一族射弹的椭圆轨迹
On the elliptic locus of a family of projectiles
摘要：
自由落体运动的一个鲜为人知的性质是，以相同的初速度从不同方向的单点发射的共面射弹所达到的最大高度的椭圆轨迹。另一个不太为人所知的、也许有些令人惊讶的特性是，所述椭圆也是射弹距发射点距离的临界点的几何轨迹。在这篇文章中，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来考虑这些轨迹。结果表明，在这个参考系中，所考虑的轨迹是一个全等圆，它们沿水平方向相切，内相切于另一个圆，这个圆是弹丸射程轨迹的两倍大。利用这种简化的几何描述，在实验室坐标系中恢复了椭圆轨迹，并进一步探讨了其性质。
英文摘要：
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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