We present a sample-based motion planning algorithm specialised to a class of underactuated systems using path parameterisation. The structure this class presents under a path parameterisation enables the trivial computation of dynamic feasibility along a path. Using this, a specialised state-based steering mechanism within an RRT motion planning algorithm is developed, enabling the generation of both geometric paths and their time parameterisations without introducing excessive computational overhead. We find with two systems that our algorithm computes feasible trajectories with higher rates of success and lower mean computation times compared to existing approaches.