Roller Coaster Energy Simulator
Drag the position slider along the track. Energy conservation determines the speed at every point.
Hamiltonian Mechanics 🖖
The system evolves along contours of constant total energy...
Speed depends only on height 🖖
On a frictionless track the ball's speed at any point is fixed entirely by how far it has dropped from its start — the shape of the track in between makes no difference. A near-vertical plunge and a long gentle slope to the same depth deliver exactly the same speed. That's why the ball can never climb higher than where it began: it would need energy it doesn't have. Add friction and each metre of track quietly skims energy away as heat.
Real loops aren't circular 🖖
A roller-coaster loop looks round but almost never is — it's a teardrop-shaped clothoid, a curve whose radius tightens toward the top. A truly circular loop fast enough to stay on the track at the crest would crush riders with punishing g-forces at the bottom. German engineer Werner Stengel introduced the clothoid loop in 1976, giving a small radius up top (little speed needed) and a gentle radius below. Same energy conservation — smarter geometry.
Example problems
- Classic - Classic
- Failed loop - Failed loop
- Half pipe - Half pipe
- With friction - With friction