  We study a stochastic process where an active particle, modeled by a
one-dimensional run-and-tumble particle, searches for a target with a finite
absorption strength in thermal environments. Solving the Fokker-Planck equation
for a uniform initial distribution, we analytically calculate the mean
searching time (MST), the time for the active particle to be finally absorbed,
and show that there exists an optimal self-propulsion velocity of the active
particle at which MST is minimized. As the diffusion constant increases, the
optimal velocity changes from a finite value to zero, which implies that a
purely diffusive Brownian motion outperforms an active motion in terms of
searching time. Depending on the absorption strength of the target, the
transition of the optimal velocity becomes either continuous or discontinuous,
which can be understood based on the Landau approach. In addition, we obtain
the phase diagram indicating the passive-efficient and the active-efficient
regions. Finally, the initial condition dependence of MST is presented in
limiting cases.
