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By Rüdiger Hohmann, Nico Hagemann and Nicolas Kaiser


The numeric treatment of boundary- and optimization problems is carried out by iterations. With an air resistance proportional to the square of the velocity, the projectile motion does not have any elementary analytic solution. In case of no air resistance however those solutions exist and they may be taken as tests for the numeric methods. The projectile motion with air resistance is characterized by a more steeply falling trajectory. Depending on the starting angle at constant initial velocity, the projectile motion as a boundary value problem and the maximization of the trajectory range are treated. The target deviation forms the functional of the boundary value problem. The algorithms consist of a modified Newton‟s Method, which is used to search the zero of the deviation, as well as of an angle correction proportional to the deviation. Optimization methods are the Three-Points Plan and the Method of the Golden Ratio. This method needs only a single new run for the comparison. During the maximization the uncertainty interval of the starting angle decreases iteratively by comparisons of the trajectory ranges. When the object hits the ground, i.e. the trajectory crosses the threshold zero, the simulation run is finished by a state event. The calculated range is then used as input parameter of the iteration. The algorithms are implemented in the TERMINAL section of the simulation system ACSL

Topics: Projectile motion, Education, Quadratic friction, Modified Newton’s
Year: 2012
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