Same rotovator, now rigorous: velocities from vis-viva, the tip's true path from actual orbital and spin rates, the payload's real transfer orbit after release, and the tether's mass from the constant-stress taper equation — so the materials question stops being a hand-wave. Pick a material and watch the whole thing become buildable or impossible.
Every velocity is v=√(GM(2/r−1/a)). The catch speed is the station's orbital speed minus the tip speed at the low point (tip speed measured relative to the station); release adds them at the high point. The violet curve is the payload's actual orbit after release, drawn from its release state.
A stress-tapered tether needs mass Mₜ/Mₚ = √π·x·e^{x²}·erf(x), where x = tip-speed ÷ the material's characteristic velocity. Because of the e^{x²}, everything hinges on the material: Kevlar caps you at partial tip speeds; nanotube makes a full-speed hook trivial.
Raise the tip speed and the catch gets easier (lower relative velocity) while the tether mass explodes. Switch materials and the whole curve shifts. This is the structure↔handshake tradeoff from the console, now in real kilograms and kilometers per second.