AcademyRelativity
Academy
Relativistic Work and Energy
Level 1 - Physics topic page in Relativity.
Principle
Relativistic work changes total energy, while rest energy remains \(mc^2\).
Notation
\(E\)
total relativistic energy
\(\mathrm{J}\)
\(E_0\)
rest energy
\(\mathrm{J}\)
\(K\)
kinetic energy
\(\mathrm{J}\)
\(m\)
rest mass
\(\mathrm{kg}\)
\(p\)
relativistic momentum magnitude
\(\mathrm{kg\,m\,s^{-1}}\)
\(\gamma\)
Lorentz factor
Method
Derivation 1: Split total energy
A particle has rest energy even when its momentum is zero. Kinetic energy is the excess above rest energy.
Total energy
\[E=\gamma mc^2\]
Rest energy
\[E_0=mc^2\]
Kinetic energy
\[K=E-E_0=(\gamma-1)mc^2\]
Derivation 2: Relate energy and momentum
The invariant energy-momentum relation works in every inertial frame.
Invariant relation
\[E^2=(pc)^2+(mc^2)^2\]
Massless limit
\[m=0\Rightarrow E=pc\]
Derivation 3: Interpret work
Net work increases kinetic energy. At high speed, extra work mainly increases \(\gamma\), not speed toward or beyond \(c\).
Rules
Total energy
\[E=\gamma mc^2\]
Rest energy
\[E_0=mc^2\]
Kinetic energy
\[K=(\gamma-1)mc^2\]
Energy momentum
\[E^2=(pc)^2+(mc^2)^2\]
Examples
Question
An electron with rest energy
\[0.511\,\mathrm{MeV}\]
has \[\gamma=3.00\]
Find its kinetic energy.Answer
\[K=(\gamma-1)mc^2=(3.00-1)(0.511)=1.02\,\mathrm{MeV}\]
Checks
- Rest energy is not kinetic energy.
- Use \(K=(\gamma-1)mc^2\), not \(\frac12mv^2\), at high speed.
- Energy and momentum conservation must be applied together in relativistic collisions.
- For photons, \(m=0\) and \(E=pc\).