Roller Coasters: The Physics of a Ride With No Engine

One number decides the whole ride: total energy

A roller coaster has no engine after the initial lift hill — every hill, loop and drop for the rest of the ride runs entirely on conservation of energy. At the top of the lift hill, the cart has a fixed amount of total mechanical energy, split between gravitational potential energy (height) and kinetic energy (speed). From that point on, the track can only trade one for the other: go higher and the cart must slow down, go lower and it speeds up, and — ignoring friction and air resistance for a moment — the total never changes.

E_total = ½ m v² + m g h        // kinetic + potential, m cancels out
v(h) = √( v_top² + 2g(h_top - h) )   // speed at any height, from energy conservation alone

This is why a well-designed coaster's very first hill is always its tallest point (barring a later powered launch) — the ride can never regain energy it loses to friction and drag, so every subsequent hill has to be shorter than the last to guarantee the cart still has enough speed to clear it.

Why loops are teardrop-shaped, not circular

A perfectly circular loop demands the same centripetal force at the top as at the bottom, but the cart is moving much slower at the top of a loop (higher, so more of its energy has become potential rather than kinetic) — a true circle would need unrealistically high speed at the bottom just to keep enough speed at the top to maintain contact with the track, translating into extreme, uncomfortable and dangerous g-forces at the bottom. Real coaster loops use a clothoid (tightening-radius) shape instead: a larger radius at the bottom, where speed is highest, and a tighter radius at the top, where speed is lowest — this keeps the g-force the rider feels roughly constant through the whole loop instead of spiking dangerously at the bottom.

G-force: what riders actually feel

Riders don't feel speed, they feel acceleration — specifically, the combination of gravity and the track's normal force pushing back on them, measured in multiples of Earth's gravity (g). At the bottom of a loop or a dip, both gravity and the seat pushing you upward add together, so you feel heavier than normal (positive g, often 3–4g on real coasters). Cresting a hill fast enough, gravity briefly outpaces the seat's support and you feel weightless or even lifted from your seat (negative or near-zero g) — the "airtime" hill sensation, which is really just a controlled, brief approach to freefall.

g_felt = (normal_force_from_track) / (m * g)     // 1.0 = feels normal
at loop bottom:  N = m*(g + v²/r)   →  g_felt > 1  (pressed into seat)
at loop top:     N = m*(v²/r - g)   →  must stay ≥ 0 to keep contact with track

Where the energy actually leaks

No real coaster is frictionless — wheel friction against the rail and air resistance both quietly drain the total energy budget throughout the ride, which is exactly why a real coaster's later hills must be built noticeably shorter than pure energy conservation alone would require, giving a margin of safety in case of a windy day or extra passenger weight changing the drag slightly. This simulation lets you compare a frictionless run against a run with realistic friction so the gap is visible directly, rather than hidden inside a single fixed physics model.

Designing your own track

Drawing a custom track and launching a cart down it is really just choosing a height function h(x) and letting the energy equation above compute the resulting speed at every point automatically — which is exactly why an unrealistically tall drop followed immediately by a tight loop can produce g-forces that would be unsafe on a real ride, and why real coaster designers spend enormously more effort on the transitions between elements (easing curvature in and out smoothly) than on the elements themselves.

Frequently asked questions

Try it live

Everything above runs in your browser — open Roller Coaster and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.