Mathematics 11–12 · Year 12

A lift ride: integrating measured acceleration to find speed and height

Applications of calculus: Rates of change (Mathematics Advanced, Year 12); Integral calculus: Areas and the definite integral (Mathematics Advanced, Year 12)

Practical, model not builtLow risk

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

The area under an acceleration-time graph is the change in velocity and the area under the velocity-time graph is the change in height, which a phone in a lift measures directly.

Safety card

Low riskAn adult supervises

Hazards

crowding in a lift
holding the doors

Controls

ride only when the lift is not needed by others
follow the building's lift rules; never hold the doors

Note

No chemicals and no heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; a teacher or other adult supervises, the school decides whether students may use its lift, and a public building's lift is an excursion run under the school's excursion risk assessment.

What you need

A phone running the phyphox Acceleration (without g) experiment, and a second phone (or a second run) with the phyphox Elevator experiment, which uses the pressure sensor
A lift in the school travelling at least three floors, used with the school's permission (a public building's lift is off-site work and is used only on an excursion covered by the school's risk assessment)
A spreadsheet

How to do it

Place the phone flat on the lift floor and start recording before the doors close; ride up three or more floors without stopping; stop recording after the doors open.
Export the vertical acceleration and plot it against time; identify the speeding-up, steady and slowing-down phases.
Integrate the acceleration numerically with the trapezoidal rule to get velocity, and integrate the velocity to get height; correct any velocity drift so that the lift ends at rest.
Read the height change from the Elevator experiment's pressure-based height and compare it with your integral.
Repeat on the way down and explain the signs of the areas.

What you should see

Worked check with test inputs: a lift that speeds up at 0.8 m/s^2 for 1.5 s, cruises for 8 s and slows at 0.8 m/s^2 for 1.5 s reaches 1.2 m/s and rises 1.8 + 9.6 = 11.4 m; near sea level that height corresponds to a pressure drop of about 134 Pa (air density 1.2 kg/m^3, g = 9.80 m/s^2). The learner's acceleration graph shows two opposite-signed pulses of equal area, the velocity graph a trapezium, and the height graph an S-shape. The learner knows it worked when the integrated height and the pressure-based height agree within a few per cent and the velocity returns to zero at the end.

What changes

What you change: time
What you measure: vertical acceleration, velocity and height
What you keep the same: phone flat and still on the floor
no stops between the first and last floor
same lift for both directions

Common misconceptions

Each of these ideas is wrong, and the activity is a chance to test it.

Zero acceleration means the lift is stopped; during the cruise it moves at a steady speed with zero acceleration.
The height comes from the largest acceleration; it comes from the area under the velocity graph.

Curriculum references

The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.

Mathematics Advanced 11–12 Syllabus (2024), Year 12 focus area Applications of calculus; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22MAV-12-06
Mathematics Advanced 11–12 Syllabus (2024), Year 12 focus area Integral calculus; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22MAV-12-05
Australian Curriculum v9No Australian Curriculum v9 code is listed.

Sources

The pages the author read to write this activity.

All Concept Studio activities