Mathematics 11–12 · Years 11–12

Projectile motion: a marble off the table and a thrown ball on video

Introduction to vectors: Projectile motion (Mathematics Extension 1, Year 12); Further work with functions: Parametric form of a function or relation (Mathematics Extension 1, Year 11); Further calculus skills: Further derivatives of functions (Mathematics Extension 1, Year 12)

Practical, model not builtLow risk

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The idea

Horizontal motion at constant velocity and vertical motion under constant gravity combine into a parabola whose range and flight time follow from the launch speed and angle.

What you need

A marble, a smooth table 0.80 m high, a short grooved ramp taped to the tabletop at least 0.50 m back from the edge so the marble rolls across the flat top before it leaves
Carbon paper on white paper on the floor, a metre rule, a plumb line
A phone camera and Tracker software for the thrown-ball part; a tennis ball

How to do it

Roll the marble from a fixed mark on the ramp so it rolls off the table horizontally; film its last 0.50 m on the table to find its speed u.
Drop the plumb line to mark the table edge on the floor and measure the horizontal range to the carbon mark for five trials.
Compute the flight time t = sqrt(2h/g) from the table height and predict the range u t; compare with the mean measured range.
Film a gentle throw across the room beside a 2 m scale, mark the ball in each frame in Tracker and fit x = u cos(theta) t and y = u sin(theta) t - 4.9 t^2.
Find the path's gradient dy/dx = (dy/dt)/(dx/dt) at launch and at landing. Measure the release height h above the floor, predict the landing point by solving h + u sin(theta) t - 4.9 t^2 = 0 for t, and compare that prediction and the level-ground range u^2 sin(2 theta)/g with the measured landing point.

What you should see

From a 0.80 m table the flight time is 0.404 s; with u = 1.50 m/s the range is 0.606 m and the marble lands at 4.23 m/s at 69.3 degrees below the horizontal. A ball thrown at 20 m/s at 45 degrees from ground level would range 40.8 m in 2.89 s with a maximum height of 10.2 m; at 30 degrees the range falls to 35.3 m. An indoor throw leaves the hand above the floor, so the level-ground formula does not apply: thrown at 5.0 m/s and 45 degrees from 1.20 m up, the ball is in the air for 0.973 s and lands 3.44 m away, while u^2 sin(2 theta)/g gives only 2.55 m. The learner knows it worked when the mean measured range sits within 5 per cent of u t, the five marks cluster within a few centimetres, and the thrown ball lands nearer the prediction that includes the release height than the level-ground range.

What changes

What you change: launch speed u (ramp release point) and, for the throw, launch angle
What you measure: horizontal range and flight time
What you keep the same: table height
same marble
ramp fixed
g

Common misconceptions

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

The marble stops moving forward once it is past the table edge; its horizontal velocity stays at u until it lands.
45 degrees always gives the longest range; it does only when launch and landing heights are equal.
A faster horizontal launch keeps the marble in the air longer; the flight time depends only on the drop height.

Safety card

Low riskLearners carry it out

Hazards

marbles on the floor underfoot
a thrown ball indoors

Controls

collect the marble after each trial
throw only along a cleared line

Note

No chemicals and no heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; ordinary classroom supervision.

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 Extension 1 11–12 Syllabus (2024), Year 12 focus area Introduction to vectors; 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 2026-09-22ME1-12-02
Mathematics Extension 1 11–12 Syllabus (2024), Year 11 focus area Further work with functions; 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 2026-09-22ME1-11-01
Mathematics Extension 1 11–12 Syllabus (2024), Year 12 focus area Further calculus skills; 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 2026-09-22ME1-12-04
Australian Curriculum v9No Australian Curriculum v9 code is listed.

Sources

The pages the author read to write this activity.

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