From the perspective of physics, this article analyzes the various factors that affect the distance of shot put in the process of actual shot put. Based on the actual situation, it uses fractional calculus and physical formulas to calculate the shot throw distance and to conduct a comprehensive analysis. The experiment confirms that the three factors that affect the shot flying distance are the shot angle, shot speed, and shot height and analyzes the relationship between these three factors and the impact of each univariate on the shot flying distance. The analysis results show that among the three factors, the shot speed has the greatest impact on the flight distance, followed by the shot angle, and the shot height has the least effect; moreover, the optimal shot angle is not a specific value but a general range.
Keywords
 fractional differential equation
 shot put motion
 optimal path
 maximum flight distance
MSC 2010
 12H99
Shot put in China is the main throwing item in trackandfield teaching and national physical exercise standards. It is one of the key physical education courses in China’s sports colleges and universities. It is also an earlier sport that we encountered after entering school. It can exercise people’s muscle’s explosive power, improve the coordination in the movements of various parts of the body, so that the upper and lower limb muscles can develop symmetrically; improve the central nervous system’s regulating function and response speed of action. The heavy shot equipment determines that this sport requires fast movements and high strength. From a physics perspective, shot motion can be regarded as oblique throwing. The flight distance is determined by three variables: shot speed, shot angle, and shot height. Therefore, it is necessary to increase the shot. The process of movement must start from increasing the shooting speed, as well as adopting suitable throwing angles and higher shooting heights [1]. At present, the general literature only considers the insitu throwing when discussing the shot flying distance and does not discuss slipping. Shot put has an initial speed and air resistance, which is like the actual training game. The definition of each mechanical variable is ambiguous, and the mechanical relationship between the shot speed, the shot angle, and the shot height cannot be fully considered. At the same time, on consulting some university textbooks, the discussion of the parabola is only limited. In the ideal case, the initial velocity of the projectile is decomposed into its horizontal and vertical components for consideration. Such a discussion is obviously not comprehensive. In practice, the three are influential and interrelated. This article will comprehensively consider the various factors mentioned above. The effect specifically of the independent mechanical variables that affect the shot flying distance and the degree of influence of each independent mechanical variable on the flying distance of shot put are studied in detail [2].
Knowing that
Let us look at the theorem that there is uniqueness. The theorem is as follows: if
There is a unique solution
Use function monotonicity to prove that
Because 0 <
Another method: use the proof method to prove that
Suppose there are
From the condition
The general steps to discuss the number of equation roots are as follows: first, find the inflection point of
Discuss the solution of equation
When
The above formula is the maximum value of
The equation has no real roots; when
Prove that equation
Judging extreme points
A positive number  0  Negative number  
Increasing 

Decrement 
From the table above, we know that
It can be known that
When researching the sports biomechanics of throwing shot technology, the shot is generally regarded as a mass point to study the mechanical variables of the time of shot release. From a kinematics perspective, since the shot is of high mass but small in volume, spherical and smooth, within a certain error range, the air resistance and the influence caused by the shot rotation can be ignored. The actual shot projectile motion is simplified to a mechanical model as shown in Figure 1.
Discuss the oblique motion of the mass point. To clearly describe the motion of the shot, simplify Figure 1 and establish the coordinate system with O as the origin, horizontal axis as the
Let the mass of shot put be
When the shot is put on the ground,
The throw distance can be expressed as
Because the horizontal distance between
The following considers the resistance of air from the perspective of theoretical mechanics to derive the expression of the shot flying distance.
Suppose that the resistance during the shot motion is
Let
Integrating again,
So
So, we have
The air drag coefficient of the shot throw is generally 0.15 × 10^{−3}. When the shot weight is 6 kg, the shot angle is 36°, the shot speed is 11 km/s, and the shot distance is 2 m, the flight distance is 14.044 m. Substituting the same data into Eq. (2), the flight distance is 14.044 m. The effect of the two parameters on the flight distance due to the presence of air resistance is 0.02 m, which is compared with the total flight distance of 14.04 m. The ratio is 0.001, and this ratio is relatively small; so, the effect of air resistance on the shot flying distance has been ignored in many documents. According to the distance expression of the shot flying movement, the research on shot putting movement often focuses on the independent factors such as shot speed, shot height, and shot angle. The following will first analyze the impact of each variable on the shot flying distance, ignoring air resistance, and then comprehensively analyze the relationship between these variables.
The release speed refers to the speed at which the shot is shot relative to the ground and is set to
In the above formula,
The shot angle refers to the angle between the shot’s shot speed
And then, we get
Then, the best shot angle can be found as
First, we can see what range of hand angles are constrained according to the actual situation. Table 1 contains the data on worldclass shot putters’ shot results and shot angles [7, 8].
In traditional sports training theories and textbooks, the best angles for putting shots are set in the range of 38° ∼ 42°, and <45°, because the shot point is higher than the landing point. From Table 1, we can the shooting angle of worldclass athletes is roughly in the range of 35° ∼ 42°. Considering the athletes’ nervousness and other factors on the playing field, the author believes that the range basically meets the traditional definition. From Table 2 [9], we can see that the reasonable range points out that the hand angle conforms to the principles of human anatomy and mechanics conforms to the motion technology model of the optimal combination of various factors that determine the throwing distance. The shot angle is dynamic.
Shot put results and shot angles of worldclass shot putters
Li Meisu  20.3  38.69 
20.95  37  
21.76  35.13  
Huang Zhihong  20.76  37.75 
21.28  41.3  
21.52  36.9  
Slupianek  21.28  41.3 
21.41  40  
22.45  36  
Sui Xinmei  21.66  39 
Lisovskaya  20.86  42.6 
Bayer  21.02  34.1 
Gunther  22.23  35.5 
Timmerman  21.35  35.8 
The shot height refers to the height of the shot from the ground. It depends on the height and arm length of the athlete. As shown in Figure 1, a person’s shoulder height is
From Eqs. (8)–(15), it is known that the greater the angle of action, the smaller the shot speed [note
The following text uses Eq. (2) to calculate the flight distance to analyze the mutual influence of the three parameters on the flight distance. The calculated specific data are shown in Table 3.
Combination modes of shot flying distance, shot speed, shot angle, and shot height
11  2  14.044  14.102  14.192  14.181  14.2  14.207 
2.05  14.094  14.15  14.237  14.226  14.245  14.25  
12  2  16.33  16.409  16.54  16.521  16.554  16.571 
2.05  16.382  16.459  16.586  16.569  16.6  16.616  
13  2  18.802  18.904  19.08  19.054  19.102  19.131 
2.05  18.855  18.956  19.129  19.103  19.15  19.177  
14  2  21.461  21.589  21.781  21.781  21.845  21.887 
2.05  21.516  21.642  21.831  21.831  21.894  21.935  
15  2  24.308  24.464  24.747  24.703  24.785  4.841 
2.05  24.364  24.519  24.797  24.754  24.835  24.89 
(Because the influence of the shot angle on the shot flying motion is not the main factor, 1° is used as a unit of measurement when making this table.)
It can be seen from Table 2 that when the fixed shot height is 2.05 m, the shot distance and shot speed are linearly related, while the shot angle and flight distance are obviously not linearly related. That is, as the shot speed increases, the shot flying distance will increase correspondingly but will not increase indefinitely. This is because the human physiological factors determine that the shot speed will have an extreme value. With the increase of the shot angle, the flight distance image is like a quadratic function image, i.e., the flight distance has a maximum value under the condition that the shot speed and the shot height are constant. Currently, the shot angle is often the best shot angle. Therefore, with a certain shot height, an athlete adjusts the shot angle to increase the shot speed as much as possible to increase the flight distance. At the same time, the effect of changing the shot angle on the flight distance is much smaller than the effect of changing the shot speed on the flight distance. At the same time, it is also difficult to grasp the optimal shot angle. Therefore, the shot speed is more important in affecting the shot flying distance.
It can be seen from Table 2 that when the shot height is constant, the effect of the shot speed on the flight distance is measured in meters; while when the shot speed is constant, the increase of the flight distance by changing the shot height is only a few tenths of a meter. In other words, the impact of the shooting height on the flying distance is small. If several athletes launch the shot at the same shooting speed and shooting angle, the athlete with a high shooting height will shoot a little longer, but it will not be too obvious.
At a certain speed, as the shot height increases, the flight distance is a positively correlated straight line, i.e., the higher the shot height, the farther the flight distance, but the inclination is relatively small; as the shot angle increases, the flight distance presents concavity and convexity, i.e., nonlinear correlation, which just reflects the best shot angle we often say, but the best shot angle is not a certain value but a general range; as the shot speed increases from top to bottom, the flight distance also gradually increases, and the graph gradually moves to the left. This verifies that with a certain shot height, the shot angle increases as the shot speed increases. To sum up, when the shooting speed is constant, the higher the shooting height, the smaller is the corresponding shooting angle; when the shooting height is constant, the higher the shooting speed, the larger is the corresponding shooting angle.
(1) The factors that affect the shot flying distance are shot angle, shot speed, and shot height, but the degree of influence varies. (2) The shooting speed has the greatest effect on the flight distance, so during training, the development of shooting speed is the primary goal; then the shooting angle, and the shooting height has the least impact. (3) The shot speed is linearly related to the flight distance, and the shot angle is nonlinearly related to the flight distance. (4) The best shot angle is not a fixed value, but a range.
Judging extreme points
A positive number  0  Negative number  
Increasing 

Decrement 
Shot put results and shot angles of worldclass shot putters
Li Meisu  20.3  38.69 
20.95  37  
21.76  35.13  
Huang Zhihong  20.76  37.75 
21.28  41.3  
21.52  36.9  
Slupianek  21.28  41.3 
21.41  40  
22.45  36  
Sui Xinmei  21.66  39 
Lisovskaya  20.86  42.6 
Bayer  21.02  34.1 
Gunther  22.23  35.5 
Timmerman  21.35  35.8 
Combination modes of shot flying distance, shot speed, shot angle, and shot height
11  2  14.044  14.102  14.192  14.181  14.2  14.207 
2.05  14.094  14.15  14.237  14.226  14.245  14.25  
12  2  16.33  16.409  16.54  16.521  16.554  16.571 
2.05  16.382  16.459  16.586  16.569  16.6  16.616  
13  2  18.802  18.904  19.08  19.054  19.102  19.131 
2.05  18.855  18.956  19.129  19.103  19.15  19.177  
14  2  21.461  21.589  21.781  21.781  21.845  21.887 
2.05  21.516  21.642  21.831  21.831  21.894  21.935  
15  2  24.308  24.464  24.747  24.703  24.785  4.841 
2.05  24.364  24.519  24.797  24.754  24.835  24.89 
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