Halfpipe Pumping Lab

A robot rider modelled as a point mass whose centre of mass can rise and sink - energy, g-forces and the physics of "pumping".

Presets

Rider technique

Ramp

World

Display

Mechanical energy of the rider (J)

kineticpotentialtotal

Normal force (g) and speed (m/s)

normal force / weightspeed

Energy ledger

Rider work is whatever the legs add or remove. Friction, drag and landing losses are tallied separately, so total = start energy + rider work - losses.

1 - How high can a given lip speed throw you?

Frictionless, no pumping. Launch speeds are swept; the maximum height of the board above the coping is measured from the integrated motion and compared with energy conservation: h = (E / m g) - y_lip.

2 - Does standing up at the bottom speed you up? (angular momentum)

The rider stands up quickly while on the circular transition. About the arc's centre, angular momentum m r v is conserved, so v_after = v_before (R - h₁)/(R - h₂). Measured by integrating the equations of motion across the stand-up, against that formula.

3 - How much does pumping really buy?

Same ramp, rider, friction and drag settings as the Setup tab, 25 s of riding: passive vs auto-pump.

The science

Model. The rider is a point mass m at the centre of mass (CoM), a height h(t) above the board's contact point on the profile curve. The curve has arc length s, tangent angle θ(s) and curvature κ(s)=dθ/ds (1/R on the transitions, 0 on the flat and the vertical wall; blended smoothly).

Equation of motion (Lagrange, one degree of freedom): with G = 1 - hκ, s'' = [ -g sinθ + 2 h' κ s' + h κ' s'² + F_t/m ] / G. The CoM speed is v = G s' (plus the h' component), so on a concave transition the CoM is closer to the curvature centre than the wheels are.

Normal force. N = m ( G κ s'² + h'' + g cosθ ): weight plus the centripetal term m v²/r. At the bottom of a 2 m transition at 7 m/s that is about 1 + 49/(9.8 x 1.4) = 4.6 g.

Pumping. The rider's work is W = ∫ N dh. Extending the legs (dh > 0) when N is large and crouching (dh < 0) when N is small gives a net positive cycle, exactly like a child pumping a swing: energy is added without any push from outside.

Losses. Rolling resistance μ N (urethane wheels, μ about 0.01) and quadratic drag 1/2 ρ C_dA v². Landing on the ramp removes the velocity component normal to the surface (inelastic).

Air. After the lip the CoM is a projectile: with no drag the board peaks at (E/(m g)) - y_lip above the coping. Experiment 1 measures this against the integrator.

Leg power limit. Leg extension speed is capped so that N h' never exceeds P_max (about 1.5 kW for a strong athlete); without it the model would pump unrealistically fast.

Simplifications. Two-dimensional motion; frictionless pivoting of the board on its wheels; no rotation or balance dynamics; rider leaves the surface only at the lip; the 'robot' is an abstract mannequin. Numerics: RK4, 0.5 ms steps; energy drift below 0.01% in the frictionless test.