This article extends the concept of input-output finite time stability (IO-FTS) to a class of hybrid systems, namely impulsive dynamical linear systems, which are systems that exhibit jumps in the state trajectory. Sufficient conditions are given for the analysis of IO-FTS and for the design of both static output and state feedback controllers, guaranteeing IO-FTS of the closed loop system. In the general case the presented results require to solve feasibility problems involving either linear matrix inequalities or bilinear matrix inequalities, which can be solved numerically in an efficient way by using off-the-shelf optimisation tools, as illustrated by the proposed examples.