  We analyze velocity-jump process models of persistent search for a single
target on a bounded domain. The searcher proceeds along ballistic trajectories
and is absorbed upon collision with the target boundary. When reaching the
domain boundary, the searcher chooses a random direction for its new
trajectory. For circular domains and targets, we can approximate the mean first
passage time (MFPT) using a Markov chain approximation of the search process.
Our analysis and numerical simulations reveal that the time to find the target
decreases for targets closer to the domain boundary. When there is a small
probability of direction-switching within the domain, we find the time to find
the target decreases slightly with the turning probability. We also extend our
exit time analysis to the case of partitioned domains, where there is a single
target within one of multiple disjoint subdomains. Given an average time of
transition between domains $\langle T \rangle$, we find that the optimal rate
of transition that minimizes the time to find the target obeys $\beta_{{\rm
min}} \propto 1/ \sqrt{\langle T \rangle}$.
