Relativistic Velocity Addition

Fire a ball forward at half the speed of light from a ship already moving at half the speed of light, and common sense says the ball moves at the speed of light. It does not. Velocities do not simply add in relativity; they combine through $w = (u+v)/(1+uv/c^2)$, and the denominator quietly holds the result below $c$ no matter how hard you push. Two halves of light speed give $0.8c$, not $c$; even $0.99c$ plus $0.99c$ falls just short. Light itself is the fixed point: add any speed to $c$ and you get $c$ back, which is exactly what makes the speed of light the same in every frame. The scene races a light pulse, the ball, and the ship from a standing start, and the ball never catches the light. Below it, the ground velocities sit on an axis that ends at the light cone, with the naive Galilean sum $u+v$ shown overshooting past the wall it can never reach. The real elegance is on the third axis: the rapidity $\phi=\operatorname{artanh}\beta$, a relabelling of velocity that runs to infinity instead of stopping at $c$. Rapidities add the ordinary way, head to tail, so the whole nonlinearity of velocity addition is just the bunching-up of $\tanh$ near its asymptote. The diagnostic plots the ground speed against the ball's speed, the relativistic curve flattening against the light ceiling while the Galilean line runs off the top.