Races and circular tracks

Learning outcome

Interpret race margins and head starts, distinguishing a circular-track meeting from a simultaneous return to the start.

Concepts and assumptions

Assume point-sized runners at constant positive speeds, without reaction delays, acceleration or stops. Straight races share a finish line; circular tracks have circumference C.

Without a head start, finishing time = race length ÷ speed. The winning margin in metres is the slower runner’s remaining distance when the winner finishes.

For a distance head start, both depart simultaneously. If the finish is R metres from A’s start and B starts h metres ahead, their distances are R and R - h. A time head start instead means an earlier departure from the same line.

On a circle, exclude time zero when runners start together at the same point. In the same direction, a faster runner must gain one lap: first later meeting time = C ÷ speed difference. Equal-speed runners remain together.

In opposite directions from the same point, their combined distances first total one lap: first later meeting time = C ÷ speed sum.

Different starting positions require the actual gap. A faster runner starting g metres behind along their common direction catches up after g ÷ speed difference, for 0 < g < C.

Returning together to the start requires complete laps by both. With whole-second lap times, take their least common multiple. Meeting elsewhere does not count.

Worked examples

Example 1 — Winning margin. A and B start a 360 m race together without a head start, at 9 m/s and 8 m/s. A finishes in 360/9 = 40 s. B covers 8 × 40 = 320 m. Margin = 360 - 320 = 40 m.

Example 2 — Distance head start. A’s finish is 600 m away; B starts 120 m ahead, simultaneously. A runs at 10 m/s and takes 600/10 = 60 s. To reach the common finish together, B must cover 600 - 120 = 480 m in 60 s: required speed = 480/60 = 8 m/s.

Example 3 — Meeting versus returning. Runners start simultaneously at the same point on a 420 m circle at 7 m/s and 5 m/s. Same-direction first meeting = 420/(7 - 5) = 210 s. Distances 7 × 210 = 1470 m and 5 × 210 = 1050 m both leave 210 m after complete laps: not the start. In a separate opposite-direction trial, first meeting = 420/(7 + 5) = 35 s. Lap times = 420/7 = 60 s and 420/5 = 84 s; their least common multiple, 420 s, gives the first simultaneous return in either trial.

Common mistakes

Distinguish distance and time head starts. Exclude the initial coincidence from later meetings. A meeting is not necessarily a return.

Practice questions

1. A and B start a 500 m race together without a head start, at 10 m/s and 8.5 m/s. Find A’s winning margin.
2. A and B run 300 m from the same line at 6 m/s and 5 m/s. How many seconds earlier must B start to finish together?
3. On a 400 m circle, runners start simultaneously in the same direction: the 7 m/s runner is 60 m behind the 5 m/s runner. Find catching time.
4. Runners start simultaneously at the same point on a 300 m circle, at 10 m/s and 6 m/s in the same direction. Find first later meeting and first simultaneous return times.

Worked answers

1. A takes 500/10 = 50 s. B covers 8.5 × 50 = 425 m. Margin = 500 - 425 = 75 m.
2. A needs 300/6 = 50 s; B needs 300/5 = 60 s. B must leave 60 - 50 = 10 s earlier, not start 10 m ahead.
3. Closing speed = 7 - 5 = 2 m/s. Use the actual gap: time = 60/2 = 30 s, not the full circumference.
4. First meeting = 300/(10 - 6) = 75 s. Lap times = 300/10 = 30 s and 300/6 = 50 s; their least common multiple gives first common return = 150 s.