Relativistic Velocity Addition — Spacetime & Rapidity Diagram

A ship races away from a station at a large fraction of the speed of light and fires a probe forward at another large fraction of light speed, measured in the ship's own frame. This 2D companion renders the encounter as a formal Minkowski spacetime diagram — worldlines plotted against position and time rather than objects animated through simulated space — with the ship's line, the probe's true relativistic trajectory, a dashed Newtonian "ghost" that visibly breaches the light cone, and the ship's own line of simultaneity. Below it, a rapidity number line shows the same physics from a second angle: hyperbolic angles φ = artanh(v/c) add exactly like plain numbers, and only the final conversion back through tanh reveals why no combination of sub-light speeds can ever reach c. Two sliders control the ship's speed and the probe's launch speed, and live readouts compare the Newtonian sum, Einstein's u = (v+u′)/(1+vu′/c²), and the rapidity-sum route to the identical answer.