Time, Speed and Distance

Concept: On a circular track, meeting depends on relative speed. In the same direction, the faster runner must gain one full lap. In opposite directions, both runners together cover one full lap.

Same direction: T = C / |SA - SB|

Opposite direction: T = C / (SA + SB)

Variables: C = track circumference, T = first meeting time, SA and SB = runner speeds.

Trap: Use the difference of speeds for the same direction and the sum for opposite directions. This gives the first meeting anywhere, not necessarily at the starting point.

Application: On a 600 m track, A runs at 8 m/s and B at 6 m/s. Same direction: T = 600 / (8 - 6) = 300 seconds. Opposite direction: T = 600 / (8 + 6) = 42.86 seconds.

Concept: Meeting anywhere and meeting at the starting point are different. To meet again at the starting point, every runner must complete a whole number of laps at the same time.

Lap time of A: TA = C / SA

Lap time of B: TB = C / SB

Two runners: Tstart = LCM(TA, TB)

Three runners: Tstart = LCM(TA, TB, TC)

Trap: Use this only when the question specifically says at the starting point. For the first meeting anywhere, use relative speed.

Application: A, B, and C complete one lap in 12, 18, and 24 minutes. Tstart = LCM(12, 18, 24) = 72 minutes.

Concept: When two runners continue running on a circular track, they repeatedly meet at fixed points. The number of distinct meeting points depends on their speed ratio.

If the speeds are in the coprime ratio m:n:

Same direction distinct meeting points = |m - n|

Opposite direction distinct meeting points = m + n

Trap: Reduce the speed ratio to its lowest terms before applying the formula. The meeting points divide the track into equal parts.

Application: If speeds are in the ratio 5:3, they meet at 2 distinct points in the same direction and 8 distinct points in opposite directions.

Concept: For three or more runners moving in the same direction, all meet together again when the fastest runner has gained a whole number of laps over every other runner at the same time.

Let SF be the speed of the fastest runner.

Gain time over runner 1 = C / (SF - S1)

Gain time over runner 2 = C / (SF - S2)

First common meeting anywhere = LCM of all one-lap gain times

Trap: This method is for same-direction running. If the question asks for the starting point, use the LCM of individual lap times instead.

Application: On a 120 m track, speeds are 6 m/s, 5 m/s, and 4 m/s. Gain times are 120 seconds and 60 seconds, so all meet together again after LCM(120, 60) = 120 seconds.

Concept: Before the first meeting on a circular track, the distances covered are in the same ratio as the speeds because both runners have travelled for the same time.

If SA:SB = m:n, then DA:DB = m:n.

Opposite direction: DA + DB = C

Same direction, with A faster: DA - DB = C

Trap: In opposite directions, split one lap in the ratio of speeds. In the same direction, the faster runner's extra distance over the slower runner is one full lap.

Application: Two runners move in opposite directions on a 900 m track with speed ratio 2:1. A covers (2/3) x 900 = 600 m and B covers (1/3) x 900 = 300 m before meeting.