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Ride
Release height, friction, air drag, mass and gravity decide whether the cart completes the loop.
Model checks
The science
Energy and the track. A cart of mass m on frictionless rails keeps kinetic plus potential energy, m v^2/2 + m g y, constant. Released from rest at height h it reaches the foot of the first drop at v = sqrt(2 g h), whatever the shape of the track. Friction and drag take energy out as heat (the sim integrates the heat, so kinetic + potential + heat stays constant).
The loop. On a circular loop of radius R the rails must supply the centripetal force. The load on the riders (their apparent weight divided by m g) is N/(m g) = cos(theta) + v^2 kappa / g. At the top this is 2h/R - 5, which is zero when h = 5R/2: below that height a cart without wheels under the rail leaves the track. At the bottom the load is 1 + 2h/R, which is why a circular loop fed from a high drop is brutal. Real loops are teardrop or clothoid shaped (curvature changes smoothly), keeping the loads near 3 to 5 g.
What is simplified. The cart is a point mass sliding without rotation on a planar track (the loop is a shallow helix of 2.4 m so the entry and exit rails do not touch; this is ignored in the dynamics). Rolling friction is a constant coefficient times the rail load, drag is quadratic. Real carts have wheel assemblies that also hold them on from above, so they never fall out of a loop; the "free cart" box switches that off. The curvature jumps at the joints between straight pieces and arcs, which would be felt as jerk.
Numerical checks (run on load): energy drift of a frictionless ride; the smallest release height that completes the loop, found by bisection, against the exact 5R/2; the rail loads at the loop bottom and top against 1 + 2h/R and 2h/R - 5; and the stopping distance and heat of a rolling cart against v0^2/(2 mu g).
What to try
- Press Just enough for the loop: a frictionless cart released at 5R/2 = 22.5 m only just stays on the rails at the top (load near zero), exactly where the bisection check lands.
- Raise rolling friction or drag: the heat bar grows while kinetic plus potential plus heat stays equal to the starting energy.
- Switch to Moon gravity: the loop condition uses g on both sides, so the minimum release height does not change, only the speeds do.