I use a similar idea for my "Hello World" program for game design.

There is a kids school yard game called Tag (or Tips in some places). The idea is simple: One player is "IT" and must touch other players to transfer the title of "IT" (being IT is bad).

Now, my computer game idea (the original I did on the Commodore Plus 4 - it was similar to the Commodore 64) is that you have 3 stats:

Speed: How fast you can move
Stamina: How long you can run for before needed to slow down
Recovery: How quickly you recover your stamina

The player could reduce the levels in one or more stats so as to raise one or more of the other stats.

The idea of my game was each player starts off with a certain number of Points to score the most points before someone was reduced to 0 points.

While you are IT you loose points, and when you are not IT you gain points. However, the closer you are to IT, the faster you gain points.

This made the players trade off between the safety of distance as compared to gaining points quickly. I didn't use a linear relationship, but an exponential relationship between the points and distance. That is, the amount of points you got rose exponentially the closer you got to IT.

This meant that it was far better to be closer than further away as once you got beyond a certain distance, you would not get enough points to score high. You might not ever get to be IT, but you would not have a chance to get enough points to have a chance of winning.

As far as playing when, you would mostly get close while you have a high stamina so as to get points, but then fall back to a safe distance to give yourself time to regain stamina.

In my game, turning rate was not factored into it and I allowed the players to change direction at any rate.