Relativistic Velocity Addition Calculator
Adds two velocities correctly under special relativity so the result never exceeds the speed of light. Essential when objects move at a significant fraction of c.
About this calculator
In Newtonian mechanics velocities add simply: v = v₁ + v₂. But this breaks down near the speed of light, c. Einstein's special relativity replaces it with the relativistic velocity addition formula: v = (v₁ + v₂) / (1 + v₁v₂/c²), where c = 2.998 × 10⁸ m/s. The denominator ensures the combined speed can never reach or exceed c, no matter how large v₁ and v₂ individually are. When both velocities are much smaller than c, the denominator is essentially 1 and the formula reduces to the classical sum. This result follows directly from the Lorentz transformation and is confirmed by countless particle-physics experiments in which near-lightspeed particles are accelerated further.
How to use
A rocket travels at v₁ = 2.4 × 10⁸ m/s (0.8c) relative to Earth. It fires a projectile at v₂ = 2.1 × 10⁸ m/s (0.7c) relative to the rocket. Classical addition would give 4.5 × 10⁸ m/s — faster than light, which is impossible. Using the relativistic formula: v = (2.4×10⁸ + 2.1×10⁸) / (1 + (2.4×10⁸ × 2.1×10⁸) / (2.998×10⁸)²). Numerator: 4.5×10⁸. Denominator: 1 + (5.04×10¹⁶ / 8.988×10¹⁶) = 1 + 0.561 = 1.561. v ≈ 4.5×10⁸ / 1.561 ≈ 2.883 × 10⁸ m/s ≈ 0.962c. The result stays below c as required.
Frequently asked questions
Why can't you just add relativistic velocities normally?
Classical addition violates the principle that the speed of light is the same in all inertial frames, a cornerstone of special relativity. If two objects each travel at 0.9c in opposite directions, simple addition suggests an observer on one sees the other at 1.8c — but experiments and theory both forbid this. The relativistic formula introduces a correction factor in the denominator that compresses the sum, keeping it below c. This has been verified in particle accelerators where proton beams are combined at near-lightspeed.
What happens to relativistic velocity addition when speeds are very small?
When v₁ and v₂ are both much smaller than c, the product v₁v₂/c² becomes negligibly small, making the denominator effectively equal to 1. The formula then reduces to v ≈ v₁ + v₂, recovering the familiar Newtonian result. This is why we don't notice relativistic effects in everyday life — car speeds are roughly 10⁻⁸ times the speed of light, making the correction utterly negligible. The relativistic formula is therefore a generalisation, not a replacement, of classical addition.
How does relativistic velocity addition apply to particle accelerators?
In collider experiments, two particle beams are accelerated to near-lightspeed in opposite directions. The relative velocity of one particle as seen from the other's rest frame must be calculated using relativistic addition, not classical subtraction. This determines collision energies and is critical for predicting reaction products. At CERN's LHC, protons travel at about 0.999999991c, and relativistic velocity addition is essential for correctly modelling their interactions in the detectors.