Relativistic Momentum Formula
Calculate the momentum of particles moving at relativistic speeds.
Shows why classical momentum breaks down near the speed of light.
The Formula
In classical (Newtonian) mechanics, momentum is simply p = mv. This works fine at everyday speeds, but breaks down as an object's velocity approaches the speed of light. Einstein's special relativity replaces this with relativistic momentum, which grows without bound as v → c — which is why no massive object can ever reach the speed of light.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| p | Relativistic momentum | kg·m/s |
| γ | Lorentz factor = 1/√(1 − v²/c²) | dimensionless |
| m | Rest mass of the particle | kg |
| v | Velocity of the particle | m/s |
| c | Speed of light = 2.998 × 10⁸ m/s | m/s |
The total relativistic energy is E = γmc², which combines rest energy (mc²) and kinetic energy. The energy-momentum relation is: E² = (pc)² + (mc²)² For photons (massless): E = pc, confirming that light carries momentum despite having no rest mass.
Example 1 — Electron at 0.9c
An electron (rest mass m = 9.109 × 10⁻³¹ kg) travels at 0.9c. Calculate its relativistic momentum.
v = 0.9 × 2.998 × 10⁸ = 2.698 × 10⁸ m/s
γ = 1/√(1 − 0.81) = 1/√0.19 = 1/0.4359 = 2.294
p = 2.294 × 9.109 × 10⁻³¹ × 2.698 × 10⁸
p ≈ 5.64 × 10⁻²² kg·m/s (compared to classical p = 2.458 × 10⁻²² — relativistic is 2.3× higher)
Example 2 — Comparing Classical vs. Relativistic
A proton (m = 1.673 × 10⁻²⁷ kg) moving at various speeds. Compare classical and relativistic momentum.
At 0.1c: γ = 1.005 → relativistic ≈ classical (0.5% difference)
At 0.5c: γ = 1.155 → relativistic is 15.5% higher than classical
At 0.9c: γ = 2.294 → relativistic is 129% higher than classical
At 0.99c: γ = 7.089 → relativistic is 609% higher than classical momentum
When to Use It
Use relativistic momentum when:
- Working with particle accelerators — LHC protons reach 99.9999991% the speed of light
- Analyzing electron beams in particle physics detectors
- Calculating radiation pressure from photons (p = E/c)
- Any situation where v/c > 0.1 (more than 10% the speed of light), where classical errors exceed 0.5%
The relativistic momentum formula ensures conservation of momentum holds in all inertial reference frames — a requirement of special relativity. Classical momentum conservation fails to hold across different reference frames at high speeds, but relativistic momentum does not have this problem.