This paper investigates feedback linearization of one degree underactuated systems the mass matrix of which does not depend on the unactuated variable. We study the dynamics of such systems in a new time scale τ. In this new time scale, we exhibit a set of coordinates for which the dynamics transverse to a particular one-dimensional manifold is dynamic feedback linearizable. To this end, we design a relative degree three output with respect to one input and use dynamic extension. We then introduce a new motion planning algorithm for this class of systems.