Relativistic Velocity Addition Calculator

Relativistic Velocity Addition Calculator | Online Free Tool | Elysia Tools

Tool usage guide

Learn when to use this tool, what it supports, and how real users apply it.

Key facts

Category: Science & Education
Input types: select, number
Output type: text
Sample coverage: 4
API ready: Yes

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.

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

1Choose your input mode: speed values as fractions of the speed of light (β) or in meters per second (m/s).
2Enter the reference frame velocity (v) and the object's relative velocity inside that frame (u′).
3Compute 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.

Examples

1. Subluminal probe launch from a high-speed vessel

Physics Student

Background

A homework problem requires finding the velocity of a probe fired at 0.5c from a spacecraft traveling past Earth at 0.5c.

Problem

Determine the probe's velocity relative to Earth and contrast it with classical Galilean mechanics.

How to use

Set speed mode to β, enter 0.5 for frame speed v, and enter 0.5 for object speed u′.

speedMode: beta, vBeta: 0.5, uPrimeBeta: 0.5

Outcome

Calculates a relativistic speed of 0.800000c (2.39834×10⁸ m/s) and a Lorentz factor γ of 1.666667, avoiding the non-physical 1.0c Galilean sum.

2. Light pulse emitted from a moving frame

Relativity Educator

Background

A lecturer needs an exact numerical demonstration showing that light emitted from a fast-moving frame still travels at c.

Problem

Show that adding 0.9c to a forward-emitted photon (1.0c) strictly results in 1.0c.

How to use

Set speed mode to β, enter 0.9 for frame speed v, and enter 1.0 for object speed u′.

FAQ

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.

speedMode: beta, vBeta: 0.9, uPrimeBeta: 1.0

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Learn when to use this tool, what it supports, and how real users apply it.

Key facts

Overview

When to use

How it works

Use cases

Examples

1. Subluminal probe launch from a high-speed vessel

2. Light pulse emitted from a moving frame

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