title: ÐÐ¿ÑÐ¸Ð¼Ð¸Ð·Ð°ÑÐ¸Ñ ÑÑÐ°ÐµÐºÑÐ¾ÑÐ¸Ð¹ Ð¿Ð¾Ð»ÐµÑÐ° ÑÑÑÐ±Ð¾Ð»ÑÐ½Ð¾Ð³Ð¾ Ð¼ÑÑÐ°
title: Ð²ÑÐ¿ÑÑÐºÐ½Ð°Ñ ÐºÐ²Ð°Ð»Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¾Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð±Ð°ÐºÐ°Ð»Ð°Ð²ÑÐ°
titleType: Subtitle
title: Optimization of the flight path of a soccer ball
Creators: name: ÐÐµÐ½ÑÐ¸ÐºÐ¾Ð², ÐÐ°Ð½Ð¸Ð¸Ð»
Publisher: name: Ð¡Ð°Ð½ÐºÑ-ÐÐµÑÐµÑÐ±ÑÑÐ³ÑÐºÐ¸Ð¹ Ð¿Ð¾Ð»Ð¸ÑÐµÑÐ½Ð¸ÑÐµÑÐºÐ¸Ð¹ ÑÐ½Ð¸Ð²ÐµÑÑÐ¸ÑÐµÑ ÐÐµÑÑÐ° ÐÐµÐ»Ð¸ÐºÐ¾Ð³Ð¾
Year: 2020
Subjects: Ð±Ð°Ð»Ð»Ð¸ÑÑÐ¸ÐºÐ°
Ð¼Ð°ÑÐµÑÐ¸Ð°Ð»ÑÐ½Ð°Ñ ÑÐ¾ÑÐºÐ°
Ð´Ð²Ð¸Ð¶ÐµÐ½Ð¸Ðµ
ÐºÑÐ¸ÑÐµÑÐ¸Ð¹ Ð¾Ð¿ÑÐ¸Ð¼Ð¸Ð·Ð°ÑÐ¸Ð¸
Ð´Ð°Ð»ÑÐ½Ð¾ÑÑÑ
ÑÐ³Ð¾Ð»
Ð²ÑÐ·ÐºÐ¾Ðµ ÑÑÐµÐ½Ð¸Ðµ
ÐºÐ²Ð°Ð´ÑÐ°ÑÐ¸ÑÐ½Ð¾Ðµ ÑÐ¾Ð¿ÑÐ¾ÑÐ¸Ð²Ð»ÐµÐ½Ð¸Ðµ
ballistics
point particle
motion
optimization criterion
range
angle
viscous friction
quadratic resistance

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description: This work is devoted to study of the flight of a soccer ball without taking into account the aerodynamic drag, taking the into account linear and quadratic dependence of resistance on speed, on the example of a point particle in the homogeneous gravitational environment of the Earth and solution of problems of the univariate and multivariate optimization. The tasks that were solved during the research: 1. Construction and analysis of the simplest optimization problem. 2. Construction and analysis of the energy-time optimization criterion for the classical problem of ballistic trajectories. 3. Investigation of ball flight in environments with linear and quadratic resistances. 4. Construction and analysis of optimization problems for movements in environments with linear and quadratic resistances. In the course of work had solved the optimization problem of a particle motion in a uniform gravitational environment of Earth without air drag, assuming a linear and quadratic dependence of resistance on speed. Calculations were performed that clearly show how the resistance value affects the optimal angle of inclination at impact to achieve the maximum range. The analysis was performed in a package of applications for solving technical computing problems MatLab. As a result of the analysis of the obtained calculations, the dependencies of optimal angles at different resistance values were obtained. It turned out that when increasing the resistance value of the optimal initial angle of attack is reduced, and that for a given value of the initial angle with increasing resistance decreases the range. The results obtained can be used for various ballistic calculations.