Mathematics 7–10 · Years 9–10
Galileo's ramp: a rolling ball's distance grows with the square of the time
Number and algebra
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The idea
A ball rolling from rest down a straight incline covers a distance proportional to the square of the elapsed time, so the distance-time graph is half a parabola, distance against time squared is a straight line, and the distances in successive equal intervals go 1 : 3 : 5 : 7.
What you need
- A smooth straight board 2.40 m long and at least 15 cm wide (a melamine shelf), or a 2.40 m length of 25 mm × 25 mm aluminium angle used as a V-channel
- Steel ball bearing 20 to 25 mm in diameter, or a solid glass marble
- Twenty wooden spacers cut 25 mm thick, checked with a steel rule, stacked into supports of 25, 50, 75 and 100 mm placed every 0.60 m under the board so it cannot sag (the last step doubles each stack)
- Spirit level
- Two metre rules or a 3 m tape fixed along the board
- Phone recording video at 30 or more frames per second on a stand, framing the whole board
- A catch box or folded towel at the lower end
How to do it
- Support the board (or the aluminium channel) every 0.60 m: the lower end on the floor or bench, then stacks of 25, 50, 75 and 100 mm at 0.60, 1.20, 1.80 and 2.40 m along it, so the upper end is 0.100 m higher and the surface is straight in between. A 2.40 m shelf held up only at its ends sags in the middle under its own weight, enough to tip its lower end uphill: for a 16 mm melamine shelf 300 mm wide, taking its modulus as 2.5 GPa, the mid-span sag 5wL⁴/(384EI) comes to about 5 cm. Stretch a string along the top surface to check it is straight, and check with the spirit level that the board is level across its width.
- Hold the ball at the zero mark against a ruler and release it by lifting the ruler, with the video running.
- Step through the video frame by frame and record the ball's position every 0.5 s (every 15 frames at 30 frames per second) until it reaches the end.
- Make three runs and take the median position at each time.
- Plot distance against time, then distance against time squared; draw the line of best fit on the second graph and find its gradient k, so that d = k t².
- Compute the distance travelled in each successive second and the ratios between them.
- Double every stack (50, 100, 150 and 200 mm) to raise the upper end by 0.200 m, predict the new k and the time to reach the end, then test the prediction.
What you should see
The distance-time points curve upwards while the distance against time-squared points lie on a straight line through the origin. For a solid ball rolling without slipping on a flat board, a = (5/7) g sin θ with g = 9.80 m/s² and sin θ = 0.100/2.40, so a = 0.292 m/s² and k = a/2 = 0.146 m/s²: the ball is at 0.146, 0.583, 1.313 and 2.333 m after 1, 2, 3 and 4 s, reaches the end of the board after 4.06 s, and covers 0.146, 0.438, 0.729 and 1.021 m in successive seconds, in the ratio 1 : 3 : 5 : 7. In a 90 degree V-channel the ball rolls on a smaller radius, so a = (5/9) g sin θ, k falls to 0.113 m/s² and the run takes 4.60 s; the quadratic shape is unchanged. Doubling the raise doubles k (to 0.292 m/s² on the flat board) and divides the run time by √2, to 2.87 s. A measured k below the prediction points to rolling resistance, a mis-measured raise or a board that is not straight. The learner knows it worked when the distance against time-squared points fall on a line through the origin and the successive-second distances follow 1 : 3 : 5 : 7 within the error of reading one video frame (1/30 s).
What changes
- What you change
- time since release (s); in the extension, the raise of the board (m)
- What you measure
- distance rolled from the start mark (m)
- What you keep the same
- the same ball
- the same board and surface
- release from rest by lifting a ruler
- board level across its width
- board supported every 0.60 m so it stays straight
Common misconceptions
Each of these ideas is wrong, and the activity is a chance to test it.
- A ball rolling down a ramp moves at a steady speed.
- Twice the time means twice the distance.
- A heavier ball of the same kind rolls down faster.
- Any upward-curving graph is a parabola.
Safety card
Hazards
- a steel ball leaving the board can land on feet or roll underfoot
- the board slipping off its supports
Controls
- catch box at the lower end
- board and spacers taped in place on the floor or a low bench
- the video frames the board and the metre rules only, with no learner in shot, and is deleted from the phone once the positions have been read
Note
No chemicals or 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 K–10 Syllabus (2022), Stage 5 Non-linear relationships B; page read 2026-09-22MA5-NLI-C-02
- Mathematics K–10 Syllabus (2022), Stage 5 Non-linear relationships A; page read 2026-09-22MA5-NLI-C-01
- Australian Curriculum v9AC9M9A04AC9M9A06
Sources
The pages the author read to write this activity.