Relativistic Velocity Calculator

Calculate relativistic velocity addition, Lorentz factor γ, time dilation, and length contraction using special relativity. Covers the Einstein velocity addition formula, twin paradox, and particle accelerator examples.

Combined Velocity u (fraction of c)
—
Combined Velocity (m/s) —
Lorentz Factor γ for u —
Extended More scenarios, charts & detailed breakdown ▾
Combined Velocity u (fraction of c)
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Classical Sum (fraction of c, non-relativistic) —
Lorentz Factor γ —
Professional Full parameters & maximum detail ▾
Lorentz Factor γ
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Relativistic KE (γ−1)m₀c² (J) —
Relativistic KE (MeV) —
Relativistic Momentum p = γm₀v (kg·m/s) —
Real-World Scenario Note —

How to Use This Calculator

  1. Enter two velocities as fractions of the speed of light c (e.g., 0.9 = 90% of c).
  2. Simple tier shows combined velocity and Lorentz factor.
  3. Use Extended tabs for time dilation or length contraction independently.
  4. Professional mode adds relativistic kinetic energy, momentum, and scenario notes.

Formula

u = (v₁ + v₂) / (1 + v₁v₂/c²)
γ = 1/√(1 − v²/c²)
Time dilation: t = γt₀ | Length contraction: L = L₀/γ
KE = (γ−1)m₀c² | p = γm₀v

Example

v₁ = v₂ = 0.9c: u = (0.9+0.9)/(1+0.81) = 1.8/1.81 = 0.9945c (not 1.8c). γ = 9.54.

Frequently Asked Questions

  • Einstein's velocity addition formula u = (v₁+v₂)/(1+v₁v₂/c²) ensures the combined speed never exceeds c. Two rockets each traveling at 0.9c relative to Earth have a combined speed of only ~0.9945c — not 1.8c.
  • γ = 1/√(1−v²/c²) amplifies time dilation, length contraction, and relativistic mass-energy. At v=0.9c, γ≈2.3; at v=0.99c, γ≈7.1; at v=0.9999c, γ≈70.7. LHC protons reach γ≈7461.
  • A moving clock runs slower. A spaceship traveling at 0.9c for 10 years ship-time will find 23 years have passed on Earth. GPS satellites at v≈3.9 km/s require relativistic corrections of ~7 μs/day.
  • One twin travels at near-c speeds and returns younger than the stay-at-home twin. The traveling twin experiences less proper time (t₀), while the stationary twin measures dilated time t = γ×t₀. The asymmetry is resolved by the acceleration of the traveler.

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Sources & References (5) ▾
  1. Einstein A – On the Electrodynamics of Moving Bodies (1905) — Annalen der Physik
  2. OpenStax University Physics Vol. 3 Ch. 5 – Relativity — OpenStax
  3. MIT OCW 8.20 – Introduction to Special Relativity — MIT OpenCourseWare
  4. NIST CODATA 2018 – Fundamental Physical Constants — NIST
  5. HyperPhysics – Special Relativity — Georgia State University