Circular Tracks • Topic 1 of 3

Meeting Point

When two bodies move on a circular track of length L, the time to their FIRST meeting depends only on relative speed. Opposite directions: they close the gap at speed (a + b) and meet after covering L between them, so t = L/(a + b). Same direction: the faster gains on the slower at (a − b), and must gain a full lap L before catching up, so t = L/|a − b|. The CAT-smart move is to never plug raw numbers blindly — read the direction first, then pick sum or difference. A useful extra: the position of the first meeting can be found by computing how far the faster (or either) body has run, then taking that distance modulo L from the start. Because the two never stop, every subsequent meeting happens after the SAME time gap, so meetings are equally spaced in time.

✅ Solved examples

1. A track is 600 m long. Ram and Shyam start together from the same point in opposite directions at 5 m/s and 7 m/s. When do they first meet?
Opposite ⇒ relative speed = 5 + 7 = 12 m/s. First meeting at t = 600/12 = 50 s.
2. On the same 600 m track, the two now run in the SAME direction at 5 m/s and 7 m/s. When do they first meet?
Same direction ⇒ relative speed = 7 − 5 = 2 m/s. The faster must gain a full lap: t = 600/2 = 300 s.
3. Two runners on a 400 m circle move in opposite directions and meet every 25 s. One runs at 9 m/s. Find the other’s speed.
a + b = L/t = 400/25 = 16 m/s. So b = 16 − 9 = 7 m/s.
4. A and B start together on a 720 m track in opposite directions at 8 m/s and 10 m/s. How far from the start (along A’s path) is their first meeting?
Time = 720/(8+10) = 40 s. A covers 8 × 40 = 320 m. 320 < 720, so the meeting is 320 m from the start along A’s direction.

✏️ Practice — try these, take hints as needed

1. A 500 m track. Two cyclists go opposite ways at 6 m/s and 4 m/s. First meeting time?
Opposite ⇒ add speeds.
Relative = 10 m/s.
t = 500/10.
50 s
2. Same 500 m track, same direction, speeds 6 m/s and 4 m/s. First meeting time?
Same ⇒ subtract speeds.
Relative = 2 m/s.
t = 500/2.
250 s
3. On a 360 m circle two runners going opposite ways meet every 15 s. One runs 14 m/s. Find the other.
a + b = L/t.
360/15 = 24.
24 − 14.
10 m/s
4. A and B run a 800 m loop in the same direction at 12 m/s and 8 m/s. First meeting time?
Gain a full lap.
Relative = 4 m/s.
800/4.
200 s
5. Opposite directions on a 240 m track at 5 m/s and 7 m/s. How far from start (along the 5 m/s runner) is the first meeting?
Time = 240/12.
= 20 s.
Distance = 5 × 20.
100 m from start

📝 Topic test — 8 questions

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First Meeting at Start →