Mathematics 7–10 · Year 10
Catch-up walkers: when and where two measured walks draw level
Number and algebra
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The idea
Two steady walks are two linear equations in time and distance, and the time and place at which the walkers draw level is their simultaneous solution, found algebraically or as the intersection of their graphs and then tested on the track.
Safety card
Hazards
- collision when walking towards each other
- tripping over cones
- sun exposure
Controls
- each walker stays in their own lane, 2 m apart
- cones at the lane edges
- hats and water
Note
No chemicals or heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; outdoor work follows the school's excursion and playground supervision rules.
What you need
- A straight, flat 50 m path or track marked every 5 m with cones, with two lanes 2 m apart
- 30 m tape measure or trundle wheel to place the cones
- Three stopwatches or phones
- Recording table and graph paper or a spreadsheet
How to do it
- Measure each walker's steady speed: walker A walks briskly and walker B slowly, each passing the 5 m cone already at pace, while a timer times the 20 m to the 25 m cone. Three trials each; speed = 20 ÷ median time.
- Write each walker's distance from the 0 m cone after t seconds when A starts at 0 m and B starts at 10 m, both setting off forwards at the same instant: d = v_A t and d = 10 + v_B t.
- Solve the pair algebraically by substitution and graphically by plotting both lines and reading the intersection, to predict the catch-up time and position.
- Run it: a starter calls go, each walker keeps the measured pace in their own lane, and an observer at the predicted position records where and when A draws level with B.
- Repeat with the walkers starting 40 m apart in their own lanes and walking towards each other: d = v_A t and d = 40 − v_B t.
- Compare predictions with observations, and compute how far each predicted point moves if A walks 0.05 m/s slower than measured.
What you should see
As a worked check, with v_A = 1.50 m/s, v_B = 1.00 m/s and a 10 m head start, 1.50t = 10 + 1.00t gives t = 20 s and d = 30 m. If A is only 0.05 m/s slower (1.45 m/s) the catch-up moves to 22.2 s and 32.2 m, because the two lines are nearly parallel (their gradients differ by only 0.5 m/s), so a small speed error moves the intersection a long way. Walking towards each other from 40 m apart the walkers meet after 40 ÷ (1.50 + 1.00) = 16 s, 24 m from A's start, and the same 0.05 m/s error moves that point by only 0.33 m. The learner knows it worked when the observed catch-up falls inside the range predicted from the speed uncertainty and the explanation of why the catch-up is more sensitive than the meeting refers to the angle between the two lines.
What changes
- What you change
- head start and walking speeds
- What you measure
- time and position at which the walkers draw level
- What you keep the same
- the same straight path
- each walker keeps the pace measured in step 1
- a simultaneous start on one call
Common misconceptions
Each of these ideas is wrong, and the activity is a chance to test it.
- The faster walker catches up at the halfway mark.
- Any two lines on a graph meet at a point that makes sense in the situation.
- A small error in a speed makes only a small error in where the walkers draw level.
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 Equations C (Path); page read 2026-09-22MA5-EQU-P-02
- Australian Curriculum v9AC9M10A02
Sources
The pages the author read to write this activity.