Calculate relativistic velocity addition u = (v+w)/(1+vw/c²). Compare Einstein vs Galilean results with visual bars and Lorentz factors.
The relativistic velocity addition calculator applies Einstein's velocity composition formula to find the combined velocity when an object moves at speed w within a reference frame that itself moves at speed v. Unlike the everyday Galilean addition (u = v + w), the relativistic formula u = (v + w)/(1 + vw/c²) ensures the result never exceeds the speed of light, no matter how fast the individual velocities are.
This fundamental result of special relativity explains why no massive object can reach or exceed the speed of light. Even if two spaceships each travel at 0.9c relative to Earth, an observer on one ship sees the other approaching at only 0.994c — not 1.8c as everyday intuition would suggest. The additional factor 1/(1 + vw/c²) becomes significant only at velocities that are a substantial fraction of c, which is why Galilean addition works perfectly for everyday speeds.
This calculator compares the relativistic and Galilean results side by side, computes the Lorentz factor for each velocity, displays visual comparison bars scaled to the speed of light, and provides a table showing how the relativistic correction grows as the frame velocity increases.
Understanding relativistic velocity addition is essential for anyone studying special relativity, particle physics, astrophysics, or modern physics courses. The visual comparison between Galilean and relativistic results makes the concept intuitive, and the parameter-sweep table shows exactly how and when the correction becomes important.
This calculator is also useful for science fiction scenario analysis — checking whether the velocities in a story are physically consistent — and for physics instructors creating examples and problem sets.
Relativistic velocity addition: u = (v + w) / (1 + vw/c²), where v = frame velocity, w = object velocity in that frame, c = speed of light. Galilean addition: u = v + w. Lorentz factor: γ = 1/√(1 − v²/c²). Speed of light: c = 299,792,458 m/s.
Result: Relativistic: 0.946c, Galilean: 1.4c (impossible)
u = (0.6 + 0.8)/(1 + 0.6 × 0.8) = 1.4/1.48 = 0.9459c. Galilean would give 1.4c, which exceeds the speed of light and is physically impossible. The relativistic formula ensures the result stays below c.
Before Einstein, Galilean relativity assumed that velocities simply add: if you throw a ball at 10 m/s from a train moving at 30 m/s, an observer on the ground sees the ball at 40 m/s. This works perfectly for everyday speeds but fails dramatically near the speed of light.
Einstein's 1905 paper "On the Electrodynamics of Moving Bodies" derived the correct velocity addition formula from two postulates: (1) the laws of physics are the same in all inertial reference frames, and (2) the speed of light in vacuum is the same for all observers. The second postulate requires that velocities cannot simply add; they must compose through the formula u = (v+w)/(1+vw/c²).
The critical feature of this formula is the denominator: 1 + vw/c². For everyday speeds (v, w << c), this factor is essentially 1.0000...001, making the correction negligible. But as either velocity approaches c, the denominator grows significantly, compressing the sum back below c.
The speed-of-light limit is not merely a mathematical consequence of the formula — it reflects a deep physical reality. As an object's velocity increases toward c, its relativistic mass (energy) increases without bound, requiring infinite energy to reach c. This is described by the relativistic energy-momentum relation E = γmc², where γ diverges as v → c.
For massless particles (photons), the situation is different: they always travel at exactly c in vacuum, regardless of the source's velocity. A flashlight on a spaceship moving at 0.999c emits photons that travel at c (not 1.999c) relative to a stationary observer. This counterintuitive result is confirmed by every experiment ever performed.
The Fizeau experiment (1851) measured the speed of light in flowing water and found a result consistent with relativistic velocity addition — decades before Einstein's theory. Modern particle accelerators confirm the formula daily: protons at the Large Hadron Collider travel at 0.999999991c, and the velocities of their collision products obey relativistic addition perfectly.
No. The relativistic addition formula ensures that the sum of any two velocities less than c always results in a velocity less than c. Only if one of the velocities is exactly c (like a photon) does the result equal c.
u = (c + c)/(1 + c×c/c²) = 2c/(1+1) = c. A photon emitted from a source moving at c still travels at c. This is the second postulate of special relativity: the speed of light is constant in all reference frames.
The correction is about 1% at v = w = 0.1c (10% of light speed, or about 30,000 km/s). Below about 0.01c, the Galilean formula is accurate to within 0.01%. For everyday speeds, the correction is immeasurably small.
The Lorentz factor γ = 1/√(1−v²/c²) quantifies time dilation and length contraction. At v = 0, γ = 1 (no effect). At v = 0.87c, γ ≈ 2 (time runs half as fast, lengths contract by half). As v → c, γ → ∞.
This formula applies to collinear (same-direction) velocity addition. For perpendicular or arbitrary-angle velocity addition, more complex vector transformations are needed, involving both longitudinal and transverse components.
Yes — u = (v+w)/(1+vw/c²) gives the same result regardless of which velocity you call v and which you call w. The formula is symmetric in v and w.