# Relativistic Kinetic Energy and Momentum

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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#### 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.