Relativistic Velocity Addition Calculator
Enter the velocity of a reference frame (such as a spaceship) and the velocity of an object relative to that frame (such as a projectile or signal). This calculator applies Einstein's relativistic velocity addition formula from special relativity, which prevents any combined speed from reaching or exceeding the speed of light. You also get the classical (Galilean) result for comparison, the Lorentz factor gamma, and the result expressed as a fraction of c. Choose from four unit systems: multiples of c, m/s, km/s, or km/h.
Formula
Worked example
A spaceship travels at 0.6c relative to Earth. It fires a probe at 0.5c relative to itself. Classical mechanics would give 0.6 + 0.5 = 1.1c, which is impossible. Einstein's formula gives u = (0.6 + 0.5) / (1 + 0.6 x 0.5) = 1.1 / 1.3 = 0.846c, safely below the speed of light.
Why you cannot simply add velocities at high speeds
In everyday life, velocities add straightforwardly: a ball thrown at 10 m/s from a train moving at 30 m/s reaches 40 m/s relative to the platform. This is Galilean or classical velocity addition. It works perfectly when all speeds are far below the speed of light (roughly 300,000 km/s). However, Einstein's special theory of relativity (1905) showed that the speed of light c is the same for all inertial observers no matter how fast they move. That constraint makes simple addition break down at high speeds. If classical addition were correct, two photons moving in the same direction from a source would each see the other at 2c relative to itself, which contradicts every known measurement.
The relativistic velocity addition formula
Einstein's relativistic velocity addition formula is: u = (v + w) / (1 + v*w/c^2). Here, v is the velocity of the moving frame relative to a stationary observer, w is the velocity of an object relative to the moving frame, and u is the resulting velocity of the object relative to the stationary observer. The denominator term (1 + vw/c^2) is always greater than 1 when both velocities are in the same direction, so u is always less than the simple sum v + w. Crucially, if w = c (a photon fired from the ship), then u = (v + c) / (1 + v/c) = c(v + c) / (c + v) = c, confirming that light always travels at c regardless of the frame. The formula also recovers Galilean addition at low speeds: when v and w are both much smaller than c, the product vw/c^2 is negligible, and u approaches v + w.
The Lorentz factor and its physical meaning
The Lorentz factor gamma = 1 / sqrt(1 - u^2/c^2) quantifies how strongly relativistic effects operate at a given speed u. At gamma = 1 (zero velocity) there are no effects; at gamma = 2 (about 0.866c) a moving clock ticks at half the rate of a stationary one and a moving ruler is compressed to half its rest length in the direction of motion. The combined velocity u returned by this calculator can be used to read off gamma directly. Ultra-relativistic particles in accelerators and cosmic rays routinely reach gammas of thousands or even millions, making the correction from Galilean addition enormous.
Opposite-direction subtraction and the general case
When the object moves opposite to the frame, the formula becomes u = (v - w) / (1 - v*w/c^2). This calculator handles that case with the direction selector. A useful consequence: even if both the frame and the object move at 0.99c in opposite directions, each sees the other at less than c (specifically about 0.9999c), not at 1.98c as classical mechanics would predict. This asymmetry is why two spaceships approaching each other at 0.9c each cannot measure a relative closing speed exceeding c.
Relativistic velocity regimes
| Speed (fraction of c) | Regime | Lorentz factor (gamma) | Time dilation |
|---|---|---|---|
| < 0.01 c | Non-relativistic | ~1.00 | < 0.005% slower |
| 0.1 c | Mildly relativistic | 1.005 | ~0.5% slower |
| 0.5 c | Moderately relativistic | 1.155 | ~13.4% slower |
| 0.8 c | Highly relativistic | 1.667 | ~40% slower |
| 0.9 c | Highly relativistic | 2.294 | ~56.4% slower |
| 0.99 c | Ultra-relativistic | 7.089 | ~85.9% slower |
| 0.999 c | Ultra-relativistic | 22.37 | ~95.5% slower |
Approximate effect of relativistic corrections at various speeds.
Frequently asked questions
What is relativistic velocity addition?
Relativistic velocity addition is the rule from Einstein's special theory of relativity that gives the correct combined speed of two objects when those speeds are a significant fraction of the speed of light. The formula u = (v + w) / (1 + vw/c^2) replaces the classical rule u = v + w and prevents any combined speed from reaching or exceeding c.
Why can nothing travel faster than the speed of light?
Special relativity requires that the speed of light c be the same for all inertial observers. If an object could reach c, its Lorentz factor would become infinite, requiring infinite energy to accelerate it further. The relativistic velocity addition formula encodes this: no matter how many velocities below c you combine using Einstein's formula, the result is always less than c.
How does the relativistic formula reduce to the classical one at low speeds?
When both v and w are much smaller than c, the product v*w is also tiny compared with c^2, making the denominator term vw/c^2 negligible. The formula then gives u = (v + w) / 1 = v + w, which is just the classical Galilean sum. This shows that Newtonian mechanics is not wrong, only incomplete: it is an extremely accurate approximation for everyday speeds.
What happens when one of the velocities equals c (speed of light)?
If w = c (for example, a photon emitted by the spaceship), the formula gives u = (v + c) / (1 + v*c/c^2) = (v + c) / (1 + v/c) = c*(v + c) / (c + v) = c. The result is exactly c regardless of the frame speed v. This is the mathematical proof that all observers measure the same speed of light.
Does the order of v and w matter?
No. The formula is symmetric: swapping v and w gives the same answer. This reflects the physical principle that you can choose which frame you call "stationary": the relative velocity between two objects is frame-independent.
What is the Lorentz factor shown in the results?
The Lorentz factor gamma = 1 / sqrt(1 - u^2/c^2) tells you the strength of relativistic effects at the combined speed u. A gamma of 1 means no effect; gamma of 2 means clocks on the object run at half speed and it is physically compressed to half its rest length. This calculator computes gamma for the combined relativistic velocity.
Can I use this calculator for opposite-direction velocities?
Yes. Select "Opposite directions" from the direction drop-down and the calculator applies u = (v - w) / (1 - vw/c^2). For example, if two spaceships approach each other at 0.9c each, the closing speed each observes is about 0.9945c, not 1.8c.