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?

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

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

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

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.

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

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.

Taking it further

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

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).