Correlation analysis on biomechanics parameters of basketball shooting based on differential equations

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1 Available online Journal of Chemial and harmaeutial Researh, 14, 6(6):86-9 Researh Artile ISSN : CODEN(USA) : JCRC5 Correlation analysis on biomehanis parameters of basketball shooting based on differential equations Jiangtian Zhu and Liang Dong Teahing and Researh Room of E, Changhun University, Changhun, Jilin, China ABSTRACT In the basketball game, the key to soring is to shoot the ball into the basket, whih involves a lot of kinematis model. In this paper, the mehanial problems involved in the proess of shooting basketball into the ring hoop are made into deep researh, in whih the shooting proess will be divided into three ases suh as forward spinning, bakward spinning, and side spinning. Respetively, these three ways were synthesized in mehanis and applied differential equations are arried out a detailed analysis. Finally, it an be onluded that greater projetion speed will be required for forward spinning ball and bakward spinning ball to ome smoothly into the basket, while side-spinning ball need higher speed and the appropriate angle of projetion, so, this paper will play guiding role on a basketball teahing. Key words: Differential Equations, movement rule, basketball-shooting analysis, mehanis model INTRODUCTION Basketball is a sport about field-goal attempts determining the sores. In the ourse of playing the game, soring beomes the key to winning hampionship, so the shooting proess is the most important in the whole math. There are many fators whih determine the basketball-hitting smoothly. The most ommon one is soring again after the ollision and the rotation of basketball [1-4]. revious studies have been done on shooting and fruitful results have been ahieved, for example, Chen Jian proposed basketball shooting model with ombination of dynamis and the probability statistis in the analysis of mehanis of rotating and aiming point in basketball. In his study, it is finally onluded that forward spinning and bakward spinning depend on basketball player s shooting ation [5-9], and on other fators like air resistane in the proess of rotation [1-1]. Researh on side spinning, Huang Shunhe raised the probability of enhaning hitting the hoop when rotating inidene angle is less than the angle of refletion as basket is hitting the ring for the purpose of improving the performane of basketball game. Based on what is found by the sholars above, this paper studies on several ases when the basketball enter the rim. The solution is onluded to inrease field goal attempts. BASKETBALL- SHOOTING AND MECHANICS ANALYSIS The ourse of the basketball into the rim an be divided into striking plate shot, rebound shot, swish shot the basket and so on. After touhing the basket, several ases may appear whih inlude forward spinning, bakward spinning, and side spinning. The basketball rotating is as shown in Figure 1: 86

2 Jiangtian Zhu and Liang Dong J. Chem. harm. Res., 14, 6(6):86-9 Figure 1: Basketball rotation figure Mehanial analysis on forward rotation of basketball in shooting The basketball bounes after the ollision with the hoop, thus, the orresponding rebound fore f from the rim will appear, so will the fore F when the ball gets the fore from the rim. The diretion of rebounding is the diretion of the resultant fore between f and F. Furthermore, the diretion of rebounding is loser to the diretion of vertial fore from the rim bouning, whih inreases the hane of basketball bumping into the hoop. Figure is shown as follows: Figure : Shemati before spinning ball boune along the ring after ollision Besides, the basketball an be more vertial as its rotations speed up so that there is muh possibility for the basketball to bump into the rim. After the basketball hits the rim, the resultant fore mentioned above will at on the bak of the rim so that the hane of ball bouning and rotating bakwards will inrease as shown in Figure : Figure : Rotating the ball hit the rim before the rebound shemati frontier Mehanial analysis on side spinning of basketball in shooting The basketball will move in a urved path if it rotates sidewise. Regardless of air resistane, aording to the law of angular-momentum onservation and the ase where the basketball rotates around OZ, the speed and fore in the horizontal diretion is orthogonal, then FM 8 = πρ wr u F m an be expressed as shown in the following: In the expression above, when the average veloity an be shown as, its orresponding veloity omponent is shown as v, therefore, the differentiation is performed on the proess of basketball movement. The formulas an be shown as follows: (1) 87

3 Jiangtian Zhu and Liang Dong J. Chem. harm. Res., 14, 6(6):86-9 In the horizontal diretion: du M ku dt = In the lumb Diretion: d M = Mg dt u 8 8 M = πρwr u u( g = πρr ) R We let t =, then z =, the orresponding elevation of = β = β an be, so: () () (4) 1 z = ( sin β) t gt t = u = u, = os β Substitude Mu u = M + ku t M ln M + s = ku t k Mu G α = w t M M u ds R = = dα ( M + ku t) into the following formulas: (5) (6) (7) (8) (9) Let speed and rotation angle be simultaneous, then, the solution is: α Mv x = v os dt = sin t M 1 ku Combine the formulas with the pratie, there is, by the simultaneous solution, the result is shown as: Mu Mu ( ) + (1 ) = 1 x y πρ To desribe the movement of basketball well, we speialize the parameters, let ; o ρ = 1.5 kg / m ( C) ; M =.6kg ; r = 1.1m, and substitute them into formula 1, so, during the ourse of flying, the exursed degree at the same level is as follows: (1) (1) 8 G = r =.18kg Mu Mu u u y = ( ) x =. (. ) x w w (1) From the formulas above, it an be known that there is a positive orrelation among the initial speed with the basketball rotating sidewise, the degree of transverse exursion and urvature of moving trak, whih is determined by the at of players throwing the ball. Mehanial analysis on bak spinning of basketball in shooting After the ollision of the ball and the ball frame, it will suffer the ation fore F opposite to the original diretion after the ball ollusive with the rim if the ball does not rotate so that the basket moves in the opposite diretion, that is, basketball is boune bak along the original diretion. In addition, the basketball also an suffer the reation fore 88

4 Jiangtian Zhu and Liang Dong J. Chem. harm. Res., 14, 6(6):86-9 from the hoop in the ring. the ation and reation work together so that the basketball has always been bak, the resultant fore will diret towards the basket, thus its possibility will inrease that the basket bump into the ring as shown in Figure 4: Figure 4: Ring does not rotate the ball is bouned bak shemati If basketball hits the forefront of the ring after hitting basket, this time it will suffer the fore of rebounding the rebound fore F from the basket, the fore in perpendiular diretion is the resultant fore, and they are very lose. Suh fores an often make basketball boune and drop into the basket afterwards, whih is shown in Figure 5. f and Figure 5: After rotating the ball hit the rim along the rebound after shemati And in this way the rotation speed is very fast before the ball drops into the basket, so the greater speed is, the more possibly the basketball drops into the basket. Therefore, there is lose onnetion between the possibilities of basketball into the hoop and the speed of rotation before the basketball enters the hoop, whih is shown in Figure 6: Figure 6: After rotating the ball hit the rim boune shemati frontier From the analysis above, it an be obtained that either forward rotating ball or bakward rotating ball; there is muh hane for the basketball to drop into the hoop. COMUTATION OF THE BASKETBALL HITTING THE RIM We let the basketball hits the upper bakporh at the speed of r and the delination is α, whih is shown in figure7: v, and its mass is M, the radius of the basketball is Figure 7: angle of delination ondition after the ball hitting the upper bakporh 89

5 Jiangtian Zhu and Liang Dong J. Chem. harm. Res., 14, 6(6):86-9 Meanwhile, let the speed of olliding point after the basketball hits the hoop reonstrut to the ontrary, in whih the angular veloity is ω, and β is the given angle At the moment of the ball hitting the hoop, the veloity of the enter of mass of the basketball is shown by uv, whose the two projetion anises an be defined as follows: = ( sinα os β + osα sin β ) = sin( α + β ) n = sinα os β osα sin β = os( α β ) z (14) (15) The angular veloity about the axis through the enter of mass is rotated ounter-lokwise and we use ω to represent it, The angular veloity after hitting the ring is expressed as between them is indiated as ϕ, as shown in Figure 8:, angle of delination along straight line Figure 8: angle of delination hange after hitting the upper bakporh Aording to Figure 8, we an get the following formulas based on the equation of impulse of the orrelative enter of mass and the law of impulse of enter of mass as follows: mz mz = Qz mn mn = Qn Jεω Jω = Qzr J = mr Wherein. (16) (17) (18) Suppose the basketball hits the point A on the rim and respetive speed before and after the ollision an be expressed as, An and, An, aording to the assumption of the fators, we an obtain the equations: An = An = 1. e = = = (19) () Regard the ourse of Collis ding the hoop as a hitting of a rigid body, when the ball hits the ring at the beginning, the uur uur uuur = + equation of kinematis orresponding with it an be as shown: A A Wherein, tangential diretion is the diretion of AC. = rω = + rω z = = ( sinα os β + osα sin β ) An n (1) () Similarly, at the moment when the ball is about to end its ollision with the ring, the equation of kinematis an be 9

6 Jiangtian Zhu and Liang Dong J. Chem. harm. Res., 14, 6(6):86-9 uur uur uuur = + expressed as: A A Wherein, tangential diretion is the diretion of A = rω = + rω An z = n () (4) ut the three formulas above into the two former formulas, it will happen like this: n z = n + rω = (5) (6) By simultaneousness, eah value for the formulas above an be shown as follows: = = sinα os β + osα sin β = sin( α + β ) n n z rω os( α β ) rω z = = 5 5 z os( α + β ) + rω ω = = r 5r Q = m = m( sinα os β + osα sin β ) = m sin( α + β ) n n Qz = m( z + rω ) = m[ os( α β ) + rω ] 5 5 Afterwards, let t he oordinates rotate, and transform then into XOY, the orresponding then the transformed formulas are shown as follows: X, Y (7) (8) (9) () (1) will hange, os( α β ) + rω X = n os β z sin β = sin( α + β ) os β + sin β 5 () os( α β ) + rω Y = n sin β + z os β = sin( α + β ) sin β os β 5 () X tgϕ = Y Besides, the enter of mass of the basketball is bound to be studied, the radius of the standard basketball is shown as L =.5m, and then, the time onsumed in horizontal diretion an be expressed as follows: L t1 = = X L α β β os( α β ) + rω 5 β sin( + )os + sin The movement in a vertial diretion an be expressed as follows: 1 Y = gt + osϕ t Then, if Y = t, the orresponding time is, so: 1 ( gt + Y ) t = (4) (5) (6) 91

7 Jiangtian Zhu and Liang Dong J. Chem. harm. Res., 14, 6(6):86-9 t = or Y sin( α + β )sin β 6 os( α β ) + 4rω t = = os β g g 5g Through alulation, it an be obtained as follows: t First, if > t1, the basketball will not enter the hoop after the ball hits the rim. t t Seondly, if 1, his basketball will enter the hoop after the ball hits the rim. CONCLUSION This paper studies on the major problems for shooting in a basketball game. The mehanial models of basketball forward spinning and bakward spinning are brought out. This problem about shooting is mainly related to the strength of basketball player throwing the ball, namely, the speed of the ball moving. Also, side rotation is related to other fators like the projetion angle. This model has a guiding signifiane for basketball players and oahes; therefore, the model is great importane to the development of basketball areer. REFERENCES [1] CHEN Jian, YAO Song-ping. Journal of Shanghai hysial Eduation Institute, 9, (5). [] Wang Luning et al. China Sport Siene, 1999, 19(),7-4. [] MAO Jie, SHAN Shuguang. Journal of Wuhan Institute of hysial Eduation, 1, 46(),7-7, 87. [4] ZHANG Lei. China Sport Siene and Tehnology, 6, 4(1),5-5. [5] FU Fan-fei, YU Zhen-feng, ZHANG Quan-ning. Shandong Sports Siene & Tehnology, 6, 8(),4-5. [6] Wang Luning et al. China Sport Siene, 1999, 19(),7-4. [7] LI Ji-hui. Journal of Shenyang Sport University, 6, 5(),6-66. [8]YANG Yue-qing, RAO Han-fu, LEN Ji-lan. Journal of Hubei Sports Siene,, (),4-5. [9] Zhang B.; Zhang S.; Lu G.. Journal of Chemial and harmaeutial Researh, 1, 5(9), [1] Zhang B.; International Journal of Applied Mathematis and Statistis, 1, 44(14), 4-4. [11] Zhang B.; Yue H.. International Journal of Applied Mathematis and Statistis, 1, 4(1), [1] Zhang B.; Feng Y.. International Journal of Applied Mathematis and Statistis, 1, 4(1), [1] Xiaomin Zhang. Journal of Chemial and harmaeutial Researh, 1, 5(1),