The lifeguard problem: what is the fastest route to a swimmer?

AA SLAA HLAI SLAI HL Solid Also in: Sport, Geometry & Voronoi

Research question

Where should a lifeguard enter the water to reach a swimmer in the least time, given measured running and swimming speeds, and does the optimal path obey Snell's law?

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

Why it makes a good exploration

Running is faster than swimming, so the straight line is not the fastest route. The calculus answer turns out to be the same law that bends light at water — a beautiful link you can test with your own speeds.

The mathematics you'll need

Distance with Pythagoras
Time as a function of the entry point
Differentiation to minimise time
Showing the optimum satisfies sin θ₁/v₁ = sin θ₂/v₂

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

Time yourself running 20 m on sand (or grass) and swimming 20 m; set a realistic beach geometry.

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

Define the geometry and variables.
Write total time T(x) and differentiate.
Solve for the optimal entry point, numerically if necessary.
Show the condition is Snell's law.
Compare with the straight-line path and the “run as far as possible” path.

Pitfalls that cost marks

Using speeds you have not measured or sourced.
Solving numerically without checking it is a minimum.
Ignoring waves and currents without comment.

Showing personal engagement

Use your own running and swimming speeds.
Test whether a dog playing fetch approximates the optimum (a famous study).
Discuss what a real lifeguard would do.

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

Taking it further

Add a current in the water and re-optimise.

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).