In the figure, you see athletes assembled at the start of a 4 × 100 m relay race. The tracks are laid out, and the athletes are all set to go racing down the tracks. Do you notice that the athletes are not at the same starting line? Those in the outer lanes seem to be starting ahead of those in the inner lanes while the finish line is the same for all of them. What could be the reason for this? The distance between the starting points of adjacent lanes is called the ‘stagger’. Notice that the stagger continues all the way to the outermost lane. Do you think the stagger gives anyone (those in the outer lanes or in the inner lanes) an unfair advantage? Why or why not? On what basis can the organisers work out the length of the stagger between lanes?

Step-by-step solution

Idea: On the straight parts all lanes are equally long. On the curved parts each lane is an arc of a circle, and a circle with a bigger radius has a bigger circumference (2πr). The stagger is there to cancel exactly that extra length.

1. Do you notice? Yes. Each runner in an outer lane starts a little ahead of the runner in the lane just inside.½ mark
2. Reason: the race begins on a bend. Each lane on a bend is part of a circle, and an outer lane is part of a bigger circle. A bigger circle has a longer edge, so if everyone started on the same line, the outer runners would have to run further.1 mark
3. Unfair advantage? No. The stagger is chosen to be exactly the extra length of the outer lane. Starting that much ahead makes every lane the same total length, so everyone runs the same distance to the same finish line.½ mark
4. Basis for the length: the difference between the lengths of the curved parts. On a semicircular bend, a lane of radius r has length πr and the next lane out, of radius r + w (w = lane width), has length π(r + w). The extra length is π(r + w) − πr = πw per semicircular bend. Add this up for all the bends that the race covers in lanes.1 mark

Check: Use the track figures given later: lane width 1.22 m. One semicircular bend adds π × 1.22 ≈ 3.83 m for the next lane, and two bends (one full lap) add about 7.67 m.

Answer to write in the exam

Common mistakes that cost marks

• Thinking outer-lane runners are being helped. The stagger only cancels the extra length of their curve.
• Saying the extra length depends on the radius of the bend. π(r + w) − πr = πw: it depends only on the lane width.
• Using the full circle 2πw for a single bend. One bend is a semicircle, so it adds πw.

How this can come in the exam

Try one yourself

On a semicircular bend the inner edges of two neighbouring lanes have radii 35 m and 36.25 m. How much farther does a runner in the outer lane go on this bend? (Use π = 227.)

More questions like this

• In my school, the playground is too small to have a 400 m track, so the school constructed a 200 m track instead. Does this mean that we need a smaller stagger for the race tracks in my school (i.e., smaller than the stagger used in the Olympics), for the same 4 × 100 m relay race?
• Here we see a circle with radius r units. What is its perimeter? How do we find out?
• What is the connection between this question and the one about the 400 m athletics track?
• What happens to the perimeter of a square if we double its side?
• What about a circle? What is its perimeter (usually called the circumference) in terms of its diameter? Is the ratio of circumference (C) to diameter (D) the same for circles of all sizes (see the figure)? What do you think?

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