Updated 12 October 2026 · By Pete Bromfield, former Head of Maths with IB examining experience

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?

Make it yours, then agree it with your teacher.

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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

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.

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.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Fits — ambitious at SL	Polynomial functions and systems of equations for coefficients; Continuity and differentiability at the joins
AA HL	Good fit	Polynomial functions and systems of equations for coefficients; Continuity and differentiability at the joins
AI SL	Fits — ambitious at SL	Polynomial functions and systems of equations for coefficients; Continuity and differentiability at the joins
AI HL	Good fit	Polynomial functions and systems of equations for coefficients; Continuity and differentiability at the joins

Level: Ambitious. Suits confident students; expect to learn some mathematics on your own. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Reflection (D)

For this idea, start with the weak points: only matching heights, not gradients, so the join has a kink; and choosing constraints with no justification. Say how each one affects your answer.

Use of mathematics (E)

SL: polynomial functions and systems of equations for coefficients; continuity and differentiability at the joins; maximum gradient using the second derivative, each explained in your own words and carried out correctly.

HL adds: curvature or matching second derivatives too, each explained in your own words and carried out correctly.

How calculus explorations reach the top bands in general: Calculus & optimisation IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

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

Extending it for HL

This idea already has HL mathematics in it: curvature or matching second derivatives too. Make that the centre of the exploration and use it with full rigour.

See a complete IA, marked

Is a hanging chain a parabola? Comparing catenary and quadratic models (AA SL) is a complete calculus exploration, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

It works for AA SL, AA HL, AI SL and AI HL. We rate it Ambitious: it suits confident students, who should expect to learn some mathematics on their own.

What mathematics does this IA use?

Polynomial functions and systems of equations for coefficients, continuity and differentiability at the joins, maximum gradient using the second derivative and hL: curvature or matching second derivatives too.

Where can I get data for this IA?

Take real heights, drops and maximum angles for a few coasters from the Roller Coaster DataBase to set your constraints. Checked sources and tools: Roller Coaster DataBase and Desmos graphing calculator.

Turn this idea into your IA

The planner opens with “Designing a smooth roller-coaster drop with piecewise functions” filled in: the question, the maths and the data sources.

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