# 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?

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.

Energy, momentum and force in special relativity

The Oh-My-God particle was an ultra-high-energy cosmic ray (most likely an iron nucleus) detected on the evening of 15 October 1991 over Dugway Proving Ground, Utah, by the University of Utah's Fly's Eye Cosmic Ray Detector. Its observation was a shock to astrophysicists (hence the name), who estimated its energy to be approximately 3×1020 [eV]. a) If the OMG particle came to a stop inside of an apple, by how much would the apple's temperature rise?

b) If the Oh-My-God particle were subjected to a uniform magnetic field of B = 1 [T], what would be the radius of particle's trajectory?

If anyone could give me any hints on how to start this problem in the comments section that would be helpful.

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