The problem of finite-time output stabilization of the double integrator is addressed applying the homogeneity approach. A homogeneous controller and a homogeneous observer are designed (for different degrees of homogeneity) ensuring the finite-time stabilization. Their combination under mild conditions is shown to stay homogeneous and finite-time stable as well. Robustness and effects of discretization on the obtained closed loop system are analyzed. The efficiency of the obtained solution is demonstrated in computer simulations.