An empirical study of parameters in different distance standing shots

5.2.1 The range of the incidence angle in swish shot

Basketball shooting belongs to the oblique throwing movement in physics. Excluding the human factors, we use mechanical reasoning. The larger the shooting angle, the higher the arc, the larger the incidence angle, the higher the field goal percentage. The field goal percentage is the highest when the incidence angle is vertical, and the allowable error range is the largest. With the arc of the shot decreasing, the incidence angle decreases, the incident cross-sectional area also decreases, and the allowable error range decreases, too.

As shown in Fig. S5, in theory, when the basketball with minimum angle enters the basket, the distance between the ends of the basket diameter and the tangent of the basketball sphere should be the diameter of the basketball. According to the mathematical theorem, we can see that the diameter of the tangent point of the circle is perpendicular to the tangent, and the foot is the tangent point. Therefore, the triangle ABC in Fig. S5 is a right triangle, and β is the angle of the basket at this time. According to the knowledge of solving the right triangle, we can get sinβ = AB/BC, where AB is the diameter of the basketball and BC is the diameter of the hoop. Then, the incidence angle β of swish shot is equal to the value of arc-sine function of the ratio of the basketball diameter (d) to the basket diameter (D), namely, β = arcsin(d/D). Because the inverse sine function ranges from −90° to 90°, the maximum shot angle of the shot is 90°.

According to the competition rules of FIBA, the standard hoop diameter is 45 cm, and the parameters for the ball model for basketball competitions in different groups shown in Table 1.

It can be seen that the male's minimum incidence angle β is 32.54°. The short axis of the cross-sectional area of the basket is equal to the diameter of the basketball. At this time, the allowable error has dropped to zero. If the incidence angle is reduced, the basketball will hit the hoop and cannot be shot directly.

In summary, the range of the incidence angle of the swish shot in the standard male game is [32.54°, 90°].

5.2.2 Theoretical parameters analysis of basketball flight trajectory when shooting from different distances

U16 male players shooting testing distance and shot height analysis.

According to the experimental test of 12 U16 male players, and high-speed image analysis of different distance shootings, we can obtain the shooting distance and the shooting height parameters. The results are shown in Table 2.

The data in Table 2 derives from the experimental test. The experimental design requires the player to follow the free throw line (4.225 m) shot, top of free-throw lane shot (5.8 m) and the three-point shot (6.75 m). The data in the above table is the actual shooting distance. Among them, the shooting distance is the horizontal distance from the geometric center of the basketball to the center of the hoop. The shooting height refers to the vertical distance from the geometric center of the ball to the ground.

2. Analysis of the relationship between minimum incidence angle, minimum shooting angle, shooting velocity and field goal percentage.

The shooting angle refers to the angle between the tangent of the flight path of the basketball center and the horizontal plane of the shot point when the ball is off the hand (Chinese Basketball Association, Basketball Rules. Beijing Sport University Press, Beijing, China, 2018). The shooting angle is positively correlated with the incidence angle. The shooting angle enlarges when the incidence angle increases (Darling and Cooke, 1987a; Darling and Cooke, 1987b; Guo et al., 2015). Theoretically the maximum incidence angle can be 90°, which is because the theoretical range of shots is the largest. However, due to the increase in the incidence angle, the shooting angle and the shooting velocity also increase. As a result, the control effect of basketball is greatly reduced, which makes the goal percentage uncontrollable (Darling and Cooke, 1987; Guo et al., 2015). According to the results of Yang et al. (2016), the minimum velocity shooting angle is the 45° together with the half of the slant angle of the shot, as shown in Fig. S7. $$\alpha =45°\frac{\varphi }{2}$$

As shown in the figure, φ = arctan(h₁/S), then there is,

Among them, h₁ refers to the vertical distance from the geometric center of the basketball to the horizontal level of the basket when shooting, that is, h₁ = 3.05-H₀, in which H₀ is the shot height, and S is the shooting distance. According to the data in Table 3, the minimum velocity shot angle parameters for different distances and shot heights can be obtained, as shown in Table 3.

The data in Table 3 shows that the shooting height H₀increases slightly, h₁ decreases, and the tilt angle φ and the shot angle α decrease as the shooting distance increases.

3. Analysis of minimum shot velocity.

According to the above trajectory theory equation, it can be known that $$y=H_0+x\tan \alpha -\frac{g{x}^2}{2v_x^2}$$

From the trajectory equation, it is not difficult to see that the shot angle and the shot velocity present nonlinear correlations when other factors are constant. For an excellent sport, the shot height of the fixed distance tends to be stable, so the equation is a function of v and α. When α is the minimum value, we bring the data in Table 2 and Table 3 into different athletes and obtain the minimum shot velocity, as shown in Table 4.

The data in Table 4 displays the theoretical shot velocity corresponding to the minimum shooting angle of the certain distance. i.e., the minimum velocity at which the distance can be swish shot. Theoretically, the shooting velocity in minimum shooting angle enlarges as the shooting distance increases. Since air resistance is neglected in the theoretical analysis, we can see the velocity should be slightly larger than the actual velocity value, but the trend is unchanged. Lower velocity shots are easier to control and can reduce motion variation, thus improving the stability of the shots (Knudson, 1993; Liu, 2004). The better the shots are, the smaller the shot velocity is, and the higher the shot hit rate is. Accordingly, top players tend to shoot with a shooting angle close to the minimum shooting velocity (Liu, 2010; Ma, 2009; Millers and Bartlettr, 1996). Therefore, the minimum entry angle and the corresponding minimum shot velocity can provide the groundwork for the athletes to weigh the three controllable factors, to determine the hand stability and the maximum controllability in order to maintain a high hit rate.

4. Calculation of the shooting velocity in different distances in theory.

According to the experimentally measured shooting distance, shot height, shot angle and other parameters into the shooting arc trajectory equation, we can calculate the theoretical velocity parameters and obtain the following results, shown in Table 5.

The data in Table 5 shows that with the increase in the distance from the penalty basket, the mid-range shot and the long-range shot, the shooting velocity increases, while the shooting angle gradually decrease.

In summary, based upon the theoretical parameters, when the same athlete shoots with a stable shot height, the same distance increases with the shot angle, and the shot velocity increases. When the same athlete shoots at three different distances, the shot angle is reduced, and the shot velocity is increased, as the shooting distance and the shot height increase.

5.3.1 Results analysis of the actual shooting angle and the angle of the basket in different distances

In the study, U16 outstanding male basketball players were subjected to on-site experimental shooting and post-analysis of the penalty shots, CIC and long-range shooting. The relevant technical analysis parameters were obtained. The results are shown in Table 6.

The data in Table 6 shows that when the athletes shoot in the field, the shot height increases slightly, and the shot angle decreases, as the shooting distance increases. This is because the farther the distance is, the higher the accuracy of the shooting technique is. The increase of the shot height plus the smaller angle of the shot can form the ability to control the ball (Ma, 2009). Combining the parameters of same distance and the height of the shot, the actual tested shooting angle of the shot is slightly larger than the minimum shot angle, and the entry angle is also greater than the minimum entry angle.

5.3.2 Analysis of actual shooting velocity parameters of different distances in standing shooting

The experiment filmed high-speed images of different distances in-standing shooting on U16 male athletes. The motion analysis software is used to analyze the parameters such as time, displacement and angle. The horizontal average velocity is obtained through calculation, and the vertical velocity and velocity parameters are obtained through calculation, too. The results are shown in Table 7.

5.3.3 Comparative analysis of in-standing shooting theory and tested parameters at different distances

1. Results of shot velocity parameters.

According to the actual shooting distance, shot angle and shot height of 12 athletes, we can calculate the theoretical velocity of the shot by the above formula. At the same time, we can also get the measured parameters of the shot velocity at different distances in the experiment. The results are shown in Table 8.

From the mean of data, we can intuitively find that the theoretical value is slightly larger than the measured value. In order to prove the accuracy of the theoretical trajectory equation (excluding the actual air resistance) and the accuracy of the test parameter analysis, we intend to do a two-sample t-test on the two sets of data.

2. Differences between the theoretical value of the original shooting and the measured value of variance (F test).

According to the statistical principle, there are two cases of Equal variance and Heteroscedastic for the two groups of data mean consistency test, so the theoretical and measured values of the shooting velocity of shots are subjected to F test to verify the differences between the two sides. The results are shown in Table 9:

The data in Table 9 are the theoretical and measured F-test results of three different distances in the penalty basket, from U16 male National Team players. The results show that the p-values of free throw is smaller than 0.05. But the other two pairs of data are all greater than 0.05.Thereby there is no significant difference between the theoretical values of two pairs of different distance data and the measured values, and the T-test of the equal-variance mean samples can be performed. And there is significant difference between the theoretical values of the free throw of different distance data and the measured values, and the T-test of the Heteroscedasticity t test mean samples can be performed.

3. Variance t-test of the theoretical and measured values of the shooting velocity of different distance shots.

According to the data results of Table 9, we can make a two-sample and variance hypothesis T test for the theoretical and measured values of the three sets of data at different distances. The results are shown in Table S1.

The data of Table 10 shows that P values of the three pairs of data are all greater than 0.05, indicating that there is no difference between the theoretical calculation parameters and the measured parameters of three different distances. The theoretical value is consistent with the mean value of the measured values. Therefore, we can calculate the velocity of the shot and the trajectory of the basketball flight based on the shooting angle of the shot, the distance of the shot and the height of the shot.

In summary, for the same elite athlete, the shot height of the same distance shot is basically stable, and the shot stability is mainly determined by the angle of the shot. The shot angle is the decisive factor for the stable shooting percentage of the same distance.