A vertical race up and back down with and without drag

Two identical objects are simultaneously projected vertically upward with the same initial speed in a uniform gravitational field and then return to their starting point. One object is subject to a resistive force proportional to the nth power of its speed where $n ≥ 0$⁠, such as linear (Stokes) drag for n = 1 or quadratic (Newtonian) drag for n = 2. The other object moves through vacuum with no resistance. Which object returns to its starting point first? It is shown analytically that the object subject to drag always wins the race if $n ≥ 1$⁠, but that either object can win otherwise, depending on the ratio of the launch speed to the terminal speed of the object subject to drag if $0 < n < 1$ or depending on the ratio of the frictional acceleration to the gravitational acceleration if n = 0. These analytical results are confirmed by Euler–Cromer numerical integration of the equations of motion.