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Trying to work out the speed of a relativistic electron

  1. May 22, 2010 #1
    1. The problem statement, all variables and given/known data

    I have a relativistic electron with energy 10^10 eV and I want to work out its speed. I'm not told if that's total energy or just kinetic energy - which is the sensible assumption to make?

    2. Relevant equations

    (E_tot)^2 = p^2*c^2 + m0^2*c^4

    p = gamma*mv

    gamma = sqrt(1/1-beta^2)

    beta = v/c

    3. The attempt at a solution

    I tried rearranging the above equations to get an expression for v, but I keep getting stupid answers (v = c, v > c).

    I have converted the energy into J and assumed that it is the total energy. Can anyone tell me what I'm doing wrong?

  2. jcsd
  3. May 22, 2010 #2


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    That's the total energy, which is much greater than the rest energy, so your answer should come out very close to c. I suggest you calculate the velocity using β=pc/E. You also might want to use a binomial expansion to approximate p since m0<<E.
  4. May 23, 2010 #3
    I think that it is better to try this (not to use p, almost the same as vela wrote)
    [tex] E = {\gamma}mc^2 = \frac{mc^2}{\sqrt{1-v^2/c^2}}[/tex]
    [tex]\frac{v^2}{c^2}=1- \frac{{E_0}^2}{E^2}[/tex]
    [tex]\frac{v}{c}=\sqrt{1- \frac{{E_0}^2}{E^2}} \approx 1-\frac{1}{2} \frac{{E_0}^2}{E^2}[/tex]
    Last edited: May 23, 2010
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