Basketball attracts many people who love sports with its unique charm. Some scholars have studied the mechanical analysis of shooting aiming points. At the same time, they gave the trajectory of the basketball shot. Some scholars have studied the influence of moment of momentum on basketball shooting techniques [1]. They applied the Momentum Momentum Theorem and the Momentum Momentum Conservation Law to basketball shooting techniques and carried out an example analysis. Some scholars have studied the shooting percentage of basketball robots based on the normal cloud model [2]. At the same time, they propose a normal cloud model. They simulate the precision error of basketball robots in manufacturing design and conduct simulation experiments with MATLAB. Some scholars have studied the application of multiple regression in basketball shooting. Some scholars have studied the factors affecting the hit rate of the basketball spin ball [3]. Some scholars have studied the computer simulation problem of basketball's one-handed shooting technique. The author studies the influencing factors of two-point shooting and three-point shooting based on mathematics and mechanics.

In the text, _{1} is the horizontal distance between the two-point line and the front edge of the rim. _{2} is the horizontal distance between the three-point line and the front edge of the basket. _{0} represents the initial speed of the ball at the time of release.

We use the shot point as the coordinate origin to establish a plane rectangular coordinate system. _{1} is the horizontal distance between the two-point shot and the front edge of the rim. The ball moves in a straight line at a constant speed in the horizontal direction, while in the vertical direction, it is in free fall due to the acceleration of gravity [4]. The trajectory equation of the center of the ball:

Eliminating the parameter

Where

We treat _{1}, _{2}]. According to the properties of a continuous function on a closed interval, we find that its maximum value must exist [5]. The minimum value of
_{0 min}. So we transform the above problem into finding the minimum value of

We assume that
_{1} sec^{2} ^{2} _{1} tan

We use sec
^{2} _{1} + 2Δ_{1} sin^{2}

We can get _{1} sec^{2} _{1} tan^{2} ^{2}

Then the minimum throwing angle is

We bring Equation (3) into Equation (2) to get the best shot speed

Where

If the ball is hit on the back edge of the basket, the trajectory equation of the basketball is:

In the same way, we get the minimum ball angle and speed

Where

So = the best angle

In the case of a two-point shot, the ball height _{1} = 4.375_{1} +

Then we get the best shot speed

Δ_{1} = 4.375_{1} + _{0} between [7.29,7.91]_{0} ∈ [7.29, 7.91].

Since the shot is at the 2-point tee, the minimum 2-point shot release speed is required in this situation [6]. In fact, because of the influence of friction and any shooting point of the two-point shot is slightly farther than the free-throw line of the two-point shot, we can adjust the shooting speed appropriately according to the situation to improve the shooting rate.

_{2} is the horizontal distance between the three-point line and the front edge of the rim [7]. If the ball hits the front edge of the basket, we also use the calculation method of Case 1 to obtain the minimum angle and speed

Where

So the best angle

The ball height h _{2} = 6. 525_{2} +

According to the formula
_{2} = 4.375_{2} + _{0} of the three-pointer between [7.98, 8.47]_{0} ∈ [7.98, 8.47].

The calculation method of the two-point shot is easy to find that the best release angle

Then we get the best shot speed.

If the shooting point is within the two-point line, the best shot angle is _{0} < [7.29, 7.91]. If the shooting point is outside the two-point line and within the three-pointer, the best shot speed is [7.29, 7.91] < _{0} < [7.98, 8.47]. If the shooting point is outside the three-point line, the best release speed is _{0} > [7.98,8.47].

If we consider air resistance, the trajectory of the two-pointer and three-pointer hitting the leading edge of the backboard is:

Where

Simplified

The second equation of equation (7) is multiplied by cos

From the square root of both sides of Equation (8), we can get

We substitute equation (9) into the first equation of equation (7) to get

We combine equations (9) and (10) to get

Take the derivative of the above formula concerning Δ

Arranged to get:

Arranged to get

This allows us to find the best shot height for two-pointers and three-pointers [9]. Because of the presence of air friction, the optimal release angle for a two-point 3-pointer requires a smaller release angle and relatively higher release speed than a frictionless 2-point 3-pointer _{0} to make a shot.

Examples of shooting angles and entry angles for different shooting speeds and shooting heights are shown in Table 1. We set the shooting height to 2.6 meters [10]. The horizontal distance is 7.3 meters. The shot speed is 8.8 m/s. The angle is 40° to 50°. At this time, the basketball trajectory we get is shown in Figure 1.

Shooting Angles and Entry Angles for Different Shooting Speeds and Shooting Heights

Shooting height | The horizontal distance between the shooting point and the center of the basket | Speed | Angle range | Hit rate |
---|---|---|---|---|

2.6m | 7.4m | 8.7m/s | 40°–50° | 60% |

2.6m | 7.4m | 8.8m/s | 40°–50° | 90% |

2.6m | 7.4m | 8.9m/s | 40°–50° | 20% |

2.6m | 7.4m | 9.0m/s | 40°–50° | 90% |

2.6m | 7.3m | 8.7m/s | 40°–50° | 80% |

2.6m | 7.3m | 8.8m/s | 40°–50° | 100 % |

2.6m | 7.3m | 8.9m/s | 40°–50° | 90% |

2.6m | 7.3m | 9.0m/s | 40°–50° | 75% |

We set the shooting height to 2.6 meters. The horizontal distance is 7.3 meters [11]. The shot speed is 8.7 m/s. The angle is 40° ~ 50°. At this point, the basketball trajectory we get is shown in Figure 2.

When the shooting height, horizontal distance, and shooting angle are the same, and the shooting speed is different, the shooting angle is between 40° and 50°. All shot trails end within the basket. The two-shot trajectories of 40° and 41° in Figure 2 end with no goal basket. The difference in shooting speed between the two pictures is only 0.1 m/s. The hit rate of Figure 2 is 20% lower than that of Figure 1. When the shooting height is 2.6 meters, the best shooting speed is 8.8 meters per second.

We looked at shooting percentages and the factors that affect them. The trajectory of a basketball depends on a certain release speed, release angle, and shot height. We have mastered the influencing factors of basketball shooting percentage, and we can improve the shooting quality by strengthening training and summing up some laws. The research conclusions of the article can provide a reference for basketball fans and basketball teaching and training.

#### Shooting Angles and Entry Angles for Different Shooting Speeds and Shooting Heights

Shooting height | The horizontal distance between the shooting point and the center of the basket | Speed | Angle range | Hit rate |
---|---|---|---|---|

2.6m | 7.4m | 8.7m/s | 40°–50° | 60% |

2.6m | 7.4m | 8.8m/s | 40°–50° | 90% |

2.6m | 7.4m | 8.9m/s | 40°–50° | 20% |

2.6m | 7.4m | 9.0m/s | 40°–50° | 90% |

2.6m | 7.3m | 8.7m/s | 40°–50° | 80% |

2.6m | 7.3m | 8.8m/s | 40°–50° | 100 % |

2.6m | 7.3m | 8.9m/s | 40°–50° | 90% |

2.6m | 7.3m | 9.0m/s | 40°–50° | 75% |

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