Random search theory has been previously explored for both continuous and intermittent scanning modes with full target detection capacity. Here we present a new class of random search problems in which a single searcher performs flights of random velocities, the detection probability when it passes over a target location being conditioned to the searcher speed. As a result, target detection involves an N-passage process for which the mean search time is here analytically obtained through a renewal approximation. We apply the idea of speed-selective detection to random animal foraging since a fast movement is known to significantly degrade perception abilities in many animals. We show that speed-selective detection naturally introduces an optimal level of behavioral intermittence in order to solve the compromise between fast relocations and target detection capability.