Question

During machine testing time, a proton in the Large Hadron Collider travels at 0.999990 c, where
c is the velocity of light.

a) Find the total energy of the proton. (ANS: 3.36144 x 10-8 J or 0.209804 TeV)
b) Find its kinetic energy. (ANS: 3.34641 x 10-8 J or 0.208866 TeV)

Answer 1

Answer on Question #42933, Physics, Relativity

During machine testing time, a proton in the Large Hadron Collider travels at 0.999990 c, where c is the velocity of light. a) Find the total energy of the proton. (ANS: 3.36144 x 10-8 J or 0.209804 TeV) b) Find its kinetic energy. (ANS: 3.34641 x 10-8 J or 0.208866 TeV)

Solution

Total energy is relativity is

$E=\frac{mc^{2}}{\sqrt{1-v^{2}/c^{2}}}$

Hence, total energy of proton is

$E=\frac{938.272\,\mbox{MeV}}{\sqrt{1-0.999990^{2}}}\approx 209804\mbox{MeV}=0.209804\,\mbox{TeV}=3.36144\cdot 10^{-8}\,J$

The rest energy is

$E_{0}=mc^{2}$

Hence, kinetic energy is

$E_{k}=E-E_{0}=mc^{2}(\sqrt{1-v^{2}/c^{2}}-1)=209804\mbox{MeV}-938.272\mbox{MeV}\approx 0.208866\,\mbox{TeV}$