Last updated: 2026-06-16

Relativistic Velocity Addition

Relativistic velocity addition is the rule for combining two speeds when at least one of them is an appreciable fraction of the speed of light $c \approx 3 \times 10^8\ \text{m/s}$. The everyday intuition — that a ball thrown forward from a moving train simply adds its speed to the train's — breaks down near light speed, because the speed of light must come out the same for every observer. Special relativity provides a formula that keeps every combined speed below $c$.

This calculator takes two velocities $u$ and $v$ along the same line and returns the combined velocity $w$ together with its value as a fraction of the speed of light, $\beta = w/c$.

How it works

Suppose a frame moves at speed $u$ relative to a reference observer, and an object moves at speed $v$ relative to that frame, both along the same direction. The object's speed relative to the reference observer is not $u + v$ but

The extra denominator is what prevents the result from ever exceeding $c$. When $u$ and $v$ are tiny compared with $c$, the term $uv/c^2$ is negligible and the formula collapses to the familiar $w \approx u + v$.

Formula

If $v = c$, the formula gives $w = c$ for any $u$ — the speed of light is invariant.

Worked example

Two objects each move at u = v = 0.75c along the same line.

Step 1 — Numerator:

Step 2 — Denominator:

Step 3 — Combined velocity:

Even though plain addition would give 1.5c, the relativistic result is only 0.96c — still below the speed of light. Enter these values in the calculator to reproduce the result.

Combining equal speeds

Real-world relevance

The velocity addition rule is confirmed every day in particle physics, where particles emitted from fast-moving sources never exceed $c$ no matter how the speeds combine. It also underlies the relativistic Doppler effect and the aberration of starlight, and it explains the classic Fizeau experiment on the speed of light in moving water — a result that puzzled physicists for decades until relativity gave it a clean explanation.

Limitation: collinear motion in special relativity

This calculator handles speeds along a single line in inertial frames. When the two velocities are not parallel, the combination is more involved and the result depends on direction, including a relativistic rotation effect known as Thomas precession. For motion along one line, the formula here is exact.