Relativistic Velocity Addition Calculator

Combine two velocities using the special relativity velocity addition formula w=(u+v)/(1+uv/c²). Enter speeds in m/s or as a fraction of light speed (β), and see the result, its c ratio, the difference from the classical sum, and the Lorentz factor.

Input

Combine two velocities using the special relativity velocity addition law. Enter each speed in m/s or as a fraction of the speed of light (β).

Unit for entering speeds

Both velocities are entered in this unit.

Velocity u

Speed of the object as seen from the reference frame. Must be below light speed.

Velocity v

The second speed measured in the object frame. Must be below light speed.

Result

Relativistic combined velocity w

264,522,757.058824m/s

The c ratio of the combined velocity is β = 0.882353.

c ratio β of result

0.882353 c

Classical sum u + v

359,750,949.6 m/s

Difference from sum

95,228,192.541176 m/s

Lorentz factor γ

2.125

This uses the relativistic velocity addition law w=(u+v)/(1+uv/c²), where c is the speed of light, 299792458 m/s. Compared with the classical sum u+v, the relativistic result is smaller and the combined velocity never exceeds the speed of light.

How it works

This uses the relativistic velocity addition formula w=(u+v)/(1+uv/c²), where c is the speed of light, 299792458 m/s.
If both input velocities are below the speed of light, the combined velocity w never exceeds the speed of light c. This is a key result of special relativity.
In classical mechanics (Galilean transformation) the combined velocity is just the sum u+v, but as speeds approach light speed the relativistic result becomes smaller than this sum.
Velocities can be entered in m/s or as a fraction of the speed of light β (β=speed/c). Values entered as β are converted to m/s internally.
The Lorentz factor γ=1/√(1−β²) is computed from the c ratio β of the combined velocity.

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