E^2=(mc^2)^2+(pc)^2
<math> E^2=(m_0 c^2)^2+(pc)^2 </math> is a formula in special relativity that relates the relativistic energy, E, rest mass, <math>m_0</math>, and momentum, p, of a particle. From it the momentum of a photon can be derived and so to the famous equation E=mc2.
Derivation
The relativistic equation for momentum is:
<math>p = \gamma m_0 v</math>
The equation for relativistic energy is:
<math>E = \gamma m_0 c^2</math>
These can be rearranged into the forms:
<math>( \frac{E}{m_0 c^2} )^2 = \gamma ^2 </math>
<math>(\frac{p}{m_0 c} )^2 = (\frac{\gamma v}{c} )^2 </math>
Subtracting the momentum equation from the energy equation an rearranging gives: <math> E^2=(m_0 c^2)^2+(pc)^2 </math>
Other Formulae
The energy of a photon can be derived by setting the rest mass, <math>m_0</math>, equal to zero, so: <math>E=pc</math>
Mass energy equivalence can be derived by setting the momentum, <math>p</math>, to zero. This gives us one of the most famous equations in physics, <math>E=m_0 c^2</math>