Velocity Addition Calculator
Combine two velocities the way special relativity requires, and see how far the everyday rule v + w drifts from the truth near the speed of light.
🚀 What is Velocity Addition?
Velocity addition is the rule for finding an object's velocity in one reference frame when you know its velocity in a second frame and the velocity of that second frame. In everyday life you simply add: a passenger walking at 1.5 m/s toward the front of a train moving at 30 m/s is travelling at 31.5 m/s relative to the ground. That rule, built from the Galilean transformation, works so well for ordinary speeds that nobody questioned it for centuries.
Special relativity replaces it, because the speed of light has the same value in every inertial frame. If simple addition were exact, a flashlight beam fired forward from a rocket would travel faster than c for a ground observer. The corrected rule, first given by Einstein in 1905, is u = (v + w) / (1 + vw/c²). It matters wherever speeds are a noticeable fraction of c: particle accelerators and cosmic ray physics, the jets of active galaxies, atomic clocks and satellite navigation corrections, and every introductory course on relativity.
Several points are easy to miss. The sign of each velocity matters, so use a minus sign for motion the other way. Velocities perpendicular to the frame motion are not unchanged, they are slowed by the Lorentz factor, which is why the direction of travel changes (the aberration of light is the best known case). Finally, the formula applies to velocities of objects, not to the rate at which two objects approach each other in a third frame. Two rockets each moving at 0.9c in opposite directions in the lab separate at 1.8c in the lab frame, but each sees the other at only 0.9945c.
This calculator handles both situations. The first mode adds two velocities along one line and compares the result with the Galilean sum, with rapidities that add exactly. The second mode takes a frame moving along x and a velocity with x and y components in that frame, and returns the full vector in the lab with its direction. Both modes draw a chart so that you can see where the classical answer breaks away from the relativistic curve and the limit set by c.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Two Fast Speeds Along a Line
A frame moves at 0.6c and an object moves at 0.8c inside it, in the same direction
Example 2 — Inputs in km/s
A frame moving at 200000 km/s and an object at 150000 km/s in that frame
Example 3 — Opposite Directions
A frame moving at 0.9c and an object moving at 0.5c the other way inside it (w = -0.5c)
Example 4 — Adding a Light Beam
A rocket at 0.9c fires a light pulse forward (w = c)
Example 5 — Sideways Motion in a Moving Frame
A frame at 0.6c along x, and an object moving at 0.8c along y in that frame
Example 6 — Moving Back and Sideways
A frame at 0.8c, and an object with wx = -0.6c and wy = 0.6c in that frame, travelling at 135 degrees
❓ Frequently Asked Questions
🔗 Related Calculators
What is the relativistic velocity addition formula?
For motion along one line, u = (v + w) / (1 + v w / c²), where v is the speed of the moving frame, w is the speed of the object in that frame and u is the speed in the original frame. At low speeds the denominator is almost 1 and it reduces to v + w.
Why can't velocities just be added in special relativity?
Simple addition predicts that 0.8c plus 0.8c equals 1.6c, which would exceed the speed of light. Experiments show that light speed is the same in all inertial frames, so the true rule gives (0.8 + 0.8) / (1 + 0.64) = 0.9756c instead.
What happens when you add the speed of light to another velocity?
The result is exactly c. Putting w = c in the formula gives (v + c) / (1 + v/c) = c for any v. This is how the formula enforces the constant speed of light in every inertial frame.
How do you add velocities in opposite directions?
Give one velocity a negative sign. For 0.9c and -0.5c the formula gives 0.4c / 0.55 = 0.7273c. If the two speeds are equal and opposite, the result is zero, which is the object being at rest.
What is rapidity and why does it add?
Rapidity is φ = artanh(v/c), a hyperbolic angle. Along a line, the rapidity of the combined velocity is the sum of the two rapidities. The calculator shows each rapidity, so you can check that 0.6c and 0.8c have rapidities 0.6931 and 1.0986 that sum to 1.7918.
How accurate is Galilean addition at 0.1c?
For two velocities of 0.1c, simple addition gives 0.2c, while the exact answer is 0.2/1.01 = 0.19802c. The classical sum is high by exactly one percent, which is why everyday engineering does not need the relativistic rule.
Can the combined speed ever exceed c?
No. If both speeds are at most c, the result is at most c. You can check this because 1 - u²/c² = (1 - v²/c²)(1 - w²/c²) / (1 + vw/c²)², which is never negative.
How do I combine a sideways velocity with a moving frame?
Use the two dimensional mode. The x component becomes (wx + v) / (1 + v wx / c²) and the y component becomes wy / (γ (1 + v wx / c²)). A frame at 0.6c and an object moving at 0.8c across the frame give 0.6c along and 0.64c across, a speed of 0.8773c.
Is velocity addition the same as the Lorentz transformation?
It follows from it. Differentiating the Lorentz transformation of position with respect to transformed time gives the velocity addition formula. The same relation applies to every inertial observer, so you can add velocities in any order and get one consistent answer in a straight line.
Which unit should I use, c, m/s or km/s?
Any of them. The unit selector only changes how your numbers are read and how the answers are shown. The formula always uses fractions of c internally, with c = 299792458 m/s, so the same inputs in km/s give the same physical result.