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?
Step-by-step solution
Idea: For a lane of width w, a half-turn (semicircular bend) adds π(r + w) − πr = πw, whatever the radius r. So the stagger depends on the number of bends in the race, not on the size of the track.
- On a semicircular bend of inner radius r, the next lane out is longer by π(r + w) − πr = πw. The radius r cancels, so a small track and a big track give the same extra length per bend.1 mark
- 400 m track: one lap = 400 m has 2 semicircular bends. Stagger between neighbouring lanes = 2 × πw = 2πw. With w = 1.22 m: 2 × 3.1416 × 1.22 ≈ 7.67 m.½ mark
- 200 m track: 400 m = 2 laps, so the race has 4 semicircular bends. Stagger = 4 × πw = 4πw ≈ 4 × 3.1416 × 1.22 ≈ 15.33 m.1 mark
- So the school does not need a smaller stagger. It needs one about twice as large, because its relay runs round twice as many bends.½ mark
Check: Each full lap on any track has one full turn, adding 2πw. Two laps add 4πw, twice one lap’s 2πw ✓.
Answer to write in the exam
Extra length per semicircular bend = π(r + w) − πr = πw (independent of r).
400 m track: 400 m = 1 lap = 2 bends ⇒ stagger = 2πw ≈ 2 × 3.1416 × 1.22 ≈ 7.67 m.
200 m track: 400 m = 2 laps = 4 bends ⇒ stagger = 4πw ≈ 15.33 m.
∴ No; the stagger is not smaller, it is about twice as large.
Common mistakes that cost marks
- Thinking a smaller track means smaller staggers. The radius cancels; only the lane width and the number of bends matter.
- Forgetting that 400 m on a 200 m track is two laps, so it has 4 bends, not 2.
- Using the lane radius instead of the lane width in πw.
How this can come in the exam
Assertion (A): On a semicircular bend, the extra length of a lane of width w over the lane just inside it is the same on a small track and on a large track.
Reason (R): π(r + w) − πr = πw, which does not contain r.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
Show answer
(A) Both A and R are true, and R is the correct explanation of A.
The difference of the two semicircle lengths is πw; the radius cancels. R explains A.
A race on a small track goes round 6 semicircular bends in lanes 1 m wide. Find the stagger between neighbouring lanes. (Use π = 227.)
Show answer
Extra per bend = π × 1 = 227 m (1 mark). Stagger = 6 × 227 = 1327 ≈ 18.86 m (1 mark).Try one yourself
A race covers 3 semicircular bends in lanes 1.5 m wide. What stagger is needed between neighbouring lanes? (Use π = 3.14.)
Show answer
3 × π × 1.5 = 3 × 3.14 × 1.5 = 14.13 m.
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