Formula and Method for Velocity Addition
In everyday life, velocities add the simple way: a person walking at 2 mph on a train moving at 60 mph appears to move at 62 mph relative to the ground. That works fine at ordinary speeds, but it quietly assumes time and distance are the same for every observer. Einstein's special relativity shows they are not — and as speeds approach the speed of light, c = 299,792,458 m/s, plain addition starts giving wrong (even impossible) answers. The relativistic velocity-addition formula fixes this: u = (v + u′) / (1 + v·u′/c²), where v is the velocity of a moving frame (say, a spaceship) relative to a stationary observer, u′ is the velocity of an object measured inside that moving frame, and u is what the stationary observer actually measures for the object's velocity. This calculator applies that exact formula for velocities along a single line.
The Einstein velocity-addition formula
The formula's denominator, 1 + v·u′/c², is the key relativistic correction. When v and u′ are small compared to c (cars, planes, even rockets), that term is essentially 1, so u ≈ v + u′ — Newtonian intuition holds. As v and u′ grow toward c, the denominator grows too, pulling the combined velocity back down so it can never reach c. A famous check: add c to c the classical way and you'd get 2c; with the relativistic formula, (c + c) / (1 + c·c/c²) = 2c / 2 = c exactly — light added to light is still just light speed, matching the postulate that c is the same for every observer. The formula is derived from the Lorentz transformation between reference frames and is confirmed experimentally, most famously in the Fizeau light-in-moving-water experiment and in particle-accelerator measurements.
Getting accurate results
- Use a consistent sign convention: pick a positive direction and stick to it. If the object in frame B moves backward relative to the direction frame B is traveling, enter u′ as negative.
- This is the 1-D (collinear) case: the formula above combines velocities along the same straight line. Velocities at an angle need the separate parallel- and perpendicular-component relativistic formulas.
- Both inputs must be less than c: the formula is undefined — and physically meaningless — for a frame or object velocity at or above the speed of light, so the calculator rejects those values.
- Don't confuse this with relativistic velocity subtraction or the Doppler effect: those use related but distinct formulas for finding a relative velocity between two moving objects, or for frequency shifts.