The article applies mathematical calculus modeling to analyze the force of shot put. We emphatically analyzed the relationship between the angle of the shot and the initial velocity and calculated the motion trajectory equation of the shot put. The above research results are used to assist in the analysis and help improve the teaching performance of sports shot put. The study found that different forces have different effects on the horizontal displacement of the shot. Under the comprehensive force analysis, the horizontal velocity gradually decreases with time, and under the gravity analysis, the horizontal velocity of the shot is constant. The horizontal displacement of the shot put is related to the shooting angle and initial speed. Under the comprehensive force analysis, the influence of air resistance and buoyancy on the throwing distance is different due to the different combinations of shooting angle and initial speed.
Keywords
 Mathematical calculus
 Dynamics
 Shot Kinematics
 Teaching achievement
MSC 2010
 53A17
We use CNKI journals as the search data source, and the search subject is mechanics. We use shot as a keyword to accurately match papers published in core journals from 2009 to 2020. It can be found that biomechanical methods mainly study 20 papers. We have carried out a detailed analysis of the various physical parameters before the shot, and there are roughly two purposes. The first is to enable readers to understand the biomechanical characteristics of outstanding athletes [1]. The second is to explore the best combination of shooting angle and initial speed. Two papers use a combination of particle dynamics and mathematics but explain the relationship between the shot angle and the initial velocity rather than analyze the force after the shot. Therefore, this article analyzes the movement trajectory of the shot from a new perspective of dynamics and comprehensively discusses its force process from a theoretical perspective.
This article takes the whole process from shot put to landing as the research object. We divide the operation process into two stages. The first stage is from the shot to the highest point. When the shot put is at the highest point, its vertical speed is zero. The second stage is from the highest point to landing. This article mainly uses the laws of kinetics, the differential equations of particle motion, and mathematical equations to analyze the entire operation process of the shot. The basic expression form of the formula is:
In Newton's classical mechanic's system, the mass of an object is constant, and time and space are also constant. However, through modern physics, people have realized that the mass, time, and space of any object change with the object's speed [3]. But when the object's speed is much lower than the speed of light, the change of the object's speed on its quality, time, and space is negligible. (See Figure 1, Figure 2, Figure 3, Table 1, Table 2, Table 3). Since the speed (
The attributes of each symbol in Figure 1
Attributes  Symbol  Attributes  

Initial shot speed  H2  Height of highest point of shot put  
Shot angle  h  The horizontal distance between the throwing point and the toeboard of the throwing circle  
The speed of the shot put at any moment  H1  Point to drop  
The horizontal distance from the shot point to the drop point  T  The trajectory of the shot put in the air 
Attributes of each symbol in Figure 2
Attributes  Symbol  Attributes  

Air buoyancy  G  Shot Put Gravity  
The vertical component of air resistance  f2  The horizontal component of air resistance  
air resistance  a  Air resistance and yaxis angle 
The attributes of each symbol in Figure 3
Attributes  Symbol  Attributes  

Acceleration of gravity  a  Air buoyancy acceleration  
Resistance horizontal acceleration  a2  Drag vertical acceleration 
When a particle is under the combined action of several forces (
Horizontal direction
In this way, we obtain the law of the change of the shot speed with time as:
In this way, we obtain the trajectories of the shot put in the direction of the
The trajectory of the shot is put in the ascending phase in the
According to the calculation process in 3.1, we divide the running process of the shot into two stages: rising and falling. In this way, two different coordinates are established respectively [9]. In this way, only the influence of gravity is considered, the horizontal direction is not affected by force, and the vertical direction is affected by gravity (see Figure 4). The law of the shot put speed changing with time in the ascending phase is:
The trajectory equation of the shot can be obtained as:
The law of the change of the shot speed with time in the descent phase is:
The trajectory equation of the shot can be obtained as:
From the trajectory equation of the shot put in the xaxis direction, it can be found that the throwing distance of the shot put is positively correlated with the initial velocity (
Considering all the forces in the ascending phase
Decline phase data
Time spent in descending phase/s  Vertical displacement during descent/m  Total horizontal displacement S1/m  

1  5.098146  20.533524  
1  5.098146  19.371923  
1  5.271457  17.945968  
1  4.240563  18.881659  
1  5.035463  17.326696  
1  4.240783  17.882375  
1  4.447863  17.719364  
1  4.751445  17.078781 
According to
Rising phase data
Rising time/s  Vertical displacement during ascent stage/m  

0.787085  3.038146  
0.787085  3.038146  
0.805437  3.181457  
0.660662  2.140563  
0.778929  2.975463  
0.675955  2.240783  
0.713678  2.497863  
0.746304  2.731445 
It can be seen from Table 6 that the athlete's shot angle is lower than 45°, so the speed
The shooting speed and angle of the top eight women's shots puts in the National Games
Grade/m  Initial shot speed/ms^{−1}  Shot angle/(°)  Shot height/m  

20.35  13.84  33.89  2.06  
19.38  13.31  35.48  1.98  
18.97  12.69  38.49  2.09  
18.41  13.09  29.72  2.1  
17.67  12.38  38.15  2.06  
17.52  12.56  31.85  2  
17.3  12.48  34.13  1.95  
17.07  12.22  36.83  2.02 
Considering that gravity is applied when the ascending phase is
Decline phase data
Time spent in descending phase/s  Vertical displacement during descent/m  Total horizontal displacement S1/m  

1.019447  5.097635  20.755531  
1.011416  5.017635  19.494315  
1.036631  5.270937  18.308807  
0.929764  4.240183  18.081877  
1.013165  5.035005  17.453717  
0.929789  4.240412  17.132075  
0.952217  4.447451  17.224113  
0.984177  4.751009  16.92287 
Ascent phase data
Rising time/s  Vertical displacement during ascent stage/m  

0.786952  3.037635  
0.786952  3.037635  
0.8053  3.180937  
0.66055  2.140183  
0.778797  2.975005  
0.67584  2.240412  
0.713557  2.497451  
0.746177  2.731009 
According to the calculation results of 3.3.1 and 3.3.2, the absolute value Δ
Difference of horizontal displacement under two different forces
Δ 
Δ 


0.222007  1.01  
0.122392  0.63  
0.362839  1.91  
0.799782  4.34  
0.127021  0.72  
0.7503  4.28  
0.495251  2.86  
0.155911  0.91 
It can be seen from Table 10 that the absolute value of the horizontal displacement difference Δ
Wilcoxon signedrank test
Z  N  Asymptotically significant. (Both sides)  

0.84  8  0.401 
1) According to the particle dynamics equation of the shot, the horizontal velocity is related to the angle of the shot. Under the comprehensive force analysis, the horizontal velocity gradually decreases with time, and under the gravity analysis, the horizontal velocity of the shot is constant. 2) The horizontal displacement of the shot put is related to the shooting angle and initial speed. Under the comprehensive force analysis, due to the different combinations of shot angle and initial speed, the influence of air resistance and buoyancy on the throwing distance may be positive or negative. 3) The ratio between the absolute value and the actual throwing distance under the comprehensive force analysis and the gravity analysis has relatively large fluctuations. There are certain errors in the shooting angle, initial speed, and the entire calculation process.
The attributes of each symbol in Figure 3
Attributes  Symbol  Attributes  

Acceleration of gravity  a  Air buoyancy acceleration  
Resistance horizontal acceleration  a2  Drag vertical acceleration 
The attributes of each symbol in Figure 1
Attributes  Symbol  Attributes  

Initial shot speed  H2  Height of highest point of shot put  
Shot angle  h  The horizontal distance between the throwing point and the toeboard of the throwing circle  
The speed of the shot put at any moment  H1  Point to drop  
The horizontal distance from the shot point to the drop point  T  The trajectory of the shot put in the air 
Rising phase data
Rising time/s  Vertical displacement during ascent stage/m  

0.787085  3.038146  
0.787085  3.038146  
0.805437  3.181457  
0.660662  2.140563  
0.778929  2.975463  
0.675955  2.240783  
0.713678  2.497863  
0.746304  2.731445 
Difference of horizontal displacement under two different forces
Δ 
Δ 


0.222007  1.01  
0.122392  0.63  
0.362839  1.91  
0.799782  4.34  
0.127021  0.72  
0.7503  4.28  
0.495251  2.86  
0.155911  0.91 
Wilcoxon signedrank test
Z  N  Asymptotically significant. (Both sides)  

0.84  8  0.401 
Ascent phase data
Rising time/s  Vertical displacement during ascent stage/m  

0.786952  3.037635  
0.786952  3.037635  
0.8053  3.180937  
0.66055  2.140183  
0.778797  2.975005  
0.67584  2.240412  
0.713557  2.497451  
0.746177  2.731009 
Attributes of each symbol in Figure 2
Attributes  Symbol  Attributes  

Air buoyancy  G  Shot Put Gravity  
The vertical component of air resistance  f2  The horizontal component of air resistance  
air resistance  a  Air resistance and yaxis angle 
The shooting speed and angle of the top eight women's shots puts in the National Games
Grade/m  Initial shot speed/ms^{−1}  Shot angle/(°)  Shot height/m  

20.35  13.84  33.89  2.06  
19.38  13.31  35.48  1.98  
18.97  12.69  38.49  2.09  
18.41  13.09  29.72  2.1  
17.67  12.38  38.15  2.06  
17.52  12.56  31.85  2  
17.3  12.48  34.13  1.95  
17.07  12.22  36.83  2.02 
Decline phase data
Time spent in descending phase/s  Vertical displacement during descent/m  Total horizontal displacement S1/m  

1.019447  5.097635  20.755531  
1.011416  5.017635  19.494315  
1.036631  5.270937  18.308807  
0.929764  4.240183  18.081877  
1.013165  5.035005  17.453717  
0.929789  4.240412  17.132075  
0.952217  4.447451  17.224113  
0.984177  4.751009  16.92287 
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