# Relativistic Velocity Addition Calculator u = (v + u′)/(1 + vu′/c²) for β or m/s pairs: relativistic sum, Galilean comparison, γ(u), and the u′ = c photon invariance case. > Canonical page: https://elysiatools.com/en/tools/velocity-addition-relativistic - **Category:** Science & Education - **Keywords:** velocity addition, relativistic velocity, einstein velocity, light speed invariant, lorentz transformation, special relativity ## Overview The Relativistic Velocity Addition Calculator computes the combined speed of two collinear velocities using Einstein's special relativity formula u = (v + u′)/(1 + vu′/c²). It determines the resulting relativistic velocity in fractions of the speed of light (β) or meters per second (m/s), calculates the corresponding Lorentz factor γ(u), and compares the output against classical Galilean addition. ## Inputs - **Speed input** (select) - **Frame speed β (v/c)** (number): e.g. 0.5 - **Object speed β (u′/c)** (number): e.g. 0.5 - **Frame speed v (m/s)** (number): e.g. 2e8 - **Object speed u′ (m/s)** (number): e.g. 1.5e8 ## When to use - Calculating the true combined velocity of high-speed objects or relativistic reference frames. - Comparing classical Galilean addition against relativistic formulas to observe speed-of-light limits. - Demonstrating light speed invariance where an emitted photon (u′ = c) remains c in all reference frames. ## How it works - Choose your input mode: speed values as fractions of the speed of light (β) or in meters per second (m/s). - Enter the reference frame velocity (v) and the object's relative velocity inside that frame (u′). - Compute the combined velocity u = (v + u′)/(1 + vu′/c²), the Lorentz factor γ(u), and the Galilean sum. ## Use cases - Physics coursework verifying relativistic kinematics and reference frame transformations. - Astrophysics calculations modeling relativistic jets and high-speed cosmic particle emissions. - Science fiction worldbuilding evaluating realistic travel times and subluminal probe launch velocities. ## Frequently asked questions ### What formula is used for relativistic velocity addition? The calculator applies Einstein's 1D addition formula: u = (v + u′) / (1 + (v · u′)/c²). ### Why don't two speeds at 0.5c add up to 1.0c? Because spacetime transforms relativistically; the denominator (1 + vu′/c²) scales the result down to exactly 0.8c. ### What happens if one object is traveling at the speed of light (c)? When u′ = c, the formula algebraically yields u = c, verifying the universal constancy of the speed of light. ### Can I enter speeds directly in meters per second (m/s)? Yes, switch the speed input mode to m/s to provide velocities in standard metric units. ### What does the Lorentz factor γ(u) represent? The Lorentz factor γ = 1/√(1 - u²/c²) indicates the magnitude of relativistic time dilation and length contraction at the resulting speed. ## Related tools - [Galvanic Cell EMF Calculator](https://elysiatools.com/en/tools/galvanic-cell-emf): E⁰cell = E⁰cathode − E⁰anode with ΔG⁰, the spontaneity verdict, and log₁₀K at 25 °C. - [AsciiMath LaTeX MathML Equation Converter](https://elysiatools.com/en/tools/asciimath-latex-mathml-equation-converter): Convert equations between AsciiMath, LaTeX and MathML: auto-detect the input format, produce canonical LaTeX, presentation MathML with an embedded TeX annotation, a best-effort AsciiMath back-translation, a screen-reader aria-label snippet (role=math), and a KaTeX HTML preview. - [Boolean Algebra Simplifier (Karnaugh Map)](https://elysiatools.com/en/tools/boolean-algebra-simplifier): Minimal SOP via Quine–McCluskey with essential prime implicants, exact minimum cover, Gray-coded Karnaugh map, and full-assignment verification. - [BOM Character Remover](https://elysiatools.com/en/tools/data-bom-remover): Remove BOM (Byte Order Mark) characters from text and file content. 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