The cart's speed comes straight from energy conservation along the track's arc length s, integrated every frame:
dv/dt = -g·(dy/ds) - μ·g·|dx/ds|·sign(v)
s(t+dt) = s(t) + v(t)·dt
The first term is gravity resolved along the local slope; the second is a simplified Coulomb-friction loss proportional to how "flat" the track is locally. A chain lift hauls the cart up the first hill at constant speed (like a real coaster motor) — everything after that crest is pure gravity and friction.
At the top of a loop of radius R, the track can only push, never pull, so the cart needs at least v ≥ √(gR) to stay on the rails (0 G at the top, all of gravity spent on centripetal force). Raise the first hill or shrink the loop to clear that bar; add friction or shrink the hill to fail it — the cart will visibly stall and roll back down when it can't make it.
- G-force — normal reaction felt by the rider, from local track curvature and speed: N/m = v²·|κ| − (gravity along the inward normal), shown in units of g.
- Energy lost to friction — cumulative, per kilogram of cart mass; mass itself cancels out of the equations of motion, same as a real coaster.