Conic Sections

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## Parabolas and ArtilleryNow that we've established the main properties and applications of conic sections known to the ancients, let's consider some results from the post-classical period. In 1638, while under house arrest, Galileo published his The first may be conceived of as straight-line motion with constant speed; the second as that of a falling object. The latter went against the commonly received notion, which said that the momentum of a falling object is proportional to the distance traversed, so that an object falling twice as far strikes the ground twice as hard and moving at twice the speed. Galileo shows this to be contrary to reason and experience. Galileo further postulates that the speeds acquired by the same body rolling down inclined planes of different grades are equal when the elevations of the starting points are equal. After discussing the reasonableness of this postulate he establishes his most important theorem, namely, that the distance traversed by a body falling from rest with a uniformly accelerated motion is proportional to the square of the elapsed time, or After developing various applications of this law through the remainder of Book III, Galileo proceeds to Book IV, where he states the result that concerns us here, namely, that a projectile subject to a uniform horizontal motion compounded with a naturally accelerated vertical motion moves in a parabolic trajectory. For instance, in the picture shown to the right, an object (say, a cannonball) is given an initial velocity of He begins with the following result from Apollonius. Suppose
To see this, consider the parabola as a section of a cone. Let
Next, the triangles This is equivalent to saying that the vertical distance to the vertex is proportional to the square of the horizontal distance to the axis, which amounts to the description of parabolas you're probably most familiar with, i.e., as graphs of equations like If we imagine firing our cannonball in the horizontal direction with initial velocity
c/v^{2}) ⋅ d_{h}^{2}It follows that the projectile follows a parabolic trajectory. Click here for a demonstration. Notice how the horizontal component of the velocity remains constant, while the vertical component increases proportionally with time. So, if after a certain unit of time the ball is dropping at a certain rate, then, after twice as much time, it will be dropping at twice the rate, and so on. If, instead of leveling the gun straight ahead, we fire at an angle, the path of motion remains parabolic, but the vertical component of the initial velocity causes the ball to move up before moving down. Galileo's laws of mechanics anticipate Newton's laws of motion and gravitation, which appeared later in the seventeeth century. It was Galileo who said:
Next let's see how conic sections can be used to model the motion of the planets. |

**Michael Ortiz, Ph.D.
Assistant Professor of Mathematics
Department of Natural and Behavioral Sciences**

Sul Ross State University Rio Grande College

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Copyright © 2013 Michael Luis Ortiz. All rights reserved.