Relativistic Momentum Calculator
Enter rest mass and velocity to get relativistic momentum p=γmv, Lorentz factor γ, and energies. v=0.8c → γ=1.667, momentum 1.667× the Newtonian value.
Input Data
Results
At a glance:Relativistic momentum p=γmv=mv/√(1−v²/c²), where γ=1/√(1−β²) is the Lorentz factor, β=v/c the speed ratio, m the rest mass and c=299792458 m/s the speed of light. At low speed (v≪c) γ≈1 and p≈mv (Newtonian limit); at high speed (v→c) γ→∞ and momentum far exceeds the Newtonian value, diverging — the fundamental reason a massive object cannot reach light speed. Total energy E=γmc², rest energy E₀=mc², kinetic energy KE=(γ−1)mc². Example: v=0.8c → β=0.8, γ=1/√(1−0.64)=1/0.6≈1.667, so momentum is 1.667× the Newtonian value. The tool takes rest mass and velocity and outputs γ, β, relativistic momentum, Newtonian momentum, total energy and kinetic energy — widely used in particle-accelerator design, cosmic-ray studies and relativistic electron microscopy.
Formula
Lorentz factor: γ = 1/√(1 − v²/c²)
Speed ratio: β = v/c
Relativistic momentum: p = γ·m·v
Newtonian momentum: p_N = m·v
Total energy: E = γ·m·c²
Kinetic energy: KE = (γ − 1)·m·c²
$$\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}, \quad \beta = \frac{v}{c}, \quad p = \gamma m v$$How to Use
- Enter rest mass m₀ (kg) and velocity v (m/s). Common: electron 9.11e-31, proton 1.67e-27.
- v must be below c=299792458 m/s, else γ is undefined (division by zero).
- The tool shows γ, β, relativistic and Newtonian momentum, total and kinetic energy.
Case Studies
Electron at 0.8c
Electron m=9.11e-31 kg, v=0.8c=2.398e8 m/s.
β=0.8, γ=1/√(1−0.64)=1.667.
Relativistic p=1.667×m×v, 1.667× the Newtonian mv.
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.