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    Relativistic Kinetic Energy and Momentum

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    An electron and a proton are each accelerated through a potential difference of 10.0 million volts.

    a) What is the kinetic energy of the electron? What is the kinetic energy of the proton?
    b) Calculate the momentum (MeV/c) of the proton using classical equations.
    c) Calculate the momentum (MeV/c) of the proton using relativistic equations.
    d) Calculate the momentum (MeV/c) of the electron using classical equations.
    e) Calculate the momentum (MeV/c) of the electron using relativistic equations.
    f) Why is there a difference?

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    https://brainmass.com/physics/relativistic-energy/relativistic-kinetic-energy-and-momentum-438313

    Solution Preview

    a) The kinetic energies of both the proton and the electron are 10.0 MeV since they each have a charge of +/-e.

    b) The classical momentum P_c of the proton is given by

    P_c = m_p v_p
    = (m_p c^2)(v_p/c)/c
    = (m_p c^2) beta_p/c
    = 938 Mev/c * beta_p.

    Now since gamma_p = 1/sqrt(1 - beta_p^2), we have

    gamma_p^2 = 1/(1 - beta_p^2)
    1/gamma_p^2 = 1 - beta_p^2
    1 - 1/gamma_p^2 = beta_p^2

    whence

    beta_p = sqrt(1 - 1/gamma_p^2).

    We can compute gamma_p from the equation

    K_p = (gamma_p - 1)(m_p c^2),

    whence

    gamma_p = 1 ...

    Solution Summary

    We calculate the relativistic kinetic energy and momentum of a proton and an electron, each accelerated through 10 million volts.

    $2.49

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