Problem:
Assume two runners $A$ and $B$ are $2h_0$ meters apart. They start running towards each other at a speed of $a$ meters per second. Assume there is a third runner that runs in between the two runners at a speed of $b$ meters per second, with $b > a$. Whenever the third runner meets $A$ or $B$, he turns around and head towards to other person, never changing his speed. What is the total distance traveled by the third runner?
My solution:
Since the third runner is always in between $A$ and $B$, his running session ends when it $A$ meets $B$, which is in the middle, i.e. at position $h_0$.
Since the third runner's position never changes, it is logically the same to ask question of where does the third runner needs to start for all three of the runners to meet in the middle?
If we treat the middle as position $0$, with $A$ starting at $-h_0$ and $B$ starting at $h_0$, then it quickly becomes apparent that the third runner needs to start at a position $\frac{bh_0}{a}$ from the middle. Indeed that's the answer.
Question:
How do I represent this in summation form without changing the question of the problem? What about if $A$ and $B$ are running at varying non-linear speed? In other words, how do I represent a piece-wise linear function that's going back and forth and converging between two intersecting monotonic functions?