Circular Tracks • Topic 2 of 3

First Meeting at Start

A subtle CAT distinction: meeting ANYWHERE on the track is not the same as meeting AT the starting point. Two bodies return together to the start only when each has completed a whole number of laps. The time for one lap is L/a and L/b, so they are both back at the start simultaneously after t = LCM(L/a, L/b). This time is INDEPENDENT of direction — same or opposite, the return-to-start time is the same, because it only asks when each individually completes full laps. A clean shortcut: with integer speeds, t = L/HCF(a, b) gives the first meeting at the start in many standard problems, since L/a and L/b share L and their LCM works out to L/HCF(a,b). Always answer the exact question asked: "meet for the first time" (use relative speed) versus "meet for the first time at the start" (use LCM of lap times).

✅ Solved examples

1. A 600 m track. Speeds 5 m/s and 7 m/s. When do they first meet at the STARTING point?
Lap times = 600/5 = 120 s and 600/7 s. t = LCM(120, 600/7). Easier: t = L/HCF(5,7) = 600/1 = 600 s.
2. Speeds 4 m/s and 6 m/s on a 240 m track. First meeting at start?
Lap times = 240/4 = 60 s, 240/6 = 40 s. LCM(60, 40) = 120 s. (Check: 120/60 = 2 laps, 120/40 = 3 laps.)
3. On a 300 m track, A does a lap in 50 s and B in 75 s. When are both first back at the start together?
t = LCM(50, 75) = 150 s. (A: 3 laps, B: 2 laps.)
4. A 420 m track, speeds 6 m/s and 9 m/s. Compare first meeting ANYWHERE (opposite) vs first meeting AT START.
Anywhere (opposite): 420/(6+9) = 28 s. At start: L/HCF(6,9) = 420/3 = 140 s. The start meeting is much later.

✏️ Practice — try these, take hints as needed

1. A 480 m track, speeds 8 m/s and 12 m/s. First meeting at the start?
Lap times = 480/8, 480/12.
60 s and 40 s.
LCM(60,40).
120 s
2. Lap times of 36 s and 45 s. When first together at the start?
Use LCM of lap times.
LCM(36, 45).
36 = 4×9, 45 = 5×9.
180 s
3. A 360 m track, speeds 5 m/s and 9 m/s (integers). First meeting at start via HCF trick?
t = L/HCF(a,b).
HCF(5,9) = 1.
360/1.
360 s
4. A 720 m track, speeds 8 m/s and 18 m/s. First meeting at start?
HCF(8,18) = 2.
t = L/HCF.
720/2.
360 s
5. Track 200 m, speeds 4 m/s and 5 m/s. How much LATER is the first start-meeting than the first opposite-direction meeting?
Opposite first meet = 200/9.
Start meet = 200/HCF(4,5) = 200.
Subtract.
200 − 200/9 = 1600/9 ≈ 177.8 s later

📝 Topic test — 8 questions

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