Designing a smooth roller-coaster drop with piecewise functions

AA SLAA HLAI SLAI HL Ambitious Also in: Modelling, Sport

Research question

Can I design a roller-coaster section from polynomial pieces that join smoothly (matching value and gradient), meets real height and angle limits from existing coasters, and keeps the steepest slope below a chosen limit?

Adapt it: change the place, the data or the comparison until the question is yours.

Why it makes a good exploration

Real coasters must join track sections without sudden changes of slope. Designing your own section with calculus constraints — informed by real coaster statistics — is creative, rigorous and very personal.

The mathematics you'll need

Polynomial functions and systems of equations for coefficients
Continuity and differentiability at the joins
Maximum gradient using the second derivative
HL: curvature or matching second derivatives too

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

Where the data comes from

Take real heights, drops and maximum angles for a few coasters from the Roller Coaster DataBase to set your constraints.

Roller Coaster DataBase — Height, drop, length, speed, inversions and g-force for 12,000+ roller coasters.
Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

A possible outline

Set design constraints from real coasters.
Choose the pieces (e.g., a cubic drop and a quadratic valley).
Solve for coefficients so values and gradients match at joins.
Check the maximum steepness and adjust.
Reflect on what the model ignores (g-forces, friction, track width).

Pitfalls that cost marks

Only matching heights, not gradients, so the join has a kink.
Choosing constraints with no justification.
Presenting a pretty graph without the algebra that produced it.

Showing personal engagement

Base the design on a coaster you have ridden.
Explain a failed first design and how you fixed it.
Compare your steepest slope with record-holding coasters.

See Criterion C: personal engagement for what examiners look for.

Taking it further

HL: match second derivatives too, and discuss why sudden changes in curvature are felt as jerk by riders.

Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Desmos workspacePlot your data, fit models and keep your graphs with your plan. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, a practising IA examiner, reads your draft and talks it through with you (€100).