What is relativistic momentum?

Relativistic momentum is p = γmv, where m is rest mass, v is velocity, and γ = 1/√(1−v²/c²) is the Lorentz factor. It replaces classical momentum mv when speeds become a significant fraction of c (roughly above 10% of c for errors greater than 0.5%).

What is the Lorentz factor γ?

γ (gamma) is the dimensionless factor that quantifies relativistic effects. At rest γ = 1; at 0.9c γ ≈ 2.29; at 0.99c γ ≈ 7.09. γ grows without bound as v → c, which is why no massive object can reach the speed of light.

Why can v not equal or exceed c?

At v = c the expression √(1−v²/c²) equals zero, making γ undefined (division by zero). For v > c the square root becomes imaginary. Physically, accelerating a massive object to c would require infinite energy.

What units should I use?

Enter mass in kilograms and velocity in m/s, or use the fraction-of-c input for convenience. The output momentum is in SI units (kg·m/s). Consistent SI units ensure γ is dimensionless and the formula works directly.

How does this differ from classical momentum?

Classical momentum is simply p = mv. Relativistic momentum multiplies by γ, so at 0.9c relativistic momentum is 2.29× larger than classical; at 0.99c it is 7.09× larger. For v < 0.1c the difference is under 0.5% and classical mechanics is a good approximation.

Relativistic Momentum Calculator

Enter rest mass and velocity to compute relativistic momentum p = γmv, Lorentz factor γ, and compare to classical momentum.

Inputs

Rest mass (m)

Proton: 1.6726e-27 kg | Electron: 9.1094e-31 kg

Velocity input mode

Velocity (v)

m/s

Speed of light c = 299,792,458 m/s

Enter mass and velocity, then click Calculate

Summary

This tool computes relativistic momentum using p = γmv, where γ = 1/√(1−v²/c²) is the Lorentz factor and c = 299,792,458 m/s. At everyday speeds γ ≈ 1 and relativistic momentum equals classical mv. As velocity approaches c, γ grows without bound — momentum diverges and no massive object can reach the speed of light. All computation happens in your browser; nothing is sent to a server.

How it works

Enter the rest mass in kilograms (kg).
Enter the velocity either in m/s or as a fraction of c (e.g., 0.9 for 90% of light speed).
Click "Calculate" — the tool computes γ = 1/√(1−v²/c²) and relativistic momentum p = γmv.
Classical momentum p_class = mv is shown alongside for direct comparison.
The ratio p_rel / p_class equals γ, illustrating how much larger relativistic momentum is at high speeds.

Use cases

Solve special relativity problems in introductory and advanced physics courses.
Verify the momentum of high-energy particles such as protons or electrons in accelerator physics.
Compare classical and relativistic predictions to understand where Newtonian mechanics breaks down.
Explore how the Lorentz factor grows as velocity approaches c.
Check textbook examples for relativistic kinematics and momentum–energy problems.

Frequently Asked Questions

Last updated: 2026-06-15 · Reviewed by Nham Vu