Two people, $A$ and $B$, are trying to paint the interval $[0,1]$. $A$ starts at the point $0$ and moves at speed $1$, while $B$ starts at the point $1$ and moves at speed $b>1$. Wherever $A$ goes is colored red, and wherever $B$ goes is colored blue. (Points are recolored every time someone visits them.)
(To avoid questions of differentiability: by moving at a given maximum speed $s$, I mean that the position over time $p(t)$ has the property that $|p(t_1)-p(t_2)|\leq s\cdot |t_1-t_2|$ for all times $t_1,t_2$.)
Each player's goal is to maximize the fraction of the interval with their color, with priority assigned to later timesteps: strategies that improve one's expected territory at $t_1$ at the cost of territory at time $t_0<t_1$ are favored. (That is, $A$ wishes to survive as long as possible, and $B$ wishes to eliminate $A$ as soon as possible.) Both players know the location of the other player at all times.
My question is: what happens at different values of $b$? How long does the game last, and how much territory is held over time?
Clearly, if $B$ ever clears out all red territory and overlaps $A$, they win forever; any speed advantage lets them follow $A$ around, making minor deviations from $A$'s path to paint over any red areas. This can always be accomplished in time $\frac{1}{(b-1)}$ by traveling to $A$'s starting point and following their path until catching up. However, this will often be suboptimal, and it's not clear that $A$ can stretch things out this long against an impatient $B$. How long do things go on under optimal play?
If there are problems with executing strategies in continuous-time environments, I'd like to understand what breaks here, and also look at the limiting case of discrete time intervals (where the same overall considerations hold, I believe).