Suppose that a sailboat $S$ moves in a straight line (geodesic) with constant speed $v$ in an infinite hyperbolic sea. You try to overtake that sailboat with a motorboat $M$ which has maximal speed $u$ and can follow any differentiable curve. Can you do it, given that $u > v$, but you cannot approach the sailboat to a distance less than $R$ ?
The hyperbolic sea has constant curvature $K = -1$ and by overtaking it is meant that there exist times $t_1 < t_2$ such that the points $M(t_1)$, $S(t_1)$, $S(t_2)$, $M(t_2)$ lie on a single straight line (geodesic) in this order. Constants $u, v, R$ are finite and positive.
I think that the answer is negative but I am not sure how to prove it.
The answer is not negative, but depends on whether $u > v \cosh R$. Can you see how the motorboat can overtake the sailboat in this case, and why the motorboat cannot overtake the sailboat otherwise?