Relativistic Velocity Addition Calculator
Combined speed of an object moving inside a moving frame, by Einstein's formula, compared with the simple sum.
Results
What this tool does
If a train moves at 0.6 c and someone on board throws a ball forward at 0.6 c, the ball does not travel at 1.2 c: special relativity gives 0.882 c, and no combination of speeds below c ever reaches c. The correction comes from Einstein's 1905 paper and shows up in the denominator 1 + vw/c², which is practically 1 at everyday speeds; for two cars at 100 km/h the difference is far below anything measurable. Fizeau's 1851 experiment with light in flowing water had already seen the effect before anyone could explain it. Enter speeds as fractions of the speed of light, with negative values for motion in the opposite direction.
Formula
u = (v + w) ÷ (1 + v·w ÷ c²)
Variables
| Symbol | Meaning | Unit |
|---|---|---|
v | Speed of the moving frame (fraction of c) | c |
w | Speed of the object inside that frame (fraction of c) | c |
U | Speed seen from outside | c |
UK | Same, in km/s | km/s |
NS | Simple sum (Newton) | c |
DF | How much relativity takes off | c |
Worked example
- Speed of the moving frame (fraction of c)0.6 c
- Speed of the object inside that frame (fraction of c)0.6 c
- Speed seen from outside0.882353 c
- Same, in km/s264,522.8 km/s
- Simple sum (Newton)1.200000 c
- How much relativity takes off0.317647 c
Limitations
- The formula assumes ideal conditions: no friction losses, no air resistance and no efficiency losses unless you enter them.