Mathematics 11–12 · Years 11–12

Average speed from 20 m splits: Bolt's 9.58 s and your own 100 m

Introduction to differentiation: Estimating change (Mathematics Advanced, Year 11); Ratios and rates: Rates (Mathematics Standard 1 and Standard 2, Year 12)

Practical, model not builtLow risk

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The idea

The gradient of a chord on a distance-time graph is an average speed, and shrinking the interval turns it into the instantaneous speed the derivative describes.

What you need

Official split times for the 2009 Berlin World Championships men's 100 m final (Biomechanics Report WC Berlin 2009, Sprint Men, page 5): Bolt reaction time 0.146 s, 20 m 2.88 s, 40 m 4.64 s, 60 m 6.31 s, 80 m 7.92 s, 100 m 9.58 s
A 100 m straight marked with cones every 20 m (six cones including the start)
Five timers with stopwatches, one at each cone, all starting on the same start signal
Graph paper or a spreadsheet

How to do it

Plot the Bolt data as distance (m) against time (s), adding the point (0.146 s, 0 m) because he is still in the blocks until his reaction time, and draw a smooth increasing curve through the points.
Compute the gradient of each 20 m segment (20 m divided by the segment time) and tabulate the five average speeds.
Compute the overall average speed 100 m / 9.58 s and compare it with the fastest segment.
Draw the chord from the origin to the 60 m point and the chord from 40 m to 60 m; decide which is the better estimate of the speed at 50 m and explain why.
Run your own 100 m with a timer at each cone (three runs, best kept) and repeat steps 1 to 3 for your data.
Describe where each graph is concave up (speeding up), close to straight (steady speed) and concave down (slowing).

What you should see

Bolt's segment average speeds are 6.94, 11.36, 11.98, 12.42 and 12.05 m/s for 0-20, 20-40, 40-60, 60-80 and 80-100 m; the overall average is 10.44 m/s, below every segment after 20 m because the first segment includes the 0.146 s reaction time and the acceleration phase. The segment speeds rise to 12.42 m/s between 60 and 80 m and fall slightly to 12.05 m/s over the last 20 m, so the graph is concave up to about 70 m and very slightly concave down at the end, with the steepest chord between 60 and 80 m. The learner's own run gives lower speeds, but hand timing at each cone can put a split out by a tenth of a second or more, about 3 per cent of a 3 s split and as large as the 3 to 6 per cent steps between Bolt's later segments; only the slow first segment is certain, and whether the learner's later segments rise, level off or fall is judged against the spread of the three runs. The learner knows the analysis worked when the segment times add to the total time and the first 20 m is clearly the slowest segment for both runners.

What changes

What you change: time interval chosen on the distance-time graph
What you measure: average speed over that interval (chord gradient)
What you keep the same: dataset (fixed splits)
cones at exactly 20 m
same runner for all three runs

Common misconceptions

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

A sprinter is fastest in the first metres; the data show the fastest segment is 60-80 m.
The average speed for the race equals the average of the five segment speeds; it does not, because the segments take different times.
A steeper line means more distance; it means more distance per second.

Safety card

Low riskLearners carry it out

Hazards

sprinting on an uneven surface
collision with cones or other runners

Controls

run on a marked, dry track
warm up before sprinting
one runner on the straight at a time

Note

No chemicals and no heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; outdoor work follows the school's grounds or excursion risk assessment (RiskAssess where the school uses it).

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 11 focus area Introduction to differentiation; 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-11-06
Mathematics Standard 11–12 Syllabus (2024), Year 12 Standard 1 focus area Ratios and rates; 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 2026-09-22MST-12-S1-05
Mathematics Standard 11–12 Syllabus (2024), Year 12 Standard 2 focus area Ratios and rates; 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 2026-09-22MST-12-S2-05
Australian Curriculum v9No Australian Curriculum v9 code is listed.

Sources

The pages the author read to write this activity.

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