Relativistic Velocity Addition Calculator (u = (v + u′)/(1 + vu′/c²))

Understanding the relativistic velocity addition calculator — concept and principle

If a spaceship moves at half light speed and fires a probe forward at three-quarters of light speed relative to itself, how fast is the probe for you? Classical physics says 125 % of c, but special relativity gives a different answer that always stays below c. This calculator applies Einstein's velocity-addition formula and compares it with the classical sum.

Speeds do not simply add; the combined speed stays below c at 90.9 %.

How it is calculated

u = (v + u′) ÷ (1 + v u′ ÷ c²)

v is the speed of the moving frame relative to you and u′ the object's speed within that frame, both as % of c along the same line. Negative values mean the opposite direction.
The combined speed is also shown in km/s (c = 299,792.458 km/s), with the Lorentz factor of the result γ = 1 ÷ √(1 − u²/c²) and the difference from the classical sum.

The principle

In everyday life, velocities simply add because the denominator 1 + vu′/c² is almost exactly 1. At high speeds, time dilation and length contraction in the moving frame change how its measurements of distance and time translate into yours. The Lorentz transformation combines these effects, and the result is the formula above. It has two key properties: combining any two speeds below c always gives a speed below c, and if either speed equals c the result is exactly c. That is the mathematical form of the postulate that light has the same speed for every observer.

Worked example

Default inputs: frame 50 % of c, object 75 % of c within that frame.

In units of c: v = 0.5, u′ = 0.75.
Numerator = 0.5 + 0.75 = 1.25 (the classical sum, 125 %).
Denominator = 1 + 0.5 × 0.75 = 1.375.
u = 1.25 ÷ 1.375 = 0.909091, or 90.9091 % of c = 272,539 km/s.
γ = 1 ÷ √(1 − 0.909091²) = 2.4004; classical sum minus result = 34.0909 % of c.

How to read the result

The gap between the classical sum and the relativistic result shows how strongly relativity matters at those speeds. At a few percent of c the gap is tiny; near c it is large. If you enter a negative object speed, for example a probe fired backward, the formula gives the correct slower result. Particle physicists and astrophysicists use this whenever two fast-moving frames are combined.

Common mistakes

Adding speeds directly at relativistic speeds.
Applying the formula to perpendicular motion; sideways components transform differently.
Mixing up which speed is measured in which frame.

Keep in mind: The formula applies to motion along one line in special relativity, without gravity or acceleration effects.

Frequently asked questions

Can two speeds below c ever add up to more than c?

No. The formula always returns a value below c when both inputs are below c.

What if the object is a light beam?

Setting u′ = 100 % gives exactly 100 % for any frame speed: light has the same speed for everyone.

Why do everyday speeds add normally?

Because vu′/c² is extremely small, so the denominator is practically 1.

What happens with opposite directions?

Enter one speed as negative; for example 50 % and −75 % give −40 % of c.

Sources: OpenStax, University Physics Volume 3 (2016), §5.6 "Relativistic Velocity Transformation"

Checked 9 Nov 2026. OpenStax, University Physics Volume 3 (2016), §5.6 “Relativistic Velocity Transformation” (u = (v + u′)/(1 + vu′/c²); Example 5.9: light from a ship gives c; Example 5.10: 0.500c and 0.750c → 0.909c, and −0.750c → −0.400c). c exact (2019 SI).

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