This paper formulates a search model that gives the optimal search plan for the problem of finding a lost target. The lost target moves according to a Brownian motion on the real line. Two searchers start the searching process for the target, the first searcher starts looking for the target from some point and the second searcher starts from some other point on the line. We find the conditions that make the expected value of the first meeting time for the target finite. Furthermore, we show the existence of the optimal search plan that minimizes the expected value of the first meeting time.