Q1.
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?
Answer
Not necessarily. The stagger depends on the width of a lane, not on the length of the track — provided each runner still goes once round a bend.
Extra distance run in the next lane out, over one full lap
= 2π(r + w) − 2πr = 2πw
The radius r has cancelled.
= 2π(r + w) − 2πr = 2πw
The radius r has cancelled.
So if your school's lanes are also 1.22 m wide, a runner who covers one complete bend needs the same 7.67 m stagger as at the Olympics, even though the track is half the size.
Why it happens: shrinking the track shrinks the radius of every lane, but it shrinks them all by the same amount — the difference between adjacent radii is still one lane width. The extra path length comes from that difference alone.
Check it yourself: what really changes is how much of a bend each leg contains. On a 400 m track a 100 m leg contains part of one bend; on a 200 m track the bends are tighter and a 100 m leg may contain a whole bend or more. The stagger must equal the extra curved length actually run in that leg, so on the smaller track the stagger for a particular changeover can even be larger. The rule to remember is: stagger = 2πw × (fraction of a full circle actually run on the bend).