A little known property of free-fall motion is the elliptic locus of the maximum heights attained by coplanar projectiles launched from a single point in different directions with the same initial speed. Another, less known and perhaps somewhat surprising, property of this family consists in the said ellipse being also the geometric locus of the critical points of the projectiles' distances from the launch point. In the article, to gain a better perspective on the geometry involved, we consider these loci from the standpoint of a free-falling frame. It is shown that, in this reference frame, the considered loci are congruent circles tangent to each other along the horizontal and internally tangent to another, twice as big a circle, which represents the locus of the projectiles' ranges. This simplified geometric description is employed to recover the elliptic locus in the laboratory frame and further explore its properties.

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关于射弹族的椭圆轨迹

自由落体运动的一个鲜为人知的特性是共面射弹从不同方向的单个点以相同的初始速度发射所达到的最大高度的椭圆轨迹。该族的另一个鲜为人知且可能有些令人惊讶的特性在于，所述椭圆也是射弹距发射点距离的临界点的几何轨迹。在本文中，为了更好地了解所涉及的几何形状，我们从自由落体框架的角度考虑这些轨迹。结果表明，在该参考系中，所考虑的轨迹是沿水平方向彼此相切且内部彼此相切的全等圆，是圆的两倍大，代表射弹射程的轨迹。