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The proper momentum P for a material particle with rest mass m and velocity V = γ (1 + v/c) is defined by
P = mcV = E/c + p
Exercise: Derive expressions for
Mass: m2c4 = E2 − p2c2
Momentum:
\[\Large{{\bf{p}}\,\,\, = \,\,m\gamma {\bf{v}}\,\,\, = \,\,\,\frac{{m{\bf{v}}}}{{\sqrt {1 - {{\bf{v}}^2}/{c^2}} }}\,\,\, = \,\,\,m\,\,\frac{{d{\bf{x}}}}{{d\tau }}\,\,\, = \,\,\,m\gamma \,\,\frac{{d{\bf{x}}}}{{dt}}}\]
Energy:
\[\Large{E\,\,\, = \,\,\,m{c^2}\gamma \,\,\, = \,\,\,\frac{{m{c^2}}}{{\sqrt {1 - {{\bf{v}}^2}/{c^2}} }}\,\,\, = \,\,\,m{c^2}\,\, + \,\,K}\]
Kinetic energy:
\[\Large{K\,\,\, = \,\,\,\left( {\gamma - 1} \right)m{c^2}\,\,\, \approx \,\,\,\frac{1}{2}\,m{{\bf{v}}^2}\,\, + \,\,\frac{3}{8}\,m\,\frac{{{{\bf{v}}^4}}}{{{c^2}}}\,\, + \,\, \ldots }\]
(Click here for solutions to this exercise.)
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