Relativistic Velocity Addition Calculator

Relativistic velocity addition is the correct formula for combining two velocities under special relativity: u' = (u+v)/(1+uv/c²). It replaces simple addition (u+v), which breaks down and can predict speeds exceeding the speed of light when u and v are both large fractions of c.

u' = (u+v)/(1+uv/c²), where u and v are two velocities being combined (for example, an object's speed relative to a moving frame, and that frame's speed relative to a stationary observer) and c is the speed of light.

Simple addition u+v has no built-in speed limit, so adding two velocities each close to c (say, 0.9c and 0.9c) would predict a combined speed of 1.8c under simple addition, violating the principle that nothing can exceed the speed of light. The relativistic formula's denominator term 1+uv/c² automatically suppresses the result to stay at or below c.

You always get exactly c back: setting u=c in the formula gives u'=(c+v)/(1+cv/c²)=(c+v)/(1+v/c)=c(c+v)/(c+v)=c for any v less than c. This confirms Einstein's second postulate, the speed of light is the same in every reference frame, regardless of the source's motion.

Yes. When both u and v are much smaller than c, the term uv/c² becomes negligibly small, so the denominator is essentially 1 and u' ≈ u+v, recovering ordinary Galilean velocity addition, which is why simple addition works perfectly well for cars, planes, and everyday motion.

Adding two velocities of 0.5c each: simple addition would give 1.0c, but the relativistic formula correctly gives 0.8c, well below the speed of light, illustrating exactly how much simple addition overestimates the combined speed as velocities approach c.

No, u' = (u+v)/(1+uv/c²) gives the same result whether you compute it as combining u with v or v with u, since both the numerator and denominator are symmetric in u and v. Order does not affect the result.

Yes, if the two input velocities have opposite signs (representing motion in opposite directions), the result can be negative, representing net motion in the opposite direction from the positive convention. This calculator assumes both inputs are non-negative speeds combined in the same direction.

It is used whenever combining velocities in different reference frames at relativistic speeds, common examples include particle physics (combining a particle's speed in an accelerator frame with the frame's own speed) and astrophysics (analyzing relativistic jets from black holes or the apparent superluminal motion sometimes observed in such jets).

The gap grows rapidly as both velocities approach c. At low speeds the overestimate is negligible, but for two velocities each at 0.9c, simple addition gives 1.8c (an impossible result) while the correct relativistic answer is about 0.9945c, an overestimate of roughly 81 percentage points of c.