RU2698333C1 - Method of aiming when performing free throws in basketball - Google Patents

Abstract

FIELD: sports.

SUBSTANCE: invention relates to sports equipment, in particular, to basketball equipment. For aiming during free throws in basketball, trajectories of ball (L) flight and its interaction with plane of shield (1) and rim of ring (3) are mathematically simulated. Using the geometrical model, in which the rim of ring (3) increases along the radius of the cross-section by the value of the ball radius and turns into a torus, and the ball accordingly by the same value decreases along the radius and turns into a material point, parameters of "flows" of possible and recommended trajectories (L) of flight of ball and corresponding parameters of "attacked" ("clean" and "expanded") targets in plane of ring (3) for no rebound and with rebound of ball from shield (1) are determined.

EFFECT: provides for individual selection of variety of execution of free throws without rebound or with rebound of ball from shield for increase of their effectiveness, as well as theoretical substantiation of modernization of basketball equipment in order to change rules of basketball.

5 cl, 31 dwg

Description

The invention relates to sports equipment, in particular to equipment for basketball.

There is a method of training athletes in the technique of performing basketball throws without rebounding the ball from the backboard (Patent 20090325739 A1 USA, А63В 43/00, G06F 19/00. Intelligent basketball / Robert S. Gold; - No. 146348/12; Filed: 06.25.08; Publ .: 12/31/09. - 11 p.). In substantiating his invention, the author cited statistics on the average team free throw percentage in the National Basketball Association, which from 1958 to 2008 was 74%. It should be noted that to date, this team result of free throws has not changed. At the London Olympics in 2012, this indicator of 12 men's teams reached 68.7% (maximum result - 74.5%, minimum - 53.2%) [URL: http://London2012.FIBA.com]. At the 2017 European Championships in Turkey, 24 men's teams showed an average team result of free throws of 75.4% (maximum value - 84.4%, minimum value - 62.4%) [URL: http://www.championat.com / basketball / _bteam / 2110 / match / 607540.html].

To increase the effectiveness of shots in basketball, the patent proposes a system that tracks the parameters of the ball with the provision of prompt feedback from the throwing player. The system includes a modified ball, in the hole of which a sensor board is mounted, containing a three-axis accelerometer, a microcontroller, and a radio module. An integral part of the system is a response radio module with a microcontroller and a USB connector for connecting to a personal computer (PC). The response radio module with the PC is at a safe distance from the ball’s flight space. The accelerometer on the sensor board continuously measures the acceleration of the basketball along three axes. This information is transmitted wirelessly to the response radio module board and processed in a PC taking into account the input data: player height, distance from the player to the target, and the height of the ring above the surface of the site. Then, the ball release parameters are promptly provided to the player, which allows you to reasonably adjust the following shots, bringing them closer to the optimal values for a particular player.

Mounting a rigid cylinder with a sensor board inside the basketball changes its weight, and also creates an eccentricity of the mass of the device relative to the center of gravity of the ball. The influence of the modification of the ball on the parameters of its flight during the throw requires its study and justification.

The well-known "Method of training and assessing the accuracy of free throws in basketball." (Pat. 2530863 C1 Russian Federation, IPC7 А63В 63/08. Method for training and assessing the accuracy of free throws in basketball / Familnikova N.V.; Rozhentsov V.V .; patentee: federal state budgetary institution of higher education "Volga State Technological University ". - No. 2013126367/12; application. 07.06.13; publ. 20.10.14, Bull. No. 29. - 6 p.).

Using 2 sensors, video cameras with a computer, the coordinates of the center of the ball in the plane of the ring are determined with respect to the coordinates of the center of the ring when the basketball player performs a free-throw.

The image of the ball on the background of the basketball hoop on the computer monitor allows the throwing athlete to quickly evaluate the result of the free-throw and in the next throw to make the necessary adjustments to the throw parameters. It is advisable to submit the image of the results of the throw on the TV screen.

In this method, there is no quantitative data on the parameters of the trajectory of the ball. These values reflect the parameters of the “hit” target in the plane of the ring, which do not coincide with the coordinates of the center of the ring.

The coordinates of the “hit” target are shifted closer to the far arc with respect to the throwing player and with respect to these values, and it is necessary to evaluate the results of the throw. This provision is set forth in detail in the present invention.

The closest way to practice basketball shots with the ball bouncing off the backboard is the invention (Patent 2386466 C2 Russian Federation, IPC7 63B 63/08. A method for determining aiming coordinates when shooting with the ball reflected from the backboard / N. Yurchenko; VN Pritykin. ; patent holder: state educational institution of higher professional education “Omsk State Medical Academy of the Federal Agency for Healthcare and Social Development” of Roszdrav. - No. 2008127598/12; filed 07.07.08; publ. 20.04.10, Bull. No. 11. - 13 p. .) in which simple visual landmarks are used as sights. Colored, mathematically calculated, aiming and reflection lines are glued on the front plane of the shield, and the vertical axis of aiming is fixed in the center of the “imaginary” ring with the help of a clamp and bracket. The axis is located at a distance of 131 mm from the front plane of the shield perpendicular to the surface of the site. The ball release point and the vertical aiming axis create a vertical plane of movement of the center of the ball, while the axis is the main visual reference.

Different parameters of the ball release require different aiming and reflection lines in coordinates, which increases the complexity of preparing equipment for training sessions.

It is difficult to visually control the area of contact of the ball with the shield plane along the reflection line, since the area of the ball print on the shield plane on average varies from 40 to 85 cm ² , which is 6-8 times larger than the area of the reflection line 1 cm wide, located under the ball in the region contact with the shield. (Morozova, N. S. Improving the accuracy of basketball shots with the ball reflected from the backboard: dis. ... Ph.D. / Morozova Natalya Sergeevna. - Omsk, 2009. - 153 p. See S. 56). This indicates the practical need for designing reflection areas on the front plane of the shield, which depend on the location coordinates of the throwing player on the court and the release parameters of the ball during the throw.

The reflection areas for shots with the ball bouncing off the backboard, established by mathematical modeling, vary on average from 312 to 605 cm ² for the recommended range of parameters of the ball’s flight paths taking into account the angle of approach to the board’s plane. (Pritykin, V.N. Modeling of targets of a complex-oriented basketball throw with the reflection of the ball from the backboard / V.N. Pritykin // Theory and practice of physical education. - 2016. No. 5. - P. 62-64.). The cross-sectional area of a basketball (large circle) passing through its center is 468 cm ² . The indicated dimensions of the ball and the established reflection areas make it possible to more accurately determine the contact coordinates when identifying a possible ball displacement relative to the center of the reflection area in comparison with the existing method for determining aiming coordinates when throwing with the reflection of the ball from the shield.

Finding the vertical axis of aiming from the back of the backboard makes it difficult to visually determine the point of intersection of the axis with the aiming line due to the partial overlap of the axis by the structural elements of the basketball equipment and the marking lines of the shield. To eliminate the noted lack of aiming, a theoretical solution is required to justify the transfer of the vertical aiming axis to the front plane of the shield.

Functional requirements, safety requirements and test methods of modern basketball equipment are presented in detail in the following regulatory documents:

- GOST R 56434-2015. National standard of the RUSSIAN FEDERATION. “Equipment for sports games. BASKETBALL EQUIPMENT. " EN 1270: 2005 (NEQ). Approved and put into effect for the first time by Order of the Federal Agency for Technical Regulation and Metrology of June 15, 2015 No. 652 of Art. - Moscow: Publishing House of Federal State Unitary Enterprise “Standardinform”. - 20 s.

- The Official Basketball Rules 2014 - Basketball Equipment: Approved by the FIBA Central Bureau, Barcelona, Spain, February 2, 2014. The Russian text of the document “Official Basketball Rules 2014 - Basketball Equipment” was prepared by the RBF Training and Methodological Commission. Effective October 1, 2014 - 23 p.

In these documents, the layout and dimensions of the playing court, basketball backboard, as well as the dimensions of the ring and ball are presented in detail. In literature, no justification was found for the size of the ring and the ball, the distance from the center of the ring to the front plane of the shield, and no scientific data were found on the dependence of the effectiveness of shots on the size of standard basketball equipment. It can be assumed that the theoretically justified modernization of basketball equipment will provide an increase in the effectiveness of basketball shots without a rebound and with a rebound of the ball from the backboard.

The objective of the method.

The objective of the method is the training and improvement of free throws in basketball without a rebound and with the ball bouncing off the backboard using standard and modernized basketball equipment, reasonable visual-spatial landmarks and objects of aiming and reflection.

The technical result of the invention is the individual choice of a variety (without a rebound or with a rebound of the ball from the backboard) of free throws to increase their effectiveness, as well as the theoretical justification for the modernization of basketball equipment in order to change the rules of the game of basketball. The specified technical result is achieved by the fact that in the aiming method the forced union of “flows” of productive trajectories is carried out, reduction and / or elimination of “flows” of ineffective flight paths of the ball, as well as an increase in the total area of “attacked” goals of free throws without rebound and with the ball bouncing shield by upgrading standard basketball equipment. The modernization of equipment and the use of reasonable visual-spatial landmarks and objects of aiming and reflection when practicing free throws of two varieties increases their effectiveness.

When performing free throws in basketball, the ball’s flight paths and their interactions with the shield plane and the ring rim are mathematically modeled. The plane of the “imaginary” shield is determined, relative to which the main and “imaginary” basketball rings are symmetrically located, and the objects (points, lines, circles, squares, “posts”) are set for aiming and reflection for basketball shots without rebound and with the ball bouncing off the board . The "imaginary" shield is located at a distance of the radius of the ball inside the playing court parallel to the plane of the main shield.

The effectiveness of hitting the ball in the goal depends on the size and on its removal. To determine the parameters of the flight paths of the ball and the boundaries of the “attacked” goal, a geometric model of the interaction of the basketball with the rim of the ring is proposed. The rim of the ring increases along the radius of the section by the value of the radius of the ball and turns into a torus, and the ball, respectively, by the same value decreases in radius and turns into a material point. A geometric model of the interaction of the center of a basketball with a torus simulating a ring using the Basket program (Calculation of the parameters of basketball shots without reflecting the ball from the Basket shield: Certificate of state registration of the computer program No. 2014611231. / Pritykin V.N., Petrushov I.V., Bokov I.S., Kuznetsova N.S. (Russia) - No. 2013661152; decl. 03.12.13; publ. 28.01.14.) Allowed to determine the parameters of the “flows” of the possible and recommended flight paths of the ball and the corresponding parameters of the "hit" targets in the plane of the ring for shots without ball rebound from the shield. A “hit” goal in basketball does not have fixed parameters. The size, shape and location of the target relative to the geometric center of the ring (the target can be shifted up to 15 cm to the far arc of the ring) depends on the change in the parameters of the ball’s flight paths, which are created by the athlete during the throw. The target has an oval shape, varying from peaked to rounded (elliptical). The plane of the ring is divided into three areas of the ball arrivals: the area of near and far arcs and the area of "clean" hit. The obtained results of the “attacked” goals for shots without a ball bounce from the shield apply to shots with the ball bounce off the shield, only the direction of the ball’s flight and, accordingly, the location of the ball’s arrival areas are reversed with respect to these characteristics for shots without the ball bounce from the shield.

For throws without rebounding the ball from the shield, the aiming point is determined and justified, which is located in the center of the far arc of the ring in the plane of flight of the center of the ball. The specified plane passes through the center of the ring and the point of release of the ball when throwing perpendicular to the surface of the playing court. The recommended range is indicated with the help of visual-spatial reference points for the maximum coordinates of the ball flight paths: ( aa ) - along the upper side of the shield; (rr) - along the top line of the aiming rectangle. An athlete performing a throw without bouncing the ball from the backboard needs to aim at the center of the far arc of the ring rim and achieve the ball flying at or above the top of the backboard. Touching the ball near the arc of the ring during the throw is considered a gross mistake.

For the shots with the ball bouncing off the backboard, which are performed on standard basketball equipment, the center of the “imaginary” ring and simple visual aiming targets are defined (patent 2386466 C2 Russian Federation, IPC7 A63B 63/08). In addition, ellipse-like reflection areas (“intermediate” targets) and peculiar spatial “columns” of aiming and reflection of productive throws on the front plane of the shield were installed.

For the shots with the ball bouncing off the backboard, performed on the modernized basketball equipment, the center of the “imaginary” ring is determined, which practically coincided with the board's plane. From the center of the “imaginary” ring, a sticky colored tape 1 cm wide is applied to the front plane of the shield, which is perpendicular to the surface of the court and is the vertical axis of aiming - the main visual reference point for shots with the ball bounce off the board. The established ellipse-like reflection areas (“intermediate” targets) are applied with colored adhesive tape to the front plane of the shield and for a fixed angle of approach of the ball to the plane of the shield (in this case, it is 90 °) when the angle of release of the ball changes, the reflection areas change shape, size and coordinates positioning above each other, forming a kind of spatial "columns" of aiming and reflection of productive shots. An athlete performing a throw with the ball bouncing off the board must aim at the intersection of the vertical axis with the center of the aiming area, which belongs to the recommended ball release angle. It is necessary to direct the center of the ball during the throw in the plane formed by the release point and the vertical axis of aim. The control of the height of the ball’s flight path is required to be carried out visually using spatial landmarks (the level of the plane of the basketball hoop, the top side of the backboard and / or the top line of the aiming rectangle) and the coordinates of the ball’s contact in the corresponding reflection area.

To train and improve free throws without a rebound and with the ball bouncing off the backboard, basketball equipment is being modernized, consisting of the following changes:

1. Transfer of the front plane of the basketball backboard (rejection of the horizontal plate of the mounting L-shaped bracket) closer to the center of the basketball hoop by 15 cm.

In this case, the marking of the free-throw line remains unchanged. The center of the “imaginary” ring will approach 15 cm to the center of the main ring, which improves the conditions of the shots with the ball bouncing off the backboard by reducing the distance from the free-throw line to the projection of the center of the “imaginary” ring on the court surface. We remind you that a throw with a ball rebound from the shield is a throw without a rebound into the "imaginary" ring. A positive fact of the modernization of basketball equipment is that the aiming axis coincides with the front plane of the shield, ensuring the quality of aiming when performing free-throw shots in basketball. Moving the center of the upgraded ring 15 cm closer to the front plane of the backboard requires a shift of the free-throw line and the line of three point-throw to the specified distance closer to the front line of the basketball court.

2. Determination of parameters and location coordinates of visual-spatial landmarks and aiming and reflection objects for basketball shots without a rebound and with the ball rebound from the backboard.

By changing the size of the structural elements of standard basketball equipment, they reduce and eliminate the “flows” of ineffective flight paths of the ball, and also increase the total area of the “attacked” goals in the ring plane by combining the “hit” goals of basketball shots without rebound and with the ball rebound from the shield. Combining the goals of the rolls of two varieties provides a gradual transition from a horizontal “attacked” target in the plane of the ring to a vertical “intermediate” target on the front plane of the shield during the practice of free throws. This approach provides a theoretically sound development of shots with the ball bouncing off the shield, contributing to the high-quality individual selection of a more productive kind of free throw for use in competitive activities.

Modernization of basketball equipment improves the quality of training and improving the technique of free-throwing and increases the efficiency of shots without a rebound and with the ball bouncing off the backboard.

The results of mathematical modeling of basketball shots without bouncing the ball from the backboard on standard basketball equipment and the theoretical justification for the modernization of basketball equipment are illustrated by figures 1-31.

The figure 1 shows the main parameters of the flight paths of the ball, which are used in mathematical modeling and in determining the effectiveness of basketball shots. Parameters F, L and Ω are shown conditionally.

The figure 2 shows a geometric representation of the "attacked" goals at the level of the middle plane of the basketball hoop.

The figure 3 presents the creation of a geometric model of the interaction of a basketball with a rim of a ring.

Figures 4, 5, 6, 7, 8, 9, 10, and 11 reveal the evolution of the “attacked” target in the profile projection of the geometric model of the interaction of the center of the ball with the surface of the torus when the entry angle changes from 0 to 90 degrees on standard basketball equipment.

Figure 4 shows the "flow" of the flight paths of the ball for an entry angle of 0 degrees of a free throw without bouncing the ball from the backboard on standard basketball equipment.

Figure 5 indicates the appearance of an effective trajectory (angle of entry of 20 degrees) of a free-throw without rebounding the ball from the backboard on standard basketball equipment.

Figure 6 reveals the appearance of the “hit” target (angle of entry 23 degrees) of a free throw without rebounding the ball from the backboard on standard basketball equipment.

Figure 7 shows the appearance of the “clean” hit path (entry angle of 35 degrees) of a free throw without rebounding the ball from the backboard on standard basketball equipment.

The figure 8 shows the "flow" of economical trajectories (E _to - the minimum value) of the flight of a free throw basketball (angle of entry 39 degrees) without rebounding the ball from the backboard on standard basketball equipment.

Figure 9 reveals the "flow" of the trajectories of the maximum volumetric-angular size of the target Ω (entrance angle 54 degrees) of a free throw without rebounding the ball from the backboard on standard basketball equipment.

Figure 10 shows the "flow" of the flight paths of the ball for an angle of entry of 75 degrees of a free throw without rebounding the ball from the backboard on standard basketball equipment.

Figure 11 discloses the "flow" of the flight paths of the ball for an angle of entry of 90 degrees of a free throw without bouncing the ball from the backboard on standard basketball equipment.

Figures 12, 13, 14 and 15 show the main parameters of the characteristic flight paths of the ball and the corresponding parameters of the “attacked” targets in the plane of the free throw ring without rebounding the ball from the backboard performed on standard basketball equipment.

In figure 12, the main parameters of the characteristic flight paths of the ball and the corresponding parameters of the “attacked” targets in the plane of the ring when the target appears are indicated.

The figure 13 shows the main parameters of the characteristic flight paths of the ball and the corresponding parameters of the “attacked” targets in the plane of the ring when the area of “clean” hit appears.

The figure 14 shows the main parameters of the economical flight paths of the ball and the corresponding parameters of the “attacked” goals in the plane of the ring for a free throw without bouncing the ball from the backboard, performed on standard basketball equipment.

Figure 15 discloses the main parameters of the characteristic flight paths of the ball and the corresponding parameters of the “attacked” goals in the plane of the ring for the maximum volumetric and angular size of the goal for a free throw without rebounding the ball from the backboard performed on standard basketball equipment.

Figures 16 and 17 show a horizontal and profile projection of the flight paths of the basketball ball with its rebound from the backboard to determine the location parameters of the “imaginary” backboard and the ring, as well as the aiming and reflection coordinates of these shots performed on standard basketball equipment.

Figures 18 and 19 show geometric models of a free throw with the reflection of the ball from the backboard for a range of recommended ball flight paths performed on standard basketball equipment.

In figures 20 and 21 shows the geometric construction of the areas of aiming and reflection on the front plane of the shield using the functionality of the CAD system "COMPASS".

In figures 22 and 23 shows the area of aiming and reflection of free throws with the ball rebound from the shield, performed on standard basketball equipment for the recommended range of flight paths of the ball.

In figures 24 and 25 presents a geometric model of a free throw with the reflection of the ball from the shield for the range of recommended flight paths of the ball, performed on the upgraded basketball equipment.

In figures 26 and 27 shows the area of aiming and reflection of free throws with the ball rebound from the shield, performed on the upgraded basketball equipment for the recommended range of flight paths of the ball.

The figure 28 shows the combined “attacked” goals in the plane of the modernized ring of 4 free throws without rebounding the ball from the shield and in the plane of the “imaginary” ring 5 '(imaginative) of free throws with the rebound of the ball from the shield for the maximum volumetric and angular size of the target. In addition, projections of the actual location of the “attacked” targets of free throws of two varieties in the plane of the modernized ring 4, the areas of which are reduced (“cut off”) by the plane of the “imaginary” shield 2, are shown.

The figure 29 shows the combined “attacked” goals in the plane of the upgraded ring of 4 free throws without rebounding the ball from the shield and in the plane of the “imaginary” ring 5 '(imaginative) of free throws with rebounding the ball from the shield for economical ball flight paths. In addition, projections of the actual location of the “attacked” targets of free throws of two varieties in the plane of the modernized ring 4, the areas of which are reduced (“cut off”) by the plane of the “imaginary” shield 2, are shown.

In figures 30 and 31 shows the "columns" of aiming and reflection for training and improving free throws with the ball rebound from the shield on standard and upgraded basketball equipment.

Designations and names of assembly units, parts, fasteners, mounting elements and visual landmarks in the figures are as follows:

About _vyp - the point of release of the ball during a free throw; V _o - the initial speed of the ball; U _o - the horizontal component of the initial speed of the ball; W _o - the vertical component of the initial speed of the ball; α _o - the angle of the ball during the throw; Ω is the objective function, or the volumetric-angular size of the goal, on which the accuracy of hitting the ball in the basketball hoop depends (goal size formula: Ω = F × sin α _in / L ² , presented in the monograph on page 35 - Pritykin, V.N. Unconventional approaches to improving the accuracy of the free throw in basketball: monograph / V.N. Pritykin. - Omsk: Publishing House of Omsk State Medical University, 2015. - 175 p.); R _m - the radius of the basketball; X _max - horizontal coordinate of the maximum point of the ball’s flight path from the release point; Z _max - the vertical coordinate of the maximum point of the ball’s flight path from the release point; H is the height from the point of release of the center of the ball to the level of the upper plane of the ring; h is the height from the level of the middle plane of the ring to the basketball located at the maximum point of the ball’s flight path; L is the length of the trajectory of the center of the ball to the center of the area of the “attacked” target F; F is the area of the “attacked” goal, located in the middle plane of the basketball hoop and providing an effective throw when the center of the ball passes through any point of this area; α _I - the angle of entry of the ball into the ring (the angle between the middle plane of the ring and the tangent to the path of the center of the ball at the point of its passage to the plane of the ring); V _to - the final speed of the ball; U _to - the horizontal component of the final speed of the ball; W _to - the vertical component of the final speed of the ball; S is the distance from the projection of the center of the basketball hoop to the surface of the basketball court to the projection of the ball release point onto the surface of the court; X and Z are the directions of the reference coordinates selected in figure 1; parameters Ω, L and F are shown conditionally;

1 - standard basketball backboard;

2 - "imaginary" shield;

3 - standard basketball hoop;

4 - an upgraded basketball hoop;

5 - “imaginary” ring; 5 '- the "imaginary" ring of the upgraded equipment;

6 - a basketball;

7 - the center of the basketball;

8 - “pure” goal;

9 - "extended" goal;

10 - aiming point for free throws without rebounding the ball from the shield;

11 - "reduced" goal;

12 - "contact" ring;

13 - model of a ring in the form of a torus;

«атакуемая» цель в баскетболе (синоним - «поражаемая» цель) - объект для попадания мячом; «атакуемая» цель - это геометрическая фигура, расположенная в плоскости баскетбольного кольца и имеющая три характерные составляющие области: область «чистого» попадания, области ближней и дальней дуг результативных бросков; каждая область и их сумма не имеют фиксированных значений; овальная форма площади «атакуемой» цели изменяется от остроконечной до округлой (эллиптической); она зависит от параметров траектории полета мяча, которые создает бросающий спортсмен при выпуске мяча; каждой траектории соответствует конкретная по параметрам и геометрическому расположению в баскетбольном кольце «атакуемая» цель; смещение центра «атакуемой» цели к дальней дуге от геометрического центра кольца может достигать 15 см; центр области «чистого» попадания совпадает с геометрическим центром кольца; для рекомендуемого диапазона траекторий полета мяча при броске размеры площадей для области «чистого» попадания и суммы трех составляющих областей соответственно принимают значения: для экономной траектории - 4 и 22% и для оптимальной - 14 и 28% от площади баскетбольного кольца;

“Attacked” goal in basketball (synonym - “hit” goal) - an object for hitting the ball; The “attacked” target is a geometric figure located in the plane of the basketball hoop and having three characteristic component areas: the area of “clean” hit, the area of the near and far arches of successful throws; each region and their sum do not have fixed values; the oval shape of the area of the “attacked” target changes from spiky to round (elliptical); it depends on the parameters of the ball’s flight path that the throwing athlete creates when releasing the ball; each trajectory corresponds to a specific “attacked” target in terms of parameters and geometric location in the basketball hoop; the displacement of the center of the “attacked” target to the far arc from the geometric center of the ring can reach 15 cm; the center of the “clean” hit area coincides with the geometric center of the ring; for the recommended range of ball flight paths during a throw, the sizes of the areas for the “clean” hit area and the sum of the three component areas, respectively, take values: for the economical path - 4 and 22% and for the optimal - 14 and 28% of the area of the basketball hoop;

баскетбольный бросок без отскока мяча от щита - бросок мяча в кольцо (синонимы:

basketball throw without rebounding the ball from the backboard - throwing the ball into the ring (synonyms:

- a basketball throw without reflecting the ball from the backboard;

- a basketball throw right into the ring;

- direct basketball throw);

баскетбольный бросок с отскоком мяча от щита - бросок мяча в щит для попадания в кольцо

basketball throw with the ball bouncing off the board - throwing the ball into the board to hit the ring

(synonyms:

- a basketball throw with the reflection of the ball from the backboard;

- a complex-oriented basketball throw;

- shield basketball throw);

«мнимый» баскетбольный щит - воображаемый баскетбольный щит; «мнимый» щит - это совокупность геометрических точек, в которых центр баскетбольного мяча изменяет направление движения при контакте мяча с лицевой плоскостью щита при выполнении бросков с отскоком мяча от щита; «мнимый» щит воображаемо, расположен на расстоянии радиуса мяча к центру игровой площадки и параллельно лицевой плоскости основного щита;

"Imaginary" basketball backboard - an imaginary basketball backboard; An “imaginary” shield is a set of geometrical points at which the center of the basketball changes direction of movement when the ball makes contact with the front of the board when throwing with the ball bouncing off the board; The "imaginary" shield is imaginary, located at a distance of the radius of the ball to the center of the playing court and parallel to the front plane of the main shield;

«мнимое» баскетбольное кольцо - воображаемое баскетбольное кольцо; «мнимое» кольцо - это воображаемое кольцо, расположенное симметрично основному кольцу относительно «мнимого» щита; центр «мнимого» кольца образован пересечением траекторий полета центра мяча за плоскость основного щита результативных бросков с отскоком от щита; в центр «мнимого» кольца устанавливается вертикальная ось прицеливания на расстоянии 131 мм за лицевой плоскостью щита; броски с отскоком мяча от щита - это броски прямо в «мнимое» кольцо, а вертикальная ось - это основной ориентир для их качественного выполнения;

"Imaginary" basketball hoop - an imaginary basketball hoop; “Imaginary” ring is an imaginary ring located symmetrically to the main ring relative to the “imaginary” shield; the center of the “imaginary” ring is formed by the intersection of the flight paths of the center of the ball beyond the plane of the main shield of productive throws with a rebound from the shield; in the center of the "imaginary" ring, a vertical aiming axis is set at a distance of 131 mm behind the front plane of the shield; throws with the ball bouncing off the backboard are throws right into the "imaginary" ring, and the vertical axis is the main reference point for their high-quality performance;

«чистая» цель - это геометрическая фигура, контур которой образован пересечением средней плоскости кольца совокупностью предельных результативных траекторий полета центра мяча, когда мяч при проходе в кольцо касается обода, но не меняет направления своего движения;

A “clean” goal is a geometric figure, the contour of which is formed by the intersection of the middle plane of the ring with the totality of the ultimate resultant flight paths of the center of the ball when the ball touches the rim when it enters the ring but does not change its direction of movement;

«расширенная» цель - это геометрическая фигура, контур которой образован пересечением средней плоскости кольца совокупностью предельных результативных траекторий полета центра мяча при его взаимодействии с ободом кольца. Контур «расширенной» цели - это линия, разделяющая штрафные броски на результативные и нерезультативные после отскока (отскоков) мяча от обода баскетбольного кольца;

An “extended” goal is a geometric figure whose contour is formed by the intersection of the middle plane of the ring with the totality of the ultimate resultant flight paths of the center of the ball during its interaction with the ring rim. The contour of the “extended” goal is the line dividing the free throws into effective and ineffective after the ball bounces (rebounds) from the rim of the basketball hoop;

F _m - the cross-sectional area of a basketball (large circle) passing through its center is 468 cm ² ; F _to - the area of the basketball hoop with a diameter of 45 cm is equal to 1590 cm ² ; F _hours - the maximum area of the “clean” goal with a diameter of 20.6 cm, created by the center of the basketball ball when it is run in the inner surface of the ring, is 333 cm ² ; passage of the center of the ball through any point of the circle (the area of the "clean" goal) leads to the ball falling into the ring without touching its rim; F _bp - the area of the "reduced" goal with a diameter of 73.4 cm, created by the center of the basketball ball when it was run in the outer surface of the ring, is 4231 cm ² ; F _c.k. - the area of the "contact" ring, in which the inner diameter is 20.6 cm (diameter of the "clean" target), and the outer diameter is 73.4 cm (diameter of the "reduced" target), is 3898 cm ² ; the arrival of the center of the ball in the space of the “contact” ring leads to the interaction of the ball with the rim of the basketball hoop; after a rebound from the rim, the ball can immediately hit the ring or after two or three rebounds from the rim inside the ring; a significant shift in the coordinates of the center of the ball from the geometric center of the ring, after it bounces off the rim of the ring leads to a miss; the border on the “reduced” goal, which divides the basketball shots into effective and ineffective, forms an “extended” (real) goal compared to the “clean” one; F _r.c. - the area of the "expanded" target, in which the diameter is 31.2 cm, is 764.5 cm ² .

L _o - short (not more than the diameter of the ball) the final segment of the trajectory of the ball center before interacting with the torus (ring rim) for throws without rebounding the ball from the shield; L ₀ '- the same value as L ₀ - only for throws with the ball rebound from the shield; in the following notations in the text and in the figures, the “” sign placed after the parameter indicates that it belongs to the shots with the ball bouncing off the backboard; L ₁ , L ₂ and L ₃ - the trajectory of the center of the ball after the first, second and third rebounds of the ball from the rim of the basketball hoop; L _K - the straight-line final segment of the trajectory of the center of the ball, at which the ball touches the rim of the ring without changing the direction of its movement; L _1p - the first successful trajectory of the center of the ball when it bounces from the rim of the far arc of the ring; L _1h - the first "clean" trajectory of the ball center, at which the ball touches the rim of the near and far arcs of the ring without changing the direction of its movement; L _m - ineffective trajectories of the center of the ball before interacting with the near and far arcs of the ring;

14 - coordinate of the boundary point of the axis of the ellipse-like "extended" target, located in the direction of the ball’s flight in the area of the far arc of the basketball hoop;

15 - the coordinate of the bounce of the center of the ball from the surface of the torus at the level of the middle plane of the ring with a successful throw to the far arc of the ring has a constant value of 10.3 cm from the geometric center of the ring;

16 - coordinate of the center of the "extended" target in the middle plane of the ring;

17 - coordinate of the point of intersection of the first "clean" trajectory with the plane of the ring in the profile projection of the flight of the ball; this point belongs to an ellipse-like contour dividing the “flow” of resultant trajectories into the areas of “clean” hit and the far arc;

18 - the center of the ring;

19 - coordinate of the boundary point of the axis of the "clean" target, located in the direction of flight of the ball in the area of the near arc of the ring;

20 - coordinate of the boundary point of the axis of the "extended" target, located in the direction of flight of the ball in the area of the near arc of the ring;

PB - “beam” of the flight paths of the center of the ball to the arc (B ₁ -B ₂ ) of the torus 13, after interaction with which, a miss occurs from the nearest arc of the ring; PD - “beam” of the flight paths of the center of the ball to the arc (D ₁ -D ₂ ) of the torus 13, after interaction with which, a miss occurs from the far arc; PN - combined “beam” (“beam” overlay) of the flight paths of the center of the ball, aimed at the near and far arcs and which disappears when the path of a “clean” hit appears; B - “bundle” of productive trajectories of the ball center, directed to the nearest arc of the ring (yellow); D - “bunch” of effective trajectories of the ball center, aimed at the far arc of the ring (blue color); H - “beam” of the trajectories of the center of the ball with a “clean” hit in the ring (red color); b — sector of productive trajectories with the ball bouncing off the surface of the torus in the region of the near arc of the profile projection of the ball’s flight; ∂ — sector of productive trajectories with the ball bouncing off the torus surface in the region of the far arc of the profile projection of the ball’s flight; ( aa ) - the horizontal plane on the upper side of the shield 1 - a visual-spatial reference point for determining the maximum point of the ball's flight path when performing a free throw in basketball; (rr) - the horizontal plane along the top line of the aiming rectangle - a visual-spatial reference point for determining the maximum point of the ball's flight path when performing a free throw in basketball; P _to - the horizontal middle plane at the level of the basketball hoop 3, in which the "attacked"goal; х _с - displacement of the center of the “attacked” target from the geometric center of the ring; E _to - kinetic energy of the ball at the time of its release;

α is the angle of approach of the center 7 of the ball 6 to the front plane of the shield; α _pad and α _neg - the angles of incidence and reflection of the ball during its interaction with the front plane of the shield in the horizontal projection of the flight of the ball (Fig. 16);

21 - vertical axis of aiming; t21 - the center of the "imaginary" ring;

22 is a tangent line to the resultant trajectory L _h at point 18 located in the plane of ring 3 (Fig. 9);

T23 - reflection point on the front plane of the shield 1; m23 - the point of change in the direction of flight of the center 7 of the ball 6 in the plane of the "imaginary" shield 2; p23 is the aiming point on the front plane of the shield 1 (figures 16 and 17);

24 and 25 - the coordinates of the centers of the standard and "imaginary" shields at the level of the upper plane of the ring;

D _VN - the inner diameter of the ring; ₁ and ₂ are the horizontal coordinates of the aiming and reflection points on the front plane of the shield; l _r and l _in - horizontal and vertical distances between the aiming and reflection points on the front plane of the shield; l _to - the smallest distance from the front plane of the shield to the nearest point of the inner diameter of the ring; z ₁ and z ₂ are the vertical coordinates of the aiming and reflection points on the front plane of the shield;

26 - sighting rectangle on the front plane of the shield 1;

27 is the aiming area on the front plane of the shield 1 of standard basketball equipment for the maximum volumetric and angular size of the goal Ω;

p27 is the center of the aiming area 27 in the form of a colored circle with a diameter of ~ 4 cm;

27.1 - the coordinate of the center 18 'of the ring 3 in the aiming area 27 (figures 19 and 22);

27.2 - coordinate of the center of the "intermediate" target on the front plane of the shield 1, duplicating the center 16 of the "extended" target 9 in the middle plane P _{to the} standard ring for the maximum value of Ω;

27.3 and 27.5 are the coordinates of the horizontal axis of the ellipse-like aiming area 27 for the maximum value of Ω standard equipment;

27.4 and 27.6 are the coordinates of the vertical axis of the ellipse-like aiming area 27 for the maximum value of Ω standard equipment;

28 - area of reflection of the "flow" of the trajectories of the maximum volumetric-angular size of the target ("intermediate" goal for throws with a rebound of the ball from the standard shield);

p28 is the center of reflection area 28 in the form of a colored circle with a diameter of ~ 4 cm;

28.1 - coordinate of the center 18 of the ring 3 of the vertical axis of the "intermediate" target 28;

28.2 - the coordinate of the center of the "intermediate" goal 28, duplicating the center 16 of the "extended" goal 9;

28.3 and 28.5 are the coordinates of the boundary points of the horizontal axis of the ellipse-like reflection area 28;

28.4 and 28.6 are the coordinates of the boundary points of the vertical axis of the ellipse-like reflection area 28;

29 - the area of aiming on the front plane of the standard shield 1 "flow" of economical ball flight paths when performing free throws with the ball rebound from the shield;

p29 is the center of the aiming area 29 in the form of a colored circle with a diameter of ~ 4 cm;

29.1-29.6 - the coordinates of the centers of the "clean" and "expanded" areas, as well as the coordinates of the boundary points of the vertical and horizontal axes of the ellipse-like aiming area 29;

30 - area of reflection on the front plane of the standard shield 1 of the “flow” of economical ball flight paths when performing free throws with the ball bouncing off the shield;

p30 is the center of the reflection area 30 in the form of a colored circle with a diameter of ~ 4 cm;

30.1-30.6 - the coordinates of the centers of the "clean" and "expanded" areas, as well as the coordinates of the boundary points of the vertical and horizontal axes of the ellipse-like reflection area 30;

31 is the aiming area on the front plane of the shield 1 for the maximum volumetric and angular size of the goal Ω of the upgraded basketball equipment;

p31 is the center of the aiming area 31 in the form of a colored circle with a diameter of ~ 4 cm;

31.1-31.6 - the coordinates of the centers of the "clean" and "expanded" areas, as well as the coordinates of the boundary points of the vertical and horizontal axes of the ellipse-like aiming area 31 of the upgraded basketball equipment;

32 - the reflection area on the front plane of the shield 1 for the maximum volumetric and angular size of the goal Ω of the upgraded basketball equipment;

p32 is the center of reflection area 32 in the form of a colored circle with a diameter of ~ 4 cm;

32.1-32.6 - - the coordinates of the centers of the "clean" and "expanded" areas, as well as the coordinates of the boundary points of the vertical and horizontal axes of the ellipse-like reflection area 32 of the upgraded basketball equipment;

33 - aiming area on the front plane of the shield 1 of the “stream” of economical ball flight paths when performing free throws with the ball rebound from the shield of the upgraded basketball equipment;

p33 — center of aiming area 33 in the form of a colored circle with a diameter of ~ 4 cm;

33.1-33.6 - the coordinates of the centers of the "clean" and "expanded" areas, as well as the coordinates of the boundary points of the vertical and horizontal axes of the ellipse-like aiming area 33 of the upgraded basketball equipment;

34 - reflection area on the front plane of the shield 1 of the "flow" of economical ball flight paths when performing free throws with the ball rebound from the shield of the upgraded basketball equipment;

p34 is the center of the reflection area 34 in the form of a colored circle with a diameter of ~ 4 cm;

34.1-34.6 - the coordinates of the centers of the "clean" and "expanded" areas, as well as the coordinates of the boundary points of the vertical and horizontal axes of the ellipse-like reflection area 34 of the modernized basketball equipment;

35 - vertical “columns” of aiming and reflection for free throws with the ball rebound from the standard shield;

36 - vertical “columns” of aiming and reflection for free throws with a ball rebound from the shield of the modernized basketball equipment;

37 - vertical axis of aiming in the form of a colored self-adhesive tape on the front plane of the shield;

38 - a single area of aiming and reflection - an ellipse-like segment belonging to the reflection areas 28 and 30 of standard basketball equipment;

39 - contours of the "reduced" goals 11 of the modernized 4 and "imaginary" 5 'rings.

For the theoretical justification of the method, the above-described geometric model of interaction of the basketball 6 with the rim of the standard ring 3 and the modernized ring 4 for shots without rebound and with the ball rebound from the shield 1 is used. The rim of the ring 3 increases along the radius of the section by the value of the ball radius R _m and turns into a torus 13, and ball 6, respectively, by the same amount decreases in radius and turns into a material point 7 (center of the basketball) - figure 3.

The figure 1 presents the main parameters of the flight paths of the ball, and the figure 2 - "attacked" targets in the middle plane of the ring. The presented parameters and designated goals were used in mathematical modeling of the ball’s flight and its interaction with the rim of a standard or upgraded ring and with the face of the shield when performing a free throw without rebound or with the ball rebound from the shield.

To study the proposed model, the following assumptions are made:

- The “flow” of the flight paths of the center (material point) 7 of the ball 6 is replaced by the “flow” of tangents to these paths at the point of intersection of the ring plane 3. The path lines of the flight center of the ball and their tangents practically coincide in a section of no more than the ball radius from the average the plane of the ring (Fig. 9).

- The maximum size of 73.4 cm of the "flow" of the flight paths of the center 7 of the ball 6 along the normal is taken equal to the diameter of the "reduced" target 11 (Fig. 2).

- The rebound of the ball 6 (material point 7) from the rim of the ring 3 (the surface of the torus 13) is considered an absolutely elastic rebound occurring in a straight line, which ensures the fulfillment of the law "the angle of incidence is equal to the angle of reflection".

Figures 4-11 show the simulation results that reveal the evolution of the “attacked” target in the profile projection of the geometric model of the interaction of the center 7 of the ball 6 with the surface of the torus 13 when the entry angle is changed from 0 to 90 degrees on standard basketball equipment. The calculations and geometric constructions of the motion parameters of the center of the ball and the corresponding parameters of the “clean” and “extended” goals are presented for the following initial parameters of the ball release: - S - the distance from the projection onto the surface of the basketball court of the center of the basketball hoop to the projection onto the surface of the court of the ball release point is 4.225 m (free throw line); - H is the height from the point of release of the center of the ball to the middle plane of the level of the ring is 1 m; - V ₀ - the initial speed of the ball in the recommended range reached 7-8 m / s; - α ₀ - the angle of release of the ball when performing a free throw in the recommended range of variation of the flight paths of the ball took values from 51 ° to 60 °.

After carrying out these calculations and geometric constructions, it was established:

- the width of the "flow" of the flight paths of the ball in the profile projection varies from 26.4 cm for α _in = 0 ° (the "flow" of non-productive flight paths of the ball L _m in Fig. 4) to 73.4 cm for α _in = 90 ° (the sum of the “flows” of the ineffective flight paths of the ball PB to the near arc and the AP to the far arc plus the sum of the “flows” of the effective flight paths of the ball: - H of the “clean” hit; - B and D for the near and far arcs of the ring with a bounce of the center of the ball from the surface of the torus in figure 11); the width of the "flow" of the flight paths of the ball in the profile projection for an entry angle of 90 ° is 2.78 times greater than the width of the "flow" for 0 °;

- the area of the "reduced" target 11 in the plane of the ring in figure 2 is F _pc. = 4231 cm ² , and the area of the "flow" PB of ineffective flight paths of the center of the ball in figure 4 is 1788 cm ² , which means that they differ by 2.37 times.

In this range of changes in the area of “flows” of productive and ineffective flight paths of the ball, the parameters and parameters of the “clean” and “expanded” targets in the middle plane of the ring level are determined.

At an angle of entry of 20 degrees, the first resultant trajectory L _1p appears, at which the center of the ball bounces off the surface of the torus at point D _{1 of the} far arc along the path L ₁ , then is repeatedly reflected from the surface of the torus 13 of the nearest arc at point B _1-2 and passes into the ring along the path of L ₂ (Fig. 5). It should be noted that the value for this trajectory is a constant for various coordinates of the ball release point - N m (Fig. 1). This constant defines the lower boundary of the possible parameters of the ball flight paths and is used in the design of technical means of throwing basketball players. The coordinate of the intersection point 14 of the ball’s effective trajectory with the middle plane of the ring P _k is located at a distance of about 15 cm from the geometric center 18 of the ring 3. In this case, the coordinate of point 14 coincides with the coordinate of the center of the “extended” target 16 in the plane of ring 3. With a gradual increase α _{in the} coordinate of the center of the “expanded” target 16 moves to the center of the ring 18 (decrease x _s ) and coincides with it at α _in = 90 ° (achieving symmetry of the areas near and far arcs of the “extended” target 9).

The “flow” of the PB of non-productive trajectories of the ball to the near arc has a maximum width of 26.4 cm. The “flow” of the PD of non-productive trajectories of the ball to the far arc for a given entry angle is 16.1 cm, and the rest of the “flow” forms a combined “stream” of overlap PN, which is cut off by the surface of the torus 13 of the near arc and has a width of 10.3 cm. Elimination of ineffective flight paths of the ball by structural elements of basketball equipment and elements of the devices created is successfully used in the design of technical FIR means to improve accuracy basketball throws [US patent 5800290 (Richard E. Barry, 1998); Patent 20140092253 (Alan W. Marty, 2014)]. The inner radius of the torus has a constant value of 10.3 cm for the maximum value of the “clean” target 8 in the middle plane of ring 3 with α _in = 90 ° (figures 2 and 11).

A further increase in the angle of entry to α _in = 23 ° leads to the disclosure of the target in the form of a pointed ellipse-like figure with a width of 1.9 cm "flow" D of the productive flight paths of the ball in the profile projection (Fig. 6). An ellipse-like figure refers to the area of the far arc and is one of the three components of the “expanded” target of the ring. The width of the "flow" of the PB (26.4 cm) remained unchanged, and the width of the "flow" to the far arc is the sum of the "flows" of ineffective trajectories of the PD (16.5 cm) and PN (8.0 cm) and the "flow" of the resultant ball flight paths D (1.9 cm), which is 26.4 cm in total. A characteristic feature of the described interaction is that the boundary effective trajectory tangent to the surface of the torus 13 of the near arc at point B _k bounces off the surface of the torus of the far arc at the level of the middle plane P _{to the} ring 3 at point 15. The coordinates of point 15 have a constant different values: the point is located at a distance of R _m = 12.2 cm from the rim of the far arc. An additional confirmation of the constancy of the coordinates of point 15 is a segment of a straight line from point 15 to the center 18 of ring 3, which is 10.3 cm.

A further increase in α _in to 35 ° leads to the appearance of the first "clean" path of the ball center L _1h (Fig. 7). The path of a “clean” hit passes through the center of the ring 18 with touching the surface of the torus of the near and far arcs at points B _k and D _k, respectively, without changing the direction of flight of the center 7 of ball 6 (the appearance of the coordinate of point 17, which in this case coincides with the coordinate of the center of the ring 18). The boundary path L _{0 of the} effective “flow” D after the interaction of the center of the ball with the surface of the torus of the far arc at point D ₁ changes its direction of flight to the path L ₁ with subsequent reflection from the surface of the torus of the nearest arc at point B _1-2 . After the second bounce, the center of the ball moves along the path of L ₂ , and after the third bounce at point D _1-k , it passes into the ring along the path of L ₃ . The width of the “flow” of the PB remains unchanged at 26.4 cm. The width of the “flow” to the far arc is divided: - into the “flow” D of the effective paths of the ball center 8.5 cm wide, aimed at the arc (D ₁ -D _to ) of the surface torus 13; - and on the “flow” of PD of non-productive trajectories with a width of 17.9 cm, directed to the arc (D ₂ -D ₁ ) of the surface of the torus 13. In this “stream” of PD there is a “stream” of non-resultant trajectories directed to the arc (D ₂ -D _2-1 ). The center 7 of the ball 6 of these paths does not reach the surface of the torus 13 of the arc (D ₂ -D _2-1 ), but changes the direction of its movement in the plane of the "imaginary" shield 2. This effect indicates that the main (1) and "imaginary" (2) the shields act as a corrector-cutter of the ball's flight paths. The sector ∂ of productive trajectories with the ball bouncing off the surface of the torus in the far-field region of the profile projection of the ball’s flight was 69 °.

The displacement of the center 16 of the "extended" target relative to the center 18 of the ring 3 decreased and amounted to x _c = -7.3 cm.

An increase in the entrance angle to 39 ° is accompanied by the disclosure of a “clean” target 8 and the formation of three component areas of the “expanded” target 9: - area of a “clean” hit (H); - the region of the near (B) and far (D) arcs of the ring (Fig. 8). The center of the “clean” goal coincides with the geometric center 18 of the basketball hoop. The boundary paths tangent to the surface of the torus 13 at points B _k and D _k intersect with the middle plane P _{k of the} ring 3 at points 17 and 19, symmetrically located relative to the center 18 of the ring 3. The distance between these points shows what amount of “clean »The goal in the direction of flight of the ball economical trajectories (E _to - minimum; Fig. 14). The “flow” of non-productive paths of the PB to the region of the near arc has decreased, and its width is 25.9 cm. The “flow” B of 0.5 cm wide represents the productive paths to the surface of the torus of this region, which after reflection pass into the ring. The productive sector b, arcs (B _to -B ₁ ) is 16 °. The “flow” of ineffective trajectories of the PD to the region of the far arc increased slightly, compared with the same “flow” for α _in = 35 ° and its width is 18.5 cm. In this “flow” there are trajectories directed to the arc (D ₂ - D _2-1 ) and the center of the ball in which it changes its direction of movement in the plane of the shield 2. Shields 1 and 2 act as a corrector-cutter of the ball's motion paths. The width of the “flow” D of productive trajectories decreased, which is 7.9 cm. The sum of “flows” (D + PD) = (B + PB) = 26.4 cm. The productive sector ∂ of the arc (D ₁ -D _k ) decreased and is 66 °. The displacement of the center 16 of the "extended" target 9 relative to the center of the ring 3 decreased and amounted to x _c = -6.0 cm

An increase in the entrance angle from 39 ° to 54 ° is accompanied by a further increase in “clean” 8 and “expanded” 9 areas: from 64 to 226 cm ² and from 350 to 454 cm ^2, respectively (figures 2, 14 and 15). The value α _in = 54 ° of the productive paths of the ball is constant for different coordinates of the ball release point - N m (Fig. 9). At a given entry angle, the volumetric-angular size of the target takes the maximum value for the indicated free throw parameters. The constancy of the angle of entry is used in the design of technical means and the development of methodological techniques, methods and techniques for increasing the effectiveness of free throws in basketball. According to the results of theoretical and experimental studies (Pritykin, V.N. Non-traditional approaches to improving the accuracy of the free-throw in basketball: monograph / V.N. Pritykin. - Omsk: Publishing House of Omsk State Medical University, 2015. - 175 p. See S.99 .) the indicated range of ball flight parameters is justified as the recommended of all possible parameters of the ball movement when performing free throws in basketball. The width of the "flow" of ineffective trajectories of the PB is 25.0 cm, and the width of the "flow" of productive trajectories B is only 1.4 cm (Fig. 9). The relevant parameters for the far arc: - PD = 19.4 cm; - D = 7.0 cm. These values of the "flows" indicate the numerical advantage of the far arc of the basketball hoop over the nearest arc. The region of the far arc of the productive trajectories is 4 times larger than the region of the near arc for α _in = 54 ° and 13 times for α _in = 39 °. Pedagogical observations confirmed the theoretical calculations: the number of hits after the ball bounces from the rim of the far arc is 2-7 times more than after the ball bounced from the rim of the near arc. Accordingly, the sizes of productive sectors b (26 °) and ∂ (62 °) differ. The displacement of the center 16 of the "extended" target relative to the center of the ring 3 decreased and amounted to x _c = -3.5 cm.

The figure 10 shows the interaction of the center 7 of the ball 6 with the surface of the torus 13 at α _in = 75 °. The choice of this angle of entry is justified by the fact that when performing a free throw with the ball bouncing off the shield 1, only half of the reflection area is placed on the front plane of the shield. The remainder of the area notionally situated above the upper side of the shield, above the visual-spatial reference point (a-a) in Figures 12-15. This circumstance serves as evidence of the correctness of the choice of the upper limit for the practically possible parameters of the flight of the ball for free throws in basketball. A possible range of ball movement parameters during free throws is from α _in = 20 ° to α _in = 75 °, and the recommended range is from 39 ° to 54 °.

For α _in = 75 °, the width of the “streams” of productive trajectories is equal to: - D = 4.0 cm; - H = 19.0 cm; - B = 1.6 cm. The width of the "streams" of ineffective trajectories is: - PD = 22.4 cm; - PB = 24.8 cm. Dimensions of productive sectors: - ∂ = 46 °; - b = 28 °. The shift of the center 16 of the “expanded” target 9 relative to the center 18 of the ring 3 decreased and amounted to x _c = -1.3 cm.

The figure 2 presents a horizontal, and figure 11 profile projection of a geometric model of the interaction of the center 7 of the ball 6 with the surface of the torus 13 on a standard basketball equipment. The diameter of the “clean” target 8 is 20.6 cm, and the diameter of the “expanded” target 9 is 31.2 cm. The width of the ineffective “streams” is: PD = PB = 21.1 cm. The sizes of the productive sectors are: ∂ = b = 53 °. The center 18 of the ring 3 and the center 16 of the "extended" goal 9 coincided.

When throwing the ball at a target on the wall, it is advisable for the athlete to aim at the center of the target so that the aiming and defeat points coincide (a qualitative explanation is borrowed from other sports: darts - https://www.sports.ru/tribuna/blogs/sibdarts/511531. html; handball - https://infourok.ru/dbuchenie-tehnike-broskov-v-gandbole-2068303.html). In our case, the aiming point can be a “floating” center 16 of the “expanded” target 9, whose location in the horizontal middle plane of ring 3 varies within 10.3 cm from point 15 to point 18. The center of this range is 10.3: 2 = 5.15 cm from the center 18 of the basketball hoop 3. For the recommended range of changing the flight parameters of the ball without bouncing it off the backboard, the offset x _s (center of the “extended” goal) takes values from (- 3.5) cm to (- 6.0 ) cm from the center ring 18. center displacement x _from the recommended range is obtained based on prime p counts: (6.0-3.5 = 2.5 cm; 2.5: 2 = 1.25 cm; 3.5 + 1.25 = 4.75 cm or 6.0-1.25 = 4, 75 cm). The approximate coincidence of the coordinates (5.15 cm versus 4.75 cm) of the established ranges for the center 7 of the ball 6 to arrive in the middle plane P _{to the} ring 3 of productive trajectories proves the possibility of choosing one aiming point for free-throw shots in basketball without rebounding the ball from the backboard. Considering the proven advantage of the far arc over the near one, the aiming point is proposed to be a fixed point on the ring rim - the center of the far arc 10. The boundary point 10 is located in front of the mounting bracket of the ring, into which the "flow" of ineffective ball flight paths is directed. This circumstance becomes a deterrent, preventing the athlete from throwing the ball for the “attacked” goal. The concentration of attention on the center of the far arc of the ring helps the basketball player to fulfill the proposed rule: "touching the near arc with the ball is considered a gross mistake." These actions achieve the translation of the ball’s flight paths from ineffective “flows” to the near and far arcs to the productive “flows” B, C and D. During the training sessions, the center 10 of the far arc of the ring is periodically indicated by a colored self-adhesive tape, and the aiming point is clearly visible from the free throw line.

All of the above advantages of the far arc over the near arc theoretically and experimentally prove that the aiming point - the center of the far arc 10 (figures 8, 9, 14 and 15) was reasonably chosen as the object of aiming for shots without bouncing the ball from the backboard.

The modeling of the flight paths of the ball and its interaction with the ring rim showed that each trajectory has its own “hit” target in the plane of the ring. Four options were chosen from the considered options (figures 4-11) for changing the trajectory parameters, which convincingly illustrate the main dependence of the “extended” and “clean” goals on the initial parameters of the ball during a free throw in basketball (figures 12, 13, 14 and 15 )

The figure 12 shows the trajectory at which the target appears in the middle plane of the ring. The target, whose area is 6 cm ² , is located ~ 15 cm from the geometric center of the ring closer to the far arc of the ring (table 1). The maximum point of the trajectory is below the visual-spatial reference point (rr), which passes through the top line of the sighting rectangle. It is extremely difficult to make an effective free throw along such a trajectory, and a “clean” throw is impossible.

When the release angle α ₀ = 48 °, a “clean” target 8 with an area of 3.4 cm ² appears, and the area of the “expanded” target 9 is already 292.4 cm ² (Fig. 13, table 2). The center of the "extended" goal 16 is located at a distance of -7.3 cm from the center 18 closer to the far arc of the basketball hoop 3.

The figure 14 presents the economical trajectory of the ball and the corresponding "expanded" target with an area of 350.5 cm ² , while the "clean" goal in its composition is 64.4 cm ² (table 3). The shift of the center of the “extended” target to the far arc is 6.0 cm. The ball flies between the two visual-spatial landmarks ( aa ) and (rr) at the maximum point of the trajectory.

The figure 15 presents the parameters of the ball, which provide the maximum value of the volume-angular size of the target. The coordinates of the maximum point of the trajectory above the surface of the site is 4.2 m and it is located above the visual-spatial reference point ( aa ). The area of the "expanded" goal is 454.4 cm ² , and the area of the "clean" goal is half of it (226.2 cm ² ). The offset x _with is equal to 3.5 cm (table 4).

The athlete must equally skillfully control the basketball shots without a rebound and with the ball rebound from the shield.

We proceed to consider the results of modeling basketball shots with the ball rebound from the shield. Figures 16 and 17 show a horizontal and profile projection of determining the flight parameters of the ball and the coordinates of aiming and reflection when it interacts with the front plane of the shield 1 (modification of the borrowed figures 2 and 3 of patent 2386466 C2 of the Russian Federation, IPC 7A 63B 63/08). The figure 16 shows that the continuation of productive trajectories beyond the plane of the shield intersect at point t21, which is the center of the "imaginary" ring 5. The center 18 of the main ring 3 is symmetrically located to the center t21 of the "imaginary" ring 5 relative to the "imaginary" shield 2, in the plane of which the center 7 of the basketball 6 changes its direction of movement after interacting with the board 1. The distance between the centers 18 and t21 is 506 mm, and the center t21 is 131 mm from the front of the backboard 1. To the back of the backboard 1 using p Ubtsy and bracket fixed vertical axis of the aiming 21 in the center of the "imaginary" ring 5 (figure 17). The basketball player takes aim at point p23, and controls the execution of the shot by the coordinates of point t23. The ball release point _Oy and the vertical axis of aiming 21 create a vertical plane of movement of the center 7 of the ball 6, while the axis is the main visual reference. The method for determining the aiming coordinates when throwing with the reflection of the ball from the shield is described in more detail in patent 2386466 C2 of the Russian Federation, IPC7 A63B 63/08.

Figures 18 and 19 show geometric models of a free throw with the reflection of the ball from the backboard for a range of recommended ball flight paths performed on standard basketball equipment. With free throws with the ball bouncing off the shield, the near and far arcs of ring 3 change the name and purpose to the opposite. The movement of the center 7 of the ball 6 in the figure 18 passes along economical trajectories. When the ball is reflected from the front plane of the shield 1, the center 7 changes the direction of its movement in the plane of the “imaginary” shield 2 and, for example, flies along the path L ₀ 'to the point 15' to hit the ring 3 along the path L ₁ 'after bouncing from the surface torus 13 of the far arc (D ₁ '-D _to '). The width of the "flow" D 'is 7.9 cm, and the width of the "flow"B' is 0.5 cm. These values coincide with the values of the "flows" D and B when performing economical free throws without rebounding the ball from the backboard (Fig. 8 ) The “flow” of the PB 'of non-productive trajectories (25.9 cm) of the center of the ball flying to the near arc (B ₂ ' -B ₁ '; 164 °) significantly decreased by the "flow" of ineffective trajectories that should interact with the arc (B ₂ ' -B _1-2 '; 134 °). The elimination of these trajectories occurred due to the presence of the main and “imaginary” shields, which serve as a corrector-cutter when the athlete selects free throws with a ball rebound from the shield. The "flows" D ', H' and B 'of the resulting trajectories form a reflection area 30 (270.59 cm ² ) in the plane of the "imaginary" shield 2, which is projected onto the front plane of the shield 1 (Fig. 23). The productive sector of the near arc b 'is 16 °, which is 4 times less than the productive sector ∂' which is 66 °. The displacement of the center 16 'of the "extended" target relative to the center 18 of the ring 3 is 6.0 cm. The presented results for a free throw with the ball bouncing off the shield convincingly confirm the advantage of the far arc over the near arc in the increased dimensions of the "streams" of the productive flight paths of the ball. The center 30.2 of the “extended” target 30 (Fig. 23) is located in the reflection area 28 (631.30 cm ² ; Fig. 22, tables 5, 6), at the same time, the center p27 of the aiming area 27 ~ coincides with the center by 80 percent p30 of the reflection area of 30 economical trajectories (Fig. 30), which greatly facilitates the athlete to aim the free throws with the ball rebound from the shield.

An object of aiming for shots with the ball bouncing off the backboard on standard basketball equipment is recommended to use a 4-cm diameter circle p29, which is located between points 29.1 and 29.2 on the front plane of the backboard 1 (figures 18 and 23, tables 7, 8).

A basketball player performing a free throw with the ball bouncing off the backboard needs to aim at a colored circle p29 with a diameter of 4 cm, which is located on the front plane of the backboard 1 and is projected onto the vertical aiming axis 21. The maximum point of the ball’s flight path above the level of the ring should be between the visual reference points ( aa ) and (rr). Visual control of the quality of the execution of a free throw is carried out by determining the coordinates of the rebound of the ball with respect to the center p30 of the reflection area 30 (Fig. 23). Deficiencies of the ball during the shots with a rebound from the backboard, first of all, belong to the “flow” of the ball (the width of 18.5 cm is conditionally shown in figure 18) of ineffective trajectories that come into the area of the mounting L-shaped bracket of the basketball hoop 3. The indicated “stream” of the ball conditionally replaces the eliminated "flow" PB 'to the arc B ₂ ' -B _1-2 '(134 °). The presented simulation results for a free throw with the ball bouncing off the backboard convincingly confirm the advantage of the efficiency of throws with a bounce from the far arc compared with the effectiveness of throws with a bounce from the near arc. This circumstance and the presence of “flows” of ineffective flight paths of a basketball on both arcs require a theoretical justification for the modernization of basketball equipment in order to increase the efficiency of free throws.

размеров «потоков» результативных траекторий полета мяча, приходящих в область дальней дуги по сравнению с «потоками», направляемыми в область ближней дуги.Figure 19 represents the parameters of free throws with the ball bouncing off the shield, performed along the productive flight paths of the center 7 of the ball 6, which correspond to the maximum values of the volumetric-angular size Ω of the “target”. The angle of release α ₀ for these trajectories is 9 degrees higher than that of economical trajectories (figures 14 and 15). The maximum point of the ball’s flight paths is located above the visual-spatial reference point ( aa ). In the plane of the “imaginary” shield 2, a change in the direction of motion of the material point 7 occurs, which then moves to the center 18 of the ring 3. The remaining productive trajectories of the “flows” D ', Ch' and B 'create a reflection area 28 in the plane of the "imaginary" shield 2 ( Fig. 22), which is projected onto the front plane of the main shield 1. The width of the "flow" of the PD 'of ineffective trajectories is 19.4 cm. The width of the "flow"D' is 7.0 cm and the width of the "flow" B 'is 1, 4 cm. These values coincide with the values of “flows” D and B when performing free throws without rebounding the ball the shield for maximum volumetric angular dimensions "amazingly" target Ω (FIG. 9). The sector ∂ 'of the productive trajectories of the far arc is 2.38 times larger than the sector b' of similar trajectories of the near arc. The offset of the center 16 'of the "expanded" target relative to the center 18 of the ring 3 is 3.5 cm. The results of modeling free throws with the ball bouncing off the backboard for an entry angle of 54 degrees (Fig. 19), similar to the results of throwing for an entry angle of 39 degrees (Fig. 18 ) They confirm the advantage of throwing performance from a far arc based on

the dimensions of the “flows” of the productive flight paths of the ball coming into the region of the far arc compared with the “flows” directed to the region of the near arc.

The width of the “flow” of PD of non-productive trajectories for the recommended range (α _in = 39-54 °) of free throws without rebounding the ball from the backboard varies from 18.5 to 19.4 cm (figures 8 and 9) and represent the ball’s flights in terms of their parameters the relation of “attacked” targets in the plane of the ring 3. The arrival of the center of the 7 ball 6 of ineffective trajectories L _{m of} this “stream” leads to a change in the direction of its movement on the arc D _2-1 -D ₁ torus 13 when the ball bounces from the far arc of the ring and partially from the horizontal surface of the L-shaped mounting bracket and, respectively, to Romakh free throw. Trajectories of the “flow” directed to the arc D ₂ -D _2-1 lead to a change in the direction of movement of the center of the ball in the plane of the “imaginary” shield 2, rebound of the ball from the shield 1 and to the miss of the free-throw. The center of the ball 7 of this part of the “stream” does not reach the surface of the torus, which is an example of the cutting off of the paths of the “stream” of APs directed to the arc D ₂ -D _2-1 with a width of 1.9 cm by basketball equipment (main and "imaginary" shields) Fig. 8) and a width of 3.4 cm (Fig. 9).

Thus, between the productive “flows” of B, H and D free throws without rebounding the ball from the backboard (figures 8 and 9) and the resultant “flows” of B ', Ch' and D '(figures 18 and 19) of the shots with the ball bouncing off there are “streams” of non-resultant trajectories of the PD 18.5 cm wide (Fig. 8) and 19.4 cm wide (Fig. 9), which can be eliminated by constructive change of the basketball hoop. The transfer of the front plane of the shield 1 (the rejection of the horizontal plate of the mounting L-shaped bracket) closer to the center 18 of the ring 3 solves this problem. It is proposed that the aiming point 10 of the basketball hoop 4 be placed in the front plane of the shield 1.

For shots with the ball bouncing off the backboard, the basketball player needs to know the “intermediate” goals on the front plane of the backboard. Obtaining areas of aiming and reflection, centers of these areas is carried out by geometric constructions using the functionality of the CAD system "COMPASS" (figures 20 and 21). The figures show an example of a cross section of the “flow” of the trajectory of the ball into the “imaginary” ring 5 ', symmetrical to the modernized ring 4 relative to the “imaginary” shield 2. The section BB shows the aiming area 31 obtained as a result of the cross section of the considered “flow” the plane of the main shield 1. On section AA shows the reflection area 32, which is obtained by the "flow" section of the plane of the "imaginary" shield 2 and transferred to the parallel plane of the shield 1. Figures 22 and 23 show the aiming areas (27 and 29), the reflection (28 and 30) and the coordinates of the centers (p21 and p29; p28 and p30) of these areas, which are located on the front plane of shield 1. Tables 5, 6, 7 and 8 show the coordinates of the limit points along the horizontal and vertical axes with an indication of the total aiming area and reflection of the recommended range of trajectory parameters that are created by the flight of the ball with free throws on standard equipment. Combining reflection areas 28 and 30 with drawing vertical lines through the coordinates of points 28.3 (y = 11.8 cm) and 30.3 (y = 11.6 cm); 28.5 (y = -11.8 cm) and 30.5 (y = -11.6 cm) to the top of the shield 1 forms a kind of vertical “column” of aiming and reflection 35. The resulting “column” 35 is supplemented by centers p27 and p29 aiming areas 27 and 29, as well as centers p28 and p30 of reflection areas 28 and 30. In the middle of the “column”, the vertical axis of aiming 37 is applied. All elements of the aiming and reflection object are applied using a 1 cm wide self-adhesive tape. Aiming and reflection centers are made in the form of colored circles with a diameter of ~ 4 cm. Ellipse-like segment 38 (total oschad reflection objects 28 and 30) is executed in color (Fig. 30).

For a qualitative assessment of the decision on a constructive change in the basketball equipment, mathematical modeling was carried out in the recommended range of ball flight parameters when performing a free throw [figures 24, 25, 26 (tables 9, 10), 27 (tables 11, 12), 28 and 29].

The ring modernization ensured that the vertical aiming axis 37 in the form of a self-adhesive adhesive tape 1 cm wide with the front plane of the shield 1 coincided. The plane of the "imaginary" shield 2 is located at point 15 and, accordingly, the distances (10-15) and (15-18) are not equal . When determining the location coordinates of the "imaginary" ring 5 and the "imaginary" backboard 2 for standard basketball equipment, it was found that the centers of the main and "imaginary" rings are symmetrical relative to the "imaginary" backboard (Fig. 16). In figures 24 and 25, the difference in the distances (10-15) and (15-18) is: 12.2 cm - 10.3 cm = 1.9 cm. When the ball 6 hits the plane of the shield 1, the radius of the ball decreases by the amount of deflection of the shell the ball. For the recommended range of ball flight parameters and for the horizontal component of the ball flight speed U _k = 3.55-4.41 m / s, the deflection value varies from 1.1 to 1.4 cm. Calculations are presented in the article: Pritykin, V.N. The use of acoustic wave sensors to determine the coordinates of the reflection of the ball from the shield during basketball throws / V.N. Pritykin, Yu.G. Dolganev // Modern problems of science and education. - 2015. - No. 1-2; URL: www.science-education.ru/125-19893 (accessed: 06/25/2015). Thus, the distance between shields 1 and 2 decreases by the amount of deflection of the shell of the ball and therefore the difference in distances (10-15) and (15-18) takes values in the following range: - (1.9 cm - 1.4 cm = 0, 5 cm); - (1.9 cm - 1.1 cm = 0.8 cm). These values (0.5-0.8 cm) are (4.1-6.6%) of the ball radius, and therefore the obtained deviations in the symmetry of the centers 18 and t21 of rings 4 and 5 relative to the plane of the "imaginary" shield can be neglected.

The width of the effective “flows” of economical free throw trajectories without rebounding the ball from the shield onto standard equipment is 11.6 cm (Fig. 8), and the area of the “expanded” target in the ring plane is F _r.c. equal to 350.48 cm ² (figure 14 and table 3). The value of the area of the “expanded” area in the plane of the ring obtained as a result of geometric construction is 357.89 cm ² , which was taken as the main option for further calculations. The width of the indicated “streams” of shots with the ball bouncing off the backboard is also 11.6 cm (Fig. 18), and the area of the “extended” goal 9 is 357.89 cm ² , but its location is rotated 180 degrees with respect to the area for the shots without rebounding the ball from the shield. The sum of the effective “flows” of the flight paths of the throwing ball without bouncing and with the ball bouncing off the backboard for standard equipment is 23.2 cm, but these “streams” are divided by an ineffective “stream” 18.5 cm wide, which is directed to the area of the mounting bracket. The movement of the center 7 of ball 6 in this “stream” means the missile’s missile for throwing bounces and the ball’s flight when throwing without throwing the ball from the shield of the “attacked” target in the plane of the ring.

значение суммы площадей «атакуемых» целей в плоскости кольца у модернизированного оборудования (на 127,24 см²) по сравнению с баскетбольным стандартным оборудованием способствуют повышению результативности штрафных бросков на модернизированное оборудование. Дополнительным положительным фактором для штрафных бросков на модернизированное оборудование является примерное совпадение координат точки прицеливания 10 при выполнении бросков без отскока мяча от щита с координатами центра 33.2 площади прицеливания 33 (фиг. 27). Разница между вертикальными координатами данных точек составляет 0,4 см. На основании вышеизложенного можно утверждать о наличии единого круга прицеливания диаметром ~ 4 см [(р31 и р33) - зеленый цвет; (фигуры 24 и 31)].As can be seen in figure 24, the effective “flows” of the free-throw trajectories of the two varieties for the modernized basketball equipment are combined into one “stream” and its total width is (B + C + D + C '+ D') = (0.5 + 3 , 2 + 4.9 + 8.0 + 8.0) = 24.6 cm. The width of this “stream” is 1.4 cm greater than the sum of similar “streams” for standard equipment (23.2 cm). The areas of “extended” goals for the upgraded equipment have the following meanings: - for throws without rebounding the ball from the shield F _r.c. = 281.12 cm ² ; - for throws with a ball rebound from the shield F _p.c. = 561.90 cm ² (Fig. 29). Comparing the sum of these areas for standard and modernized equipment, we obtain the following results: [(281.12 + 561.90) - (357.89 + 357.89)] = 127.24 cm ² . The lack of a gap between productive "flows", as well as

the value of the sum of the areas of the “attacked” targets in the ring plane of the upgraded equipment (by 127.24 cm ² ) in comparison with the standard basketball equipment contributes to an increase in the effectiveness of free throws on upgraded equipment. An additional positive factor for free throws on upgraded equipment is the approximate coincidence of the coordinates of the aiming point 10 when throwing without bouncing the ball from the ball with the coordinates of the center 33.2 of the aiming area 33 (Fig. 27). The difference between the vertical coordinates of these points is 0.4 cm. Based on the foregoing, it can be argued that there is a single aiming circle with a diameter of ~ 4 cm [(p31 and p33) - green; (figures 24 and 31)].

Continuing to analyze the results of modeling free throws without a rebound and with the ball bouncing off the backboard performed on standard and upgraded basketball equipment, we note that the trajectories for the maximum value of the volumetric-angular size of the goal have similar data that were identified for the economical flight paths of the ball with a free throw .

значение суммы площадей «атакуемых» целей в плоскости кольца у модернизированного оборудования (на 135,98 см²) по сравнению с баскетбольным стандартным оборудованием способствуют повышению результативности штрафных бросков на модернизированное оборудование. Положительным фактором для штрафных бросков на модернизированное оборудование является примерное совпадение координат точки прицеливания 10 при выполнении бросков без отскока мяча от щита с координатами центра 31.2 площади прицеливания 31 бросков с отскоком мяча от щита (фигуры 25 и 26). Разница между вертикальными координатами данных точек составляет 1,2 см. Вышеизложенное подтверждает наличие единого круга прицеливания диаметром ~4 см [(р31 и р33) - зеленый цвет; (фигуры 25 и 31)] для отработки штрафных бросков двух разновидностей, выполняемых в рекомендуемом диапазоне траекторий полета мяча на модернизированное баскетбольное оборудование.The width of the resultant “flows” without bouncing and with the ball bouncing off the backboard for standard equipment is 20 cm, but these “streams” are separated by a non-resultant “flow” 19.4 cm wide, which is directed to the area of the mounting bracket (figures 9 and 19). The resultant “streams” of the two-shot free throw trajectories for the modernized basketball equipment are combined into one “stream” and its total width is (B + C + D + C '+ D') = (1.4 + 11.6 + 2.5 + 14.1 + 7.0) = 36.6 cm, which is 3.4 cm less than the sum of productive streams for standard equipment. The value of the area of the "expanded" area in the plane of the ring, obtained as a result of geometric construction, is 456.04 cm ² , which was taken as the main option for further calculations. The areas of “extended” goals for the upgraded equipment have the following meanings: - for throws without rebounding the ball from the shield F _r.c. = 392.32 cm ² ; - for throws with a ball rebound from the shield F _r.ts. = 655.74 cm ² (Fig. 28). Comparing the sum of these areas for standard and modernized equipment, we obtain the following results: [(392.32 + 655.74) - (456.04 + 456.04)] = 135.98 cm ² . The lack of a gap between productive "flows", as well as

the value of the sum of the areas of the “attacked” targets in the ring plane of the upgraded equipment (by 135.98 cm ² ) in comparison with standard basketball equipment contributes to an increase in the effectiveness of free throws on upgraded equipment. A positive factor for free throws on upgraded equipment is the approximate coincidence of the coordinates of the aiming point 10 when throwing without bouncing the ball from the backboard with the coordinates of the center 31.2 of the aiming area of 31 shots with the ball bouncing off the board (figures 25 and 26). The difference between the vertical coordinates of these points is 1.2 cm. The above confirms the presence of a single aiming circle with a diameter of ~ 4 cm [(p31 and p33) - green; (figures 25 and 31)] for practicing free throws of two varieties, performed in the recommended range of flight paths of the ball on the modernized basketball equipment.

The aiming method when performing a free throw in basketball is implemented as follows. Two lateral stationary stations are equipped in the gym for throwing basketball players. Recovery coefficients for the data panels of stations are approximately equal. One station is equipped with standard basketball equipment, and an upgraded basketball hoop is installed at the other station. For a station with standard equipment, visual-spatial landmarks and aiming objects are prepared, as shown in Figure 30. A station equipped with a modernized ring is designed to practice free throws using visual-spatial landmarks and aiming objects, which are shown in Figure 31.

The structure of the aiming method for practicing free throws in basketball:

1. Testing of basketball players and analysis of experimental results.

2. Theoretical exercises using colored posters, manuals (figures 1 to 31).

3. Practical training of basketball players on equipped throwing stations.

4. Repeated testing of athletes, analysis of the results and the individual choice of the type of free throw (without rebound or with the ball rebound from the shield).

5. Practical training of basketball players, taking into account the individual recommendations of the coach with constant ongoing monitoring of the effectiveness of free throws in training and competitive activities throughout the sports season.

Claims (5)

1. The method of aiming when performing free throws in basketball, characterized in that mathematically simulate the flight paths of the ball and its interaction with the shield plane and the ring rim and using a geometric model in which the ring rim increases in cross section radius by the value of the radius of the ball and turns into torus, and the ball, respectively, by the same amount decreases in radius and turns into a material point, determine the parameters of the "flows" of the possible and recommended flight paths of the ball and the corresponding pairs Meters "attacked" - "clean" and "extended" targets plane of the ring for surges without rebound and bounce of the ball on the shield. 2. The method according to p. 1, which sets the parameters of the visual-spatial landmarks - horizontal planes along the upper side of the shield and the top line of the aiming rectangle, allowing to determine the maximum coordinate of the ball's flight path above the plane of the ring. 3. The method according to claim 1, in which for the shots without rebounding the ball from the shield, the aiming point is determined and justified - the center of the far arc located in the plane of the ring and in the plane of flight of the center of the ball passing through its release point. 4. The method according to p. 1, in which for the shots with the ball bouncing off the backboard, aiming and reflection objects are determined in the form of "intermediate" targets - the vertical aiming axis, ellipse-like aiming and reflection areas, data centers, a single aiming and reflection area, " columns ”aiming and reflecting on the front plane of the basketball backboard, ensuring the quality of aiming during training and improving the technique of making free throws in basketball on standard basketball equipment. 5. The method according to p. 1, in which, by changing the size of the structural elements of standard basketball equipment, reduce and eliminate the "flows" of ineffective flight paths of the ball, and also increase the total area of the "attacked" goals in the plane of the ring by combining the "hit" basketball goals throws without a rebound and with the ball bouncing off the shield; combining the goals of two types of throws provides a gradual transition from a horizontal “attacked” target in the plane of the ring to a vertical “intermediate” target on the front plane of the shield; this approach provides a theoretically sound development of shots with the ball bouncing off the shield, contributing to the high-quality individual selection of a more productive kind of free throw for use in competitive activities.

Images