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

How fast is a sprinter at each moment? Differentiating 100 m splits

AA SLAA HLAI SLAI HL Solid Also in: Sport, Modelling, Kinematics

Research question

Can a model of distance against time fitted to 10 m split times estimate a sprinter's top speed and when it is reached, and how does my own sprint compare?

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Why it makes a good exploration

Split times record distance at each 10 m, but not speed. Fitting a model and differentiating it recovers the velocity curve — then you can compare an elite race with your own sprint filmed on a phone.

The mathematics you'll need

Fitting a model such as d(t) = v_max(t − τ(1 − e^(−t/τ)))
Differentiation to find velocity and acceleration
Least squares or parameter estimation
Comparing parameters across athletes

Where the data comes from

Film your own 60 m sprint with markers every 10 m and read times; published elite split times exist in biomechanics reports from major championships.

Tracker video analysis — Free tool to track an object frame by frame in a video and export its x–y coordinates.
World Athletics records and results — All-time lists, records and progression for every athletics event.

Real data for this idea

Primary source: World Athletics records and toplists: All-time toplists. Opens on the men's 100 m: choose your event, copy the marks you need into your own table and cite the page and date. Licence: No reuse licence found: link and cite only.
Also useful: Olympedia results. Use the event result pages; copy only the rows you need and cite each page. Licence: No reuse licence found: link and cite only.

There is no open reuse licence for these figures: copy only the values you analyse into your own table, and cite the page and the date. Links checked October 2026; we link to the data and never copy it. More free data sets.

A possible outline

Explain why a sprinter's speed rises then levels off.
Fit a model to split data.
Differentiate to get velocity and find top speed.
Compare your own sprint with an elite race.
Reflect on reaction time and deceleration at the end.

Pitfalls that cost marks

Using average speed over a split as instantaneous speed without comment.
Choosing a polynomial that behaves badly outside the data.
Ignoring the reaction time at the start.

Showing personal engagement

Race yourself and friends with consent, filming carefully.
Predict your 100 m time from your 60 m model.
Compare two different training sessions.

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	Fitting a model such as d(t) = v_max(t − τ(1 − e^(−t/τ))); Differentiation to find velocity and acceleration
AA HL	Good fit	Fitting a model such as d(t) = v_max(t − τ(1 − e^(−t/τ))); Differentiation to find velocity and acceleration
AI SL	Good fit	Fitting a model such as d(t) = v_max(t − τ(1 − e^(−t/τ))); Differentiation to find velocity and acceleration
AI HL	Good fit	Fitting a model such as d(t) = v_max(t − τ(1 − e^(−t/τ))); Differentiation to find velocity and acceleration

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

Reflection (D)

For this idea, start with the weak points: using average speed over a split as instantaneous speed without comment; and choosing a polynomial that behaves badly outside the data. Say how each one affects your answer.

Use of mathematics (E)

SL: fitting a model such as d(t) = v_max(t − τ(1 − e^(−t/τ))); differentiation to find velocity and acceleration; least squares or parameter estimation; comparing parameters across athletes, 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

Fit the model to several elite athletes and compare the acceleration constant τ; relate it to body mass or event specialism.

Extending it for HL

Add an HL technique the question genuinely needs: what an HL calculus exploration adds.

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 Solid: it needs some independent work beyond class examples.

What mathematics does this IA use?

Fitting a model such as d(t) = v_max(t − τ(1 − e^(−t/τ))), differentiation to find velocity and acceleration, least squares or parameter estimation and comparing parameters across athletes.

Where can I get data for this IA?

Film your own 60 m sprint with markers every 10 m and read times; published elite split times exist in biomechanics reports from major championships. Checked sources and tools: World Athletics records and toplists, Olympedia, Tracker video analysis and World Athletics records and results.

Turn this idea into your IA

The planner opens with “How fast is a sprinter at each moment? Differentiating 100 m splits” filled in: the question, the maths and the data sources.

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