Basketball is one of the most popular sports in our lives. Improving shooting accuracy is something we all look forward to. We found that shot accuracy was related to the angle and speed at which the basketball entered the hoop. In this paper, a dynamic model is established for the fixed-point shooting process. This paper theoretically explores the conditions for the hollow of basketball into the frame to improve the shooting accuracy in basketball. The process of basketball from the shot to the hollow into the frame can be divided into two movement processes . There is no need to consider the basketball shape during the previous exercise. We can think of basketball as a particle. At this time, the movement process of basketball can be regarded as oblique throwing movement. During the last movement, the movement process is very short. This process can be approximated as a uniform linear motion. We need to consider the impact of the actual shape of the basketball on the frame. Therefore, we first regard the basketball as a mass point and establish the trajectory model of the basketball center. We get the condition that the basketball center passes through the center of the basket. On this basis, we analyze the conditions that the incident angle should satisfy when the basketball is about to enter the frame. Finally, the analysis of the article gives the angle of the shot (lateral declination) and the allowable deviation of the speed.
Consider first the case where the center of the basketball ball hits the center of the basket. At this time, we can regard the basketball as a particle without considering the blocking of the basket. From the oblique throw motion knowledge, we can decompose the shot release speed along with the horizontal and vertical directions . At this time, we obtain the kinematic equations of the basketball in the horizontal and vertical directions and then establish the trajectory model of the basketball center. Since the center of the basket is on the trajectory of the basketball, we substitute the coordinates of the center of the basket into the trajectory equation of the basketball. At this point, we can get the functional relationship between the shot speed and the angle.
Assume that the release velocity is in the same plane as the center of the basket when shooting. There is no lateral deflection in the model at this time. Basketball only moves in the plane determined by the center of the ball, the center of the basket, and the shot’s speed. Without considering the rotation of the basketball, we regard the basketball’s motion as a two-dimensional oblique throwing motion . So we set the origin of the coordinates at the center of the ball when the basketball is shot. The horizontal direction is set to the x-axis direction, and the vertical direction is set to the y-axis. At this point, we establish a plane rectangular coordinate system, as shown in Figure 1.
The x-axis is the horizontal direction, and the y-axis is the vertical direction . The basketball is thrown at
Where g is the acceleration of gravity and M is the mass of the basketball. Substitute the initial condition
From equation (2), we can get the shooting trajectory model:
We substitute the coordinate
From the root formula we get:
Under a certain shooting height
We think of basketball as a particle. The relationship between the release angle
The basketball cannot be considered a particle when it is about to enter the basket due to its blocking. When we look at the disk on the table, the closer our eyes are to the plane of the disk, the more “flat” we see. A disk is only round when viewed directly above it. The area is also the largest at this time . The basketball’s trajectory was flying diagonally when shooting is a parabola. Still, when the basketball is close to the basket, we can think that the basketball enters the basket in a straight line at a uniform speed. The diameter of the basket is
So we make the center of the basketball pass through the center of the hoop. We must make the short axis of the incident section not less than the diameter of the basketball sphere: 2b ≥ 2
The basketball diameter is 0.246m and the diameter of the basket is 0.45m. After sorting out, we can get the basketball incident angle
The negative value of the derivative of the trajectory of the basketball center at
Since the range of the shot angle and the incident angle is (0°, 90°), we can get
If the basketball is regarded as a solid sphere, the release angle and release speed should satisfy
We project the basket to a perpendicular plane to the incident direction and pass through the front of the basket. The basket is an ellipse, as shown in figure . The basketball is tangent to this projected ellipse at maximum offset. We only need to consider what happens when the center of the basketball hits the major axis AB of the ellipse and establish the equation of the ellipse and the great circle of the basketball. At this point, we calculate the maximum deviation distance between the center of the ellipse and the center of the great circle. We divide the maximum offset distance by the distance OO’ from the release point to the center of the basket to obtain the tangent of the lateral deflection angle. Since the maximum lateral deflection angle is small, this angle is approximately equal to the tangent of this angle. From this we can find the maximum lateral deflection angle . The established coordinate system is shown in Figure 5.
The parameter of the ellipse is
When the circle and the ellipse are tangent to the ellipse vertex (
Substituting in the ellipse parameters we can get
There is a formula since the basketball incident angle is in the range of (0, 90°)
When the circle and the ellipse are not tangent to the ellipse vertex (
We use MATLAB simulation to obtain the relationship between
From the above conclusions, it can be seen that the maximum offset distance of left and right is
First, find the relationship between the release angle deviation and the landing deviation .
The basketball trajectory equation is as follows:
The above formula can be transformed into
From this, it can be deduced that the influence formula of the deviation of the release angle on the deviation of the landing point is as follows:
We look for the relationship between the release speed deviation
Taking the partial derivative of
We can analyze the size of the landing deviation caused by different ideal projection angles and projection speeds . At this time, we need to find the size of
Since the deviation of angle and speed is very low when shooting with superior release speed and angle, the following approximate relationship
Therefore, the allowable deviations of different shot angles and speeds corresponding to Δ
We can calculate the allowable deviation of Δ
When the shot angle is 60°, we set the shot speed size
When the shot speed is 9m/s, we can take out the hand angle of 55°. According to formula (7), it can be obtained that the basketball incident angle
After the above solution, it can be obtained that the shot angle is 60° when
The results of this paper have theoretical and practical application significance. We can generalize the results to a basketball player shooting drills. We generalize the theory to practical training. The basketball shooting angle and shooting force can be controlled within a certain range for training using some scientific methods. Athletes can respond quickly by flexibly applying the scientific principles of this article when preparing to shoot, depending on their position and situation. Athletes save physical strength to improve the hit rate and win the game with a more relaxed attitude.
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