Velocity Addition Calculator

In classical mechanics, velocities add linearly — if a spaceship moves at 0.6c and fires a projectile at 0.5c, classical addition gives 1.1c, which would exceed the speed of light. Einstein's special relativity forbids this. The relativistic addition formula ensures the combined velocity always remains below c, no matter how close each individual velocity is to the speed of light.

The formula is u = (v + w) / (1 + vw/c²), where v and w are the two velocities and c is the speed of light (≈ 299,792,458 m/s). When velocities are much smaller than c, the denominator approaches 1 and the result matches classical addition. As velocities approach c, the denominator grows and prevents the result from reaching or exceeding the speed of light.

Express each velocity as a fraction of c (e.g. 0.6 for 60% the speed of light). Plug both values into the formula u = (v + w) / (1 + v×w). The result is the combined velocity, also as a fraction of c. You can then multiply by 299,792,458 to get the speed in m/s.

Velocities are typically considered relativistic when they exceed roughly 10% of the speed of light (0.1c ≈ 29,979,246 m/s). Below this threshold, classical and relativistic results are nearly identical. In particle physics, electrons and protons are routinely accelerated to 99.9%+ of c, making relativistic addition essential.

Beta particles (electrons) emitted during radioactive decay can reach speeds of about 0.9c or higher — well into the relativistic regime. For example, if the nucleus is already moving at 0.5c and emits an electron at 0.9c relative to the nucleus, the relativistic addition formula gives a combined speed of approximately 0.9655c, not 1.4c as classical addition would suggest.

This calculator handles one-dimensional (parallel) velocity addition, where both velocities are along the same axis. For velocities in different directions, the formula becomes more complex and involves separate transformation equations for the perpendicular components. In 2D or 3D cases, Lorentz transformation must be applied to each component individually.

The Lorentz factor γ = 1 / √(1 - u²/c²) quantifies how much time dilation and length contraction occur at a given velocity. A Lorentz factor of 1 means no relativistic effects; as u approaches c, γ approaches infinity. The calculator shows γ for the combined velocity so you can gauge the intensity of relativistic effects in your scenario.

Even if both velocities are 0.9999c, the relativistic formula ensures their combination stays below c. For example, 0.9999c + 0.9999c ≈ 0.999999995c — extremely close to c but never equal to or exceeding it. This is a fundamental consequence of special relativity: the speed of light is an absolute upper limit for any object with mass.