Velocity Addition Calculator

Consider three inertial frames: S (the "lab" frame), S' (moving at velocity v relative to S), and an object moving at velocity u in frame S'. The velocity w of the object as seen from frame S is:

$$w = \frac{u + v}{1 + \frac{uv}{c^2}}$$

This formula is derived from the Lorentz transformation of spacetime coordinates. Key properties:

• Reduces to Galilean at low speeds: When $$uv \ll c^2$$, the denominator approaches 1 and $$w \approx u + v$$.
• Speed of light is invariant: If $$u = c$$, then $$w = \frac{c + v}{1 + v/c} = c$$, regardless of v. Light moves at c in all frames.
• Never exceeds c: If both $$|u| < c$$ and $$|v| < c$$, then $$|w| < c$$. The denominator always "compresses" the result below c.
• Commutative: $$w(u, v) = w(v, u)$$ — the order of addition doesn't matter (in one dimension).
• Associative: Adding three velocities can be done sequentially: $$w_{123} = w(w(u_1, u_2), u_3)$$.

The correction factor $$\frac{1}{1 + uv/c^2}$$ shows how much the relativistic result differs from the classical sum. When both velocities have the same sign (both moving in the same direction), this factor is less than 1, reducing the combined speed. When they have opposite signs (approaching from opposite directions), the factor is greater than 1.

For everyday scenarios — cars, airplanes, even rockets — the correction is negligible. At 1000 km/h + 1000 km/h, the relativistic correction is about 1 part in 10¹³. But for particles in accelerators, cosmic rays, or photons, the formula is essential.

The three-dimensional generalization involves both parallel and perpendicular components and is more complex, but for collinear (one-dimensional) motion, the formula above is exact.

The relativistic combined velocity is the physically correct result. The classical velocity is shown for comparison — when it exceeds c, classical physics has clearly failed. The correction factor quantifies the relativistic suppression: values near 1 mean the classical result is nearly correct; values significantly below 1 indicate strong relativistic effects. Note that negative velocities indicate motion in the opposite direction.