Resumen de An Alternate View of the Elliptic Property of a Family of Projectile Paths
Fernández-Chapou and colleagues1 analyzed projectile trajectories and showed an elliptic property hidden in them. For that analysis, they considered projectiles shot from a point with a common value of speed and different angles of projection. Such projectile paths exhibit some interesting characteristics. For example, pairs of projectiles with complementary angles of projection have common values of ranges. This is a well-known property of ideal projectile motion. However, the authors pointed out and demonstrated a little known property of these projectiles, namely, the vertices of these projectiles lie on an ellipse. That ellipse is from a family of projectiles shot from a point with common speed and different angles. We present in this article an alternate view point of observing this hidden ellipse. We emphasize that we don’t use vectors or calculus or Newton’s laws in our analysis. We follow the traditions of Galileo—we use geometry principles We consider here the semi-parabolic paths of the projectiles.
Let us consider a circle of arbitrary radius a in a vertical plane. We draw vertical and horizontal diameters A1B and CD, respectively. From points Ai on the circle in the first quadrant, we draw vertical lines AiOi to CD (see Fig.1). We place plane mirrors inclined at 45° to the horizontal, at mi, the midpoints of the lines AiOi. We let particles fall from rest vertically down from points Ai. They fall along Aimi with uniform acceleration. At mi the particles hit the mirrors and get reflected (in perfectly elastic collision with the mirrors) to move along the horizontal direction with the speed vi, acquired in falling through Aimi. As they so move with the constant speed vi along the horizontal direction, the particles are subjected to uniform acceleration in the vertically downward direction. Thus, the particles are subjected to a uniform motion of constant speed along the horizontal direction and uniformly accelerated motion along the vertically downward direction. As a result of these two simultaneous motions in orthogonal directions, the particles execute a semi-parabolic motion in the vertical plane. We can get the corresponding paths in the second quadrant from symmetry considerations (here, by reflection). The starting points of these parabolic motions (positions of mirrors) are the vertices of the parabolic paths. The locus of these vertices gives the upper half of an ellipse.
