2}) (\frac {m_0 c^2} {E} )^2, $$ in which ##E## is the total energy.


    2. Relevant equations
    I'm not sure what equations are relevant. This problem was listed at the end of a chapter that included:
    $$ m = \gamma m_0, $$
    $$ p = \gamma m_0 u, $$
    $$ E = m_0 c^2 + K, $$
    $$ E^2 = (pc)^2 + (m_0 c^2)^2, $$
    $$ dE/dp = u = \frac {pc^2} {E}. $$



    3. The attempt at a solution

    I have tried to combine/manipulate the above equations into the desired expression, or something similar that I could then use a high-speed approximation on, but I've had no luck. A hint to get me going would be appreciated.
     
  2. jcsd
  3. Jan 16, 2017 #2

    Orodruin

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    Can you show us what you tried instead of saying "a few things"?

    Clearly, you will not get the expression directly since it is a high speed approximation. You will need to extract u from somewhere and then use that some quantity is much smaller than another quantity in the relevant limit.
     
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