Circular Motion: Points on track where sprinters meet

Now before all those visitors who come here looking for fundas are scared off by the problems posted, here is a post just for them...

We will tackle the funda of identifying the points on the circular track where two or more sprinters meet, when they run with different speeds. Please make a note in your mindmap that we are not talking of how many times they meet or after how much time they met for the nth time (though this is closely related). To repeat, the following funda is about: the points on the track where the sprinters could possibly meet.

For the impatient ones, if two sprinters run on a circular track with speeds a and b (or if time taken to complete a round is a and b for the two sprinters)...... focus on the data given to capture what is given, run it through your mind once more...... consider the ratio of a and b in the lowest reduced form say m:n. Then, if they are running in the same direction they would meet only at m ~ n distinct points on the track (~ refers to absolute difference). And if in opposite direction the meeting points would be m + n distinct points.

These points will all be equidistant and the starting point will be one of them. Thus you can locate all the points on the track.

E.g. 1. Two sprinters start simultaneously from the same point on a circular track and run with speeds 10 and 25 kmph. Or else the time taken by two sprinters to run one complete round across a circular track is 35 mins and 14 minutes. In both the cases the ratio of speeds is 2:5. Thus they will meet at 3 distinct points on the track (1/3rd of track from staring point, 2/3rd of track from starting point and starting point). If they run in opposite directions, they would meet at 7 distinct points (1/7th, 2/7th, 3/7th, ..., 6/7th and starting point)

For the ones more inclined to know why... in the same time as the faster one would complete 5 rounds, the slower one would complete 2 rounds. Thus both would be together at the starting point. Consider running in same direction: The faster one would be 3 rounds ahead of the slower one. This difference of 3 rounds would have "creeped" up between them linearly i.e. the gap did not suddenly arise. Lets make the difference betwen the rounds run as 2...rewind the situation slowly, we see that when the faster one ran 2/3rd of 5 i.e. 3 & 1/3rd of round, the slower one ran 2/3rd of 2 i.e. 1 & 1/3rd of round. Thus faster one is two full rounds ahead of the slower one i.e. both are together. And they are at 1/3rd the distance from starting point.
Rewinding further we see that when faster one runs 1/3rd of 5 i.e. 1 and 2/3rd of round, slower one runs 1/3rd of 2 i.e. 2/3rd of round. Faster one is 1 round ahead of slower one i.e. again at same point and this point is 1/3rd from starting point.
Rewinding further would land us at the start of the race itself.

Additional Points to learn: If two sprinters start from same point and run in same direction...
If one of them has run 7.5 rounds and other has run 3.666 rounds, they have met a total of 3 times so far. Faster one is 3.833 rounds ahead of slower one, so would have met the slower one when he managed a gap of 1 round, then when he managed a gap of 2 rounds and finally third time when he managed a gap of 3 rounds. Right now, he is ahead of the slower one by a further of 0.833 rounds and when this increases to full round, the fourth meeting will happen.

Can you repeat the above for opposite direction? In this case they would meet whenever together they have run one full round.

Things to watch for:
1. Both sprinters should start from same point and simultaneously
2. The method just gives us the points at which they meet, and NOT the sequence i.e. as seen in e.g. 1, first they meet at 2/3rd from starting point, then at 1/3rd and finally at the starting point. Had the speeds been 5 kmph and 20 kmph, the sequence of meeting would be 1/3rd, then 2/3rd and finally starting point.

Time Aimcats are pretty found of such problems. Seen more than two of such types so far. Consider one from it (in a slightly modified manner)... Three sprinters A, B and C run on a circular track with ratio of speeds as 1 : 3 : 5. If A and C run clockwise and B runs anticlockwise, find the number of distinct points on the track where any two sprinters would meet.

Can you use the above funda to solve the following problem orally: A, B and C take 15 mins, 25 mins and 35 minutes to complete a full round around a circular track. If they start simultaneously from same point, with A and B running clockwise and C running anticlockwise, find time after which all three will meet for the first time?

Please send in comments if anyone needs the solutions to the examples at the end.

Chandra

I am just a novice in blogging. I will learn how to represent maths symbols and fractions, till then do with the above format.