Relativistic Momentum Calculator

How Relativistic Momentum Is Calculated

At everyday speeds, Newton’s momentum formula (p = mv) works perfectly. But as objects approach the speed of light, momentum increases far beyond the classical prediction. Einstein’s special relativity corrects this.

Relativistic Momentum Formula: p = γ × m × v

Where:

• p = relativistic momentum (kg·m/s)
• γ (gamma) = Lorentz factor = 1 ÷ √(1 − v²/c²)
• m = rest mass of the object (kg)
• v = velocity of the object (m/s)
• c = speed of light = 299,792,458 m/s

Lorentz Factor Values:

• At 10% of c: γ ≈ 1.005 (barely different from classical)
• At 50% of c: γ ≈ 1.155
• At 90% of c: γ ≈ 2.294
• At 99% of c: γ ≈ 7.089
• At 99.9% of c: γ ≈ 22.37

Worked Example: A proton (mass = 1.673 × 10⁻²⁷ kg) traveling at 90% of the speed of light:

• v = 0.9 × 299,792,458 = 269,813,212 m/s
• γ = 1 ÷ √(1 − 0.81) = 1 ÷ √0.19 ≈ 2.294
• p = 2.294 × 1.673 × 10⁻²⁷ × 269,813,212
• p ≈ 1.035 × 10⁻¹⁸ kg·m/s

This is 2.294× higher than the classical result — the proton behaves as if it’s much heavier.

Why It Matters: Particle accelerators like the LHC must account for relativistic momentum when steering and focusing beams. Cosmic ray particles hitting Earth’s atmosphere routinely travel at 99.99%+ of c.