AcademyRelativity
Academy
Relativistic Momentum
Level 1 - Physics topic page in Relativity.
Principle
Relativistic momentum is \(\gamma m\vec v\), which keeps momentum conservation valid at high speed.
Notation
\(\vec p\)
relativistic momentum
\(\mathrm{kg\,m\,s^{-1}}\)
\(m\)
rest mass
\(\mathrm{kg}\)
\(\vec v\)
particle velocity
\(\mathrm{m\,s^{-1}}\)
\(\gamma\)
Lorentz factor
\(E\)
total energy
\(\mathrm{J}\)
\(p c\)
momentum-energy product
\(\mathrm{J}\)
Method
Derivation 1: Replace Newtonian momentum
Newtonian momentum \(m\vec v\) works at low speed but fails to transform consistently between inertial frames.
Relativistic momentum
\[\vec p=\gamma m\vec v\]
Magnitude
\[p=\gamma mv\]
Derivation 2: Use energy units
Particle physics often reports momentum as \(pc\) or as \(\mathrm\{MeV\}\\,c^\{-1\}\).
Momentum-energy form
\[pc=\gamma mvc\]
Using beta
\[pc=\gamma\beta mc^2\]
Derivation 3: Connect to massless particles
For a photon, rest mass is zero but momentum is not. The energy-momentum relation gives \(E=pc\).
Rules
Relativistic momentum
\[\vec p=\gamma m\vec v\]
Momentum magnitude
\[p=\gamma mv\]
Momentum energy
\[pc=\gamma\beta mc^2\]
Photon momentum
\[p=\frac{E}{c}\]
Examples
Question
A particle of mass \(m\) moves at
\[0.800c\]
Find \(p\) in units of \(mc\).Answer
\[\gamma=1.67\]
\[p=\gamma mv=(1.67)m(0.800c)=1.33mc\]
Checks
- Do not use \(p=mv\) at relativistic speeds.
- Momentum increases without bound as \(v\) approaches \(c\).
- The velocity still stays below \(c\) for a massive particle.
- Momentum is a vector; signs and components still matter.