IA idea · Kinematics & physics-style modelling

Crossing a river with a current: fastest or straightest?

AA HLAI HLAA SL Solid Also in: Trig & navigation, Optimisation

Research question

If a swimmer or boat crosses a river with a current, which heading minimises the crossing time, which lands directly opposite, and how do those headings change if the current is faster in the middle?

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

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Why it makes a good exploration

A classic vector problem made genuinely interesting by a realistic current profile. The uniform case is exact; the varying case needs calculus or a numerical method.

The mathematics you'll need

Velocity vectors and components
Minimising time; minimising drift
A current that varies across the river (a quadratic profile)
Integration of drift across the river
Optimising a variable heading (numerically)

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

Measure a stream's surface speed at several points with floats and a stopwatch (safely from the bank), or use a stated profile.

Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.
GeoGebra — Free geometry and graphing software — Voronoi diagrams, loci and regression built in.

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

Solve the uniform-current case.
Measure or set a realistic current profile.
Calculate drift for a fixed heading.
Find the best fixed heading and compare with a changing one.
Reflect on safety and real swimming.

Pitfalls that cost marks

Confusing heading with track.
Measuring the current unsafely.
Not checking the uniform case against a known result.

Showing personal engagement

Use a stream you know.
Test with a toy boat.
Ask a lifeguard about crossing advice.

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

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	Velocity vectors and components; Minimising time; minimising drift
AA HL	Good fit	Velocity vectors and components; Minimising time; minimising drift
AI SL	Not a natural fit	The mathematics is mainly AA or HL (calculus or proof beyond AI SL); an AI SL version would need a data-driven, technology-based approach.
AI HL	Good fit	Velocity vectors and components; Minimising time; minimising drift

Level: Solid. Needs some independent work beyond class examples. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Personal engagement (C)

Film or record the motion yourself, choose what to vary, and say what you expected before you analysed it.

Reflection (D)

Compare your model with the data and explain each mismatch (air resistance, friction, tracking error); reflect on how noise affects numerical derivatives. For this idea, start with: confusing heading with track — say how it affects your answer.

Use of mathematics (E)

SL: Displacement, velocity and acceleration related by differentiation and integration, fitted models compared with residuals, and the physics assumptions turned into stated mathematical assumptions.

HL: Vectors for motion in two dimensions, a differential equation for a resisted motion solved and compared with data, or numerical differentiation with an error analysis.

Criteria A and B (presentation and communication) work the same way for every idea: see the guides to Criterion A and Criterion B.

Taking it further

Find the optimal heading as a function of position across the river (a calculus of variations taster).

Extending it for HL

Add air resistance or friction as a differential equation, solve it (analytically or with Euler's method) and compare with the simpler model.

See a complete IA, marked

Our annotated exemplar How long does a skydiver take to reach terminal velocity? (AA HL) asks a different question, but shows how a complete kinematics exploration is structured and marked, with an examiner's comment on every criterion. Free excerpts and the full marking table are on its page.

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Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Criteria checklistCheck your plan or draft against the A–E descriptors and self-grade before you hand it in. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, an IB examiner, reads your draft and talks it through with you (€100).