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 $n\ge 0$, 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 $n\ge 1$, 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.

中文翻译：

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有或没有阻力的上下垂直比赛

在相同的重力场中，两个相同的对象同时以相同的初始速度垂直向上投射，然后返回其起点。一个物体承受的阻力与其速度的n次方成正比，其中$ñ\ge 0$如线性（斯托克斯）拖动用于ñ  = 1或二次（牛顿）拖动为Ñ  = 2。其它对象移动通过真空，没有阻力。哪个对象首先返回其起点？从分析上可以看出，如果$ñ\ge \mathrm{1个}$，但是该对象可以以其他方式获胜，具体取决于发射速度与要拖动的对象的最终速度之比，如果 $0<ñ<\mathrm{1个}$如果n  = 0 ，则取决于摩擦加速度与重力加速度的比值。这些分析结果通过运动方程的Euler-Cromer数值积分得到证实。