Variation of the classic problem about the student and bus

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Hello, I have a question.

There is a classic problem in which a student who is at constant speed approach a bus which has begun to accelerate and is at a certain distance from the student; You need to figure out when it reaches the bus. The problem is easy to solve, if Xo is the distance between both and Vs is the speed of the student, we can find the time in which they will find matching positions "1/2 * a * t ^ 2 + Xo = Vs * t " over time you can get the place where they meet.

Now my question is this. Suppose that in fact the student and the bus are running at a roundabout, turning constantly (we can also imagine that it is a straight but closed path, ie, as in pac-man game where one goes to the right and reappears from left), in that situation, regardless of the acceleration of the bus and the distance, always will find themselves at certain points that will vary; how would the equation that would give me the points where they find? I can not find it, I'm rusty with my physics.

Thank you very much
 

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Let ds be the differential distance around the circumference of the roundabout. Then ##ds = Rd\theta##, where R is the radius of the roundabout and ##\theta## is the cumulative angle traveled by each. Just solve in terms of ##\theta##.
 
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we can find the time in which they will find matching positions "1/2 * a * t ^ 2 + Xo = Vs * t " over time you can get the place where they meet.
If they are on a loop then they will not just meet once, but an infinite number of times, once for each "lap" around the track. Can you think of a simple additional term you can add which will express that condition for lap number n where the distance for a single lap is L.
 

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