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Relativity -- Momentum and energy

  1. Sep 13, 2016 #1
    1. The problem statement, all variables and given/known data
    (a) A proton at rest has energy Eproton = mprotonc2 ≈ 938 MeV. (Its momentum is zero). The protons which circulated inside the Fermilab Bevatron had energies close to 1000 GeV (1GeV = 1000 MeV). What value of γ did a Tevatron proton have?

    (b) Somehow a Bevatron proton captures an electron, becoming a fast-moving hydrogen atom without changing its speed. An electron at rest has Eelectron = mec2 ≈ 0.511 MeV. What is the electron's energy in the rest frame of Fermilab's sedentary buffalo herd?

    2. Relevant equations
    ##\gamma = \dfrac{1}{1-\beta^2}##
    where ##β = \frac{v}{c} ##
    KE = TE - BE (TE is total energy, BE is rest energy or binding energy)

    3. The attempt at a solution
    I just want to check if my solution is correct or not.
    (a) p (momentum) = ##\sqrt{2mK}##
    K = 106 MeV - 938 MeV
    So, mv = ##\sqrt{2mK}##
    v = ##\sqrt{\frac{2K}{m}}##
    Substitute for K = 1.6 × 10-13 J and m = 1.67 × 10-27 kg and get v in m/s
    Substitute v in the equation for ##\gamma##

    (b) Energy = KE + BE
    = ##\frac{m_e v^2}{2}## + 0.511 × 106 × 1.6 × 10-19
    Substitute in v from above equation, and sub in mass of electron.

    Are these calculations right?
     
  2. jcsd
  3. Sep 13, 2016 #2

    PeroK

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    What about using ##E = \gamma mc^2##
     
  4. Sep 13, 2016 #3
    For what part? The second?
     
  5. Sep 13, 2016 #4

    PeroK

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    Why not both?
     
  6. Sep 13, 2016 #5
    So you mean,
    (a) ##\gamma = \dfrac{10^3}{938} ##

    and
    (b) ## E = \dfrac{10^3}{938} × 0.511 MeV ##

    Why? Also, is my answer wrong?
     
  7. Sep 13, 2016 #6

    PeroK

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    Those should be ##10^6##, otherwise yes.

    Note that, for a particle:

    ##\gamma = E/E_0##

    So if you have the energies, there is no need for all those intermediate calculations, which I didn't check.
     
  8. Sep 13, 2016 #7
    Oh, okay I get it!
    And yeah, that was a careless error, it is 106
    Thank you!
     
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