These notes concern a classical optimal control problem in aerospace engineering, how to land safely a spacecraft on the moon. Starting from an analysis of the original problem proposed by Miele (The calculus of variations in applied aerodynamics and flight mechanics, 1962), consisting in minimizing the consumption of fuel used for landing, we study a variant of the problem consisting in minimizing the landing time with a different thrust control. We prove that the optimal control (if it exists) is piecewise constant and provides the exact landing strategy, in dependence of the initial data (height, velocity, and fuel of the lander).