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

How long does a bottle take to drain? Testing Torricelli's law

AA SLAA HLAI SLAI HL Solid Also in: Differential equations, Calculus

Research question

Does the water height in a draining bottle follow the quadratic predicted by Torricelli's law, and how well does the model predict the total draining time for different hole sizes?

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Why it makes a good exploration

Poke a hole in a bottle and the flow slows as it empties. Torricelli's law predicts exactly how — the height should fall as a quadratic in time — and you can test it in your kitchen with a ruler and a phone.

The mathematics you'll need

Quadratic models and least-squares fitting
Square-root functions and rates of change
HL: setting up and solving dh/dt = −k√h by separation of variables
Comparing predicted and measured emptying times

Modelling the data, step by step

This idea usually needs a quadratic model. See it worked step by step, with a criterion tip at every step: Quadratic: three points, completing the square, regression.

Model your own data Paste it from Desmos, GeoGebra or a spreadsheet and get the same play-by-play with your numbers. New to modelling? Start with the modelling workflow. Writing it up? The IA modelling planner comments on each paragraph as you draft — it never writes it for you.

Where the data comes from

Film a clear bottle with a ruler taped to it draining through a hole; read the height every 5 seconds. Repeat for 2–3 hole diameters.

Tracker video analysis — Free tool to track an object frame by frame in a video and export its x–y coordinates.
Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.

A possible outline

Explain Torricelli's law and the assumptions behind it.
Collect height–time data for several holes.
Fit a quadratic and compare with a linear and an exponential model.
Relate the fitted constant to the hole area and test the relationship across holes.
Reflect on the bottle's shape (not a perfect cylinder) and on the vena contracta effect.

Pitfalls that cost marks

Bottles are not cylinders near the neck — restrict the domain and say why.
Too few repeats to separate a real effect from measurement noise.
Quoting the DE solution without deriving it (HL).

Showing personal engagement

Predict how halving the hole diameter changes the draining time, then test it.
Explain which part of the model broke down near the end and why.
Link it to something real: a water tower, a sink, an hourglass.

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	Quadratic models and least-squares fitting; Square-root functions and rates of change
AA HL	Good fit	Quadratic models and least-squares fitting; Square-root functions and rates of change
AI SL	Good fit	Quadratic models and least-squares fitting; Square-root functions and rates of change
AI HL	Good fit	Quadratic models and least-squares fitting; Square-root functions and rates of change

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: bottles are not cylinders near the neck — restrict the domain and say why; and too few repeats to separate a real effect from measurement noise. Say how each one affects your answer.

Use of mathematics (E)

SL: quadratic models and least-squares fitting; square-root functions and rates of change; comparing predicted and measured emptying times, each explained in your own words and carried out correctly.

HL adds: setting up and solving dh/dt = −k√h by separation of variables, each explained in your own words and carried out correctly.

How modelling explorations reach the top bands in general: Modelling with functions IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

HL: derive the model from conservation of volume and compare the fitted discharge coefficient with the theoretical value near 0.6.

Extending it for HL

This idea already has HL mathematics in it: setting up and solving dh/dt = −k√h by separation of variables. Make that the centre of the exploration and use it with full rigour.

See a complete IA, marked

Is it cheaper to leave the heating on low overnight? A two-temperature model of my bedroom (AI HL) is a complete modelling 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?

Quadratic models and least-squares fitting, square-root functions and rates of change, hL: setting up and solving dh/dt = −k√h by separation of variables and comparing predicted and measured emptying times.

Where can I get data for this IA?

Film a clear bottle with a ruler taped to it draining through a hole; read the height every 5 seconds. Repeat for 2–3 hole diameters. Checked sources and tools: Tracker video analysis and Desmos graphing calculator.

Turn this idea into your IA

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