Continuing with the original problem. Three friends A B & C regularly jog in circular jogging track every morning. The track is 1000 m in circumference.

A takes 8 mins to complete one lap, B takes 10 mins & 40 secs & C takes 12 mins. They all start from a point on the track but C decides to run in the opposite direction from the same point.

As they set out B suddenly has cramps (not at the starting point) and starts walking in the same direction as before. His walking speed takes him 32 mins to complete a lap. He decides to continue walking till they all meet. At what point should he have cramps such that they all meet at any point in the fastest time?

A and C meet each other every 288 seconds, as their relative angular speed is 1/480+1/720 = 1/288.

When they meet the first time, at t=288 seconds, A has traveled 3/5 of the way around the track, while C has traveled 2/5 in the other direction.  As B runs slower than A, no matter what time he could slow down, he couldn't make this first meeting.

After another 288 seconds, t=576, A has traveled 6/5 of the way around; that is, he's 200 m into his second lap. In any combination of B's walking and running paces (1/640 rev/sec and 1/1920 rev/sec), he would be between 300 and 900 m along the track at this time, so he can't make this meeting.

The next meeting between A and C, at t=864, will be where A has traveled 9/5 of the way around the track: he's 800 m into his second lap. At an all-walking pace, B would be at the 450 m mark; running, he'd be 350 m into his second lap, so some combination of running and walking will leave him at the 800 m mark.  Let the time he has cramps be called x.

x/640 + (864 - x) / 1920 = 800/1000
x/640 - x/1920 = 4/5 - 864/1920
x/960 = 7/20
x = 336

So if B gets the cramps at t=336 seconds (5 min, 36 sec), when he's 525 m along the track, he will make this meeting at t=864 (14 min, 24 sec), 800 m along the track.

Posted by Charlie on 2006-04-15 00:33:07