Velocity Addition Calculator

Background.

Walk forward at 2 m/s along the aisle of an airliner cruising at 250 m/s and, by ordinary reckoning, you are moving at 252 m/s over the ground. That answer is right to better than a millionth of a millionth. It stops being right as speeds approach light's 299,792,458 m/s, because in special relativity velocities do not simply stack: two speeds below c always combine into a speed below c.

The rule Einstein derived in 1905 is w = (u + v) ÷ (1 + uv/c²). The numerator is the everyday sum. The denominator is the correction, and its size depends on the product of the two speeds compared with c². For the airliner it exceeds 1 by about 5.6 parts in a thousand trillion; for two speeds of half the speed of light it is 1.25, cutting the sum by a fifth. However close to c the inputs come, the result can approach c but never reach it, and if either speed were c itself the answer would be exactly c — the constancy of light's speed falling straight out of the formula.

The rule had experimental support before relativity existed. In 1851 Hippolyte Fizeau sent light through flowing water and found it was carried along by only a fraction of the water's speed; in 1907 Max von Laue showed that the fraction is exactly what relativistic velocity addition predicts.

The page handles the one-dimensional case with two positive speeds below c: an object moving at u relative to a frame that itself moves at v in the same direction, or two objects closing head-on, where the speed of one as seen from the other is the same composition. Velocities at an angle, a backwards motion to be subtracted, acceleration and gravity are outside it.

What is velocity addition calculator?

Relativistic velocity addition gives the speed w of an object measured in one inertial frame when the object moves at u relative to a second frame and that second frame moves at v relative to the first, all along the same line. It replaces the Galilean sum u + v, which it matches closely whenever both speeds are tiny compared with c, and it guarantees that combining speeds below light speed never yields a speed at or above it.

How to use this calculator.

1. 01Enter the first speed u in metres per second, for example the speed of a probe relative to the ship that launched it.
2. 02Enter the second speed v, the speed of that ship relative to the frame you want the answer in, along the same direction. The order does not matter, because the formula is symmetric.
3. 03To work from fractions of light speed, multiply by 299,792,458: 0.6c is 179,875,474.8 m/s. Both inputs must be positive and below c.
4. 04For two objects approaching each other head-on, enter each one's speed in the lab frame; the result is how fast either one sees the other coming.
5. 05Divide the result by 299,792,458 to read it as a fraction of c.

The formula.

w = (u + v) ÷ (1 + uv/c²), with c = 299,792,458 m/s, the exact value fixed by the SI definition of the metre. The numerator is the classical answer. For positive speeds the denominator is always greater than 1, so the relativistic result is always smaller than the simple sum. How much smaller depends on uv/c², the product of the two speeds as fractions of c: at 0.1c each it is 0.01 and trims the sum by about 1%, while at 0.9c each it is 0.81 and cuts the sum by about 45%. An equivalent picture uses rapidity, φ = artanh(v/c): rapidities add exactly, and the combined velocity is c × tanh of their total, which can approach c but never get there. At everyday speeds the correction falls far below the 12 significant figures reported, so the page returns the plain sum.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Einstein A. On the Electrodynamics of Moving Bodies. 1905 (English translation).
2. [2]OpenStax, University Physics Volume 3, section 5.7, Relativistic Velocity Transformation.
3. [3]BIPM, The International System of Units (SI Brochure), 9th ed., version 3.01, coherent derived units and quantity equations.