This paper examines the use of output feedback to stabilize a planar vertical take-off and landing (PVTOL) aircraft in a finite time. The control of such PVTOL aircraft is already challenging due to the nonlinear cross-coupling between the aircraft axes and the issue of non-minimum phase dynamics. The results from this paper explore the control of the PVTOL aircraft model when there is the added design challenge of limited state information about the dynamics of the aircraft. A state feedback controller is designed by employing the use of a finite-time convergent control law. We then derive a finite-time observer such that the unknown states can be recovered in finite time and thus reducing the output feedback stabilization problem to the state feedback one. The finite-time observer proposed in this paper is unique due to the fact that the observer has two convergence modes to guarantee faster convergence rate for both small or large state errors.