The science
Seen from the side, a rider on a circular halfpipe is a pendulum: the centre of mass (CoM) swings about the pipe's centre of curvature at radius l = R − h, where h is the CoM height above the board.
- Equation of motion (variable length):
d(l²θ')/dt = −g l sinθ − friction, i.e. θ'' = −(g sinθ + 2 l' θ' + μ N sgnθ')/l with normal force N = m(g cosθ + l θ'²).
- Pumping is parametric driving: the rider stands up (
l gets shorter by the pump amplitude) while rolling up the wall and crouches while coming down. Angular momentum l²θ' is almost conserved while l changes, so shortening at the fast bottom raises the kinetic energy, and lengthening at the slow top costs little. Net: energy gain each cycle.
- Airs: at the lip the wall is vertical, so the rider leaves straight up with speed
v = lθ' and rises v²/2g before falling back (no drag).
- Without pumping or friction the period obeys
T = 4√(l/g) K(sin(θ0/2)) with the complete elliptic integral K; the check button compares this exactly.
Honest simplifications: point-mass rider, perfectly circular pipe (no flat bottom), Coulomb-style rolling friction, no air drag, no tricks or rotation, and the pump rule is a simple automatic one; the 3D page adds tricks and scoring that are not modelled here.