Relativistic Velocity Addition Calculator — Einstein Velocity Addition
Enter two velocities as fractions of the speed of light — the velocity of a moving frame (v) and the velocity of an object in that frame (u) — and calculate the combined velocity seen by the lab using Einstein's relativistic addition formula.
As a fraction of c — Einstein velocity addition theorem
- 1
Classical sum u + v
0.5 + 0.6 = 1.1 - 2
Relativistic denominator 1 + uv
1 + 0.5 × 0.6 = 1.3This denominator prevents the combined speed from reaching or exceeding c. - 3
Combined velocity (u + v) ÷ (1 + uv)
1.1 ÷ 1.3 = 0.846154
How does this calculator work?
u' = (u + v) / (1 + uv) in units of c. At everyday speeds the denominator ≈ 1 and this reduces to classical addition; at relativistic speeds it keeps the result below c. Enter both sub-luminal velocities as fractions of c to get the combined speed and the Lorentz factor.
Formula
How this is calculated
In classical (Galilean) mechanics, velocities simply add: an object moving at 0.6c in a frame that itself moves at 0.5c relative to the lab would appear to move at 1.1c — faster than light. Special relativity forbids this. Einstein's velocity addition theorem corrects for the relativity of simultaneity and time dilation, replacing the simple sum with u' = (u + v) / (1 + uv/c²).
At everyday speeds (v and u both much less than c), the denominator is very close to 1 and the formula reduces to ordinary addition. The difference only becomes measurable when at least one velocity approaches a significant fraction of c. Crucially, no matter what sub-luminal values of u and v you enter, the combined result u' is always strictly less than c — light-speed is an absolute ceiling.
The chart compares the relativistic result (lower line, bounded below c) with the classical sum (upper line, which would exceed c for large inputs). The Lorentz factor γ = 1/√(1 − u'²/c²) shown in the stats describes time dilation and length contraction at the combined velocity.
Frequently asked questions
Einstein's formula gives (0.9 + 0.9) / (1 + 0.9×0.9) = 1.8 / 1.81 ≈ 0.9945c — always less than 1. The denominator (1 + uv) grows as both velocities increase, shrinking the result toward (but never reaching) c. It is a built-in feature of the Lorentz transformation, not an ad-hoc patch.
The calculator handles the 1-D collinear case (both velocities along the same axis). For velocities at arbitrary angles the formula is more complex (Thomas-Wigner rotation adds a direction change), but for head-on addition and subtraction the 1-D formula is exact.
γ = 1/√(1 − v²/c²) appears in every relativistic calculation: time dilation (moving clocks run slow by factor γ), length contraction (moving rulers shrink by factor 1/γ), and relativistic mass/energy increase. At the combined velocity shown, γ tells you how strongly these effects are felt.
Also known as
TG we-Calculate Editorial Team. (2026). Relativistic Velocity Addition Calculator — Einstein Velocity Addition [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/velocity-addition-calculator
TG we-Calculate Editorial Team. "Relativistic Velocity Addition Calculator — Einstein Velocity Addition." TG we-Calculate. 2026. https://we-calculate.com/calculator/velocity-addition-calculator.
TG we-Calculate Editorial Team, "Relativistic Velocity Addition Calculator — Einstein Velocity Addition," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/velocity-addition-calculator
@misc{wecalculate_velocity_addition_calculator, title = {Relativistic Velocity Addition Calculator — Einstein Velocity Addition}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/velocity-addition-calculator}}, year = {2026}, note = {TG we-Calculate} }
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