Investigating Science 11–12 · Year 11
A draining bottle: testing a mathematical model and refining it
Module 3: Scientific Models
This site has no interactive model of its own. Where a step or a material names a Concept Studio model, simulation or tool, it has not been built; an external simulation a step names (for example PhET) is not part of this site.
The idea
A mathematical model predicts how the water level in a draining container falls; testing it against a real bottle shows where the ideal model fails and how one measured constant repairs it.
What you need
- a straight-sided clear container 10.0 cm in internal diameter (a length of acrylic pipe with a capped base, or a straight-sided bottle) with a 4.0 mm hole drilled in the centre of the base by the teacher
- ruler taped vertically with zero at the hole, phone to video the level, stopwatch, water, a tray or sink
How to do it
- Fill to 20.0 cm above the hole with a finger over the hole.
- Release and film the level falling; read the height every 10 s from the video.
- Repeat three times.
- Plot √h against time; the model predicts a straight line.
- Calculate the ideal drain time from the model and compare with the measured time; find the discharge coefficient C_d = t_ideal / t_measured.
- Use the fitted model to predict the drain time from 10.0 cm, then test the prediction.
What you should see
Torricelli’s law v = √(2gh) with an area ratio (100/4.0)² = 625 predicts an initial jet speed of 1.98 m/s and an ideal drain time from 20.0 cm of t = 625 √(2 × 0.200/9.80) = 126 s. The real container takes longer because the jet narrows as it exits a sharp-edged hole; the learner’s √h against t plot is still straight, so the square-root law holds and only the constant needs fitting. A fitted C_d of 0.61, for example, gives 207 s; with C_d fitted, the model predicts the 10.0 cm drain time (t ∝ √h₀, so 0.707 of the 20 cm time), which the learner then tests against their timing uncertainty.
What changes
- What you change
- starting height (and hole size in an extension)
- What you measure
- water height against time and total drain time
- What you keep the same
- same container and hole
- container vertical
- same water temperature
- three trials per height
Common misconceptions
Each of these ideas is wrong, and the activity is a chance to test it.
- Water drains at a steady rate.
- If a model’s number is wrong the whole model is useless.
- A bigger container drains faster because more water pushes down.
Safety card
Hazards
- water spills
- sharp edges on the drilled hole
Controls
- drain into a sink or tray
- teacher drills and deburrs the hole in advance
Note
No hazardous chemicals or heat sources: record the activity in the school's RiskAssess risk assessment, following the NSW Department of Education Science safety and compliance page; the Chemical Safety in Schools package is not triggered.
Curriculum references
The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.
- Investigating Science Stage 6 Syllabus (2017), NESA; currentINS11-10INS11/12-2INS11/12-3INS11/12-4
- Australian Curriculum v9No Australian Curriculum v9 code is listed.
Sources
The pages the author read to write this activity.