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Why do runners start in staggered positions for the 200m and 400m?

The stagger is not an advantage given to the outside lanes, it is the only way to make eight different arcs add up to the same distance.

Look at the start of a 400m and every lane begins at a different point on the track, with the outside lanes set well ahead of the inside ones. It looks like a head start. It is actually the opposite: a correction for a problem the curve creates.

Same distance, different radius

A standard track has lanes about 1.22 metres wide, and each lane out from the inside adds that width to the radius of both curves. A bigger radius means a longer way round, even though every runner still covers 400m measured down their own lane. Without a stagger, the outside runner would simply run further than the inside runner for the same race distance.

Why it still looks unfair from the stands

An outside-lane runner at the start of a 400m looks like they are 10 or more metres ahead of lane one. They are not ahead. They are standing on a line that was moved forward by exactly the amount their lane's extra curve distance would otherwise cost them. The staggered start is what makes the race fair, not what makes it unequal.

A single 2D camera filming a start can tell you plenty about a sprinter's drive phase. It cannot tell you whether the stagger itself was measured correctly, because that is a question about the track's geometry and the meet's lane markings, set long before anyone steps into the blocks.

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Try it: Sprint timer app: 40-yard dash and flying sprints with one phone