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数学建模:篮球罚球的研究

2012-08-03 24页 pdf 952KB 24阅读

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数学建模:篮球罚球的研究 SIAM REVIEW c© 2005 Society for Industrial and Applied Mathematics Vol. 47, No. 4, pp. 775–798 Modeling Basketball Free Throws∗ Joerg M. Gablonsky† Andrew S. I. D. Lang‡ Abstract. This paper presents a mathematical model for basketball free throws. It is intend...
数学建模:篮球罚球的研究
SIAM REVIEW c© 2005 Society for Industrial and Applied Mathematics Vol. 47, No. 4, pp. 775–798 Modeling Basketball Free Throws∗ Joerg M. Gablonsky† Andrew S. I. D. Lang‡ Abstract. This paper presents a mathematical model for basketball free throws. It is intended to be a supplement to an existing calculus course and could easily be used as a basis for a calculus project. Students will learn how to apply calculus to model an interesting real-world problem, from problem identification all the way through to interpretation and verification. Along the way we will introduce topics such as optimization (univariate and multiobjective), numerical methods, and differential equations. Key words. basketball, mathematical modeling, calculus projects AMS subject classifications. 00-01, 00A71, 26A06 DOI. 10.1137/S0036144598339555 1. Introduction. In these days of superstar basketball players, you would think that shooting free throws should be as much a formality, and just as exciting, as the extra point in professional football. Not so. Take for example Shaquille O’Neal, the subject of our first model, who as of the end of the 2004–2005 regular season had a career free throw percentage of 53.1%. His troubles seemed to increase during the playoffs, where he shot around 45% from the line. Shaquille is not alone in his free throw shooting troubles. In fact nearly one-third of all NBA players shoot less than 70% from the foul line. When a basketball player steps up to shoot a free throw he does not usually think (unless he also happens to be a mathematician), “I wonder if my free throw shooting percentage would improve if I changed my initial shooting angle,” or “I wonder how air resistance affects the trajectory of my shot,” or even “Should I be aiming for the back rim, front rim, or the middle of the basket?” We present here a calculus-based model for basketball free throws to show that they should address some of these musings. We begin by conjecturing that some players shoot poorly from the line because they are shooting the ball at the wrong angle. Therefore, the focus of our model will be the release angle, a simple place to start, and we will extend it later. Some of the more interesting facts that we’ll discover by refining and interpreting our model are: 1. The best way to shoot free throws depends upon the person shooting. The two most important factors are their height and ∗Received by the editors May 22, 1998; accepted for publication (in revised form) July 20, 2005; published electronically October 31, 2005. http://www.siam.org/journals/sirev/47-4/33955.html †The Boeing Company, Mathematics and Engineering Analysis, P.O. Box 3707, MC 7L-21, Seat- tle, WA 98124-2207 (joerg.m.gablonsky@boeing.com). ‡Department of Computer Science and Mathematics, Oral Roberts University, 7777 South Lewis Ave., Tulsa, OK 74171 (alang@oru.edu). 775 776 JOERG M. GABLONSKY AND ANDREW S. I. D. LANG how consistent they are in controlling both the release angle and the release velocity. 2. In general, the taller you are, the lower your release angle should be. We’ll actually see that taller players are allowed more error in both their release angles and release velocities and thus they should have an easier time shooting free throws than shorter players. 3. It is much more important to consistently use the right release velocity than the right release angle. 4. The best shot does not pass through the center of the hoop. The best trajectories pass through the hoop somewhere between the center and the back rim. Taller players should shoot closer to the center while shorter players should aim more towards the back rim. 2. Mathematical Modeling. Before we jump into modeling a basketball free throw, it would help for us to tell you exactly what we mean by mathematical mod- eling: Mathematical modeling is the process of formulating real world situations in mathematical terms. Less formally, mathematical modeling takes observed real-world behaviors or phe- nomena and describes them using mathematical formulae or equations. All the formu- lae you see in your physics, chemistry, and biology classes are mathematical models. Mathematical models can be found everywhere, not only in science, but also in the social sciences and even in business. For instance, there are people who get paid very well to model the stock market. By constructing mathematical models, we can often explain real-world behavior, predict how sensitive real-world situations are to certain changes, and even predict future behavior (very useful for the people who model the stock market). The following is a summary of the standard steps for constructing a mathematical model: 1. Identify the problem. What do you want to find out? 2. Derive the model. Identify the constants and variables involved. Make assumptions about which variables to include in the model. Determine the interrelationships between the variables. 3. Solve the equations and interpret the model. 4. Verify the model. Does it answer the original problem? Does it match up to real-world data? 5. Refine the model. If the model is not satisfactory, refine it by removing some of your earlier assumptions. We’ll discuss these steps in greater detail as we use them to model basketball free throws. 3. Our First Model: The Best Angle. It is true for most models, including ours, that trying to include every possible physical effect immediately is rather ambitious, especially if you want to be able to solve the model. The modeling process typically begins with the construction of very simple models which are easy to solve. Models are then refined to make them more realistic, which in turn requires the introduction of more powerful mathematics in order to solve them. In the end the model should MODELING BASKETBALL FREE THROWS 777 Table 3.1 The physical constants of the problem. Physical constant Symbol Value Rim diameter Dr 1.5 ft Ball diameter Db 0.8 ft Horizontal distance traversed l 13 ft 6.5 in Vertical distance traversed h 1 ft 1 34 in Acceleration due to gravity g −32 ft s−2 be refined enough to describe reality as closely as possible while still being solvable. You’ll see this refinement process in action as we go through the modeling procedure. 3.1. Problem Definition. When watching basketball players shoot free throws we notice that sometimes they make small errors and still make the basket. It seems reasonable that the amount of error that the player can make and still have the shot go in depends on the initial angle that the ball was thrown. We’ll therefore begin by defining the problem as follows: Given a basketball player of a certain height, what is the best angle for him to shoot a free throw. 3.2. Deriving Our First Model: Identify the Constants and Variables. The physical constants (the diameter of the rim, etc.) that we shall use to derive the equa- tions of motion that govern the flight of the ball can be obtained from various sources, including the Internet, books [14, 17], and actual (tape measure) measurements. The diameter of the rim, Dr, is 1.5 ft. The diameter of the ball, Db, is taken to be 0.8 ft.1 It has been observed [7] that free throws are shot from a few inches in front of the free throw line. We thus take the horizontal distance traversed, l, to be 13’6.5” rather than the total distance, 14’, from the free throw line to the center of the hoop. It has also been observed [4, 8, 19] that shooters release the ball, on average, from a height of approximately 1.25 times the shooter’s own height. For example, a 7’1” tall player releases the ball, on average, at a height of approximately 8’1014”. Thus, for a 7’1” tall player, we would take the net vertical distance traversed, h, to be 1’1 34”. See Table 3.1 and Figure 3.1. 3.3. Deriving Our First Model: Simplifying Assumptions. We make the fol- lowing simplifying assumptions for our first model: 1. Allow only “nearly nothing but net” shots. By this we mean, allow only trajectories that either (a) go directly in (nothing but net), or (b) hit the back of the rim and then go directly in. We do this to account for a large range of successful trajectories while keeping things fairly simple. To make sure that the ball actually goes in, and does not bounce out, after hitting the back of the rim, we shall consider only trajectories where the center of the ball is at or below the height of the rim when the ball hits it. See Figure 3.2. 2. Ignore air resistance. The effect of air resistance is minor compared to the mathematical complexity it adds to the model. 1The actual diameter of a basketball, Db, can vary legally from approximately 0.78 ft to 0.81 ft. We assumed an average diameter of 0.8 ft. 778 JOERG M. GABLONSKY AND ANDREW S. I. D. LANG Fig. 3.1 The conceptualization of the free throw. Fig. 3.2 Ball going in off the back of the rim. 3. Ignore any spin the ball may have. Spin becomes important if we allow the ball to bounce before it goes in. Since we are only allowing nearly nothing but net shots, and ignoring air resistance, we’ll also ignore spin. 4. There is no sideways error in the trajectory. If you want to be a good free throw shooter, you really ought to shoot straight. The benefit we get from assuming the shooter always shoots straight is that the model will be two- dimensional (constrained to a plane). If transverse error were to be included, the model would be a more realistic, but harder to solve, three-dimensional one. 5. There is no error in the initial shooting velocity. We are assuming that some basketball players have problems shooting free throws because they are shooting at the wrong angle. Therefore, our first model concentrates on errors in the release angle only. 6. The best shot is one that goes through the center of the hoop. That is, the model will be one in which the initial velocity is the velocity that would drop the center of the ball through the center of the hoop. Some coaches encourage this by placing an insert into the ring that makes the aperture smaller. 7. The shooter is 7’1” tall. After we find the best angle for Shaq, we will quickly remove this assumption and find the best angle for people of a more diminutive stature. MODELING BASKETBALL FREE THROWS 779 Fig. 3.3 Resolving the initial velocity into horizontal and vertical components. These assumptions may seem very stringent. For example, not everyone is as tall as Shaq, and basketball is not usually played in a vacuum. Remember, though, that to begin with, the model should be a simple one—one that is easy to solve and interpret. Later, in the refinement stage, the model will become more realistic and some of theses assumptions will be removed. 3.4. Deriving Our First Model: Mathematical Interrelationships between the Variables. The goal of this section is to derive a mathematical formula that expresses the amount of error a player can make in the release angle in terms of the other vari- ables identified above. We’ll do this by taking standard projectile motion equations that are derived from Newton’s second law of motion. A more in-depth discussion of these “projectile motion” equations than presented here can be found in any basic physics book [6]. Instead of finding one long formula for the amount of error that the player can make before missing the basket, it is better to break down the equation into separate parts (called submodels) and put things back together later. We begin by resolving the initial velocity v0 into horizontal and vertical components, vH = v0 cos(θ0)(3.1) and vV = v0 sin(θ0),(3.2) respectively, where θ0 is the initial release angle. See Figure 3.3. Using a subscript 0 to identify initial values of variables is a common convention in mathematical mod- eling. Horizontally, there is no acceleration due to gravity or, by assumption, any air resistance. Thus the horizontal equation of motion is x(t) = vt,(3.3) where x(t) stands for distance, v for velocity, and t for time. Substituting our initial horizontal velocity into this equation we obtain x(t) = v0 cos(θ0)t.(3.4) Using l as the horizontal distance to the center of the basket and letting T be the time it takes to get there, we substitute x(T ) = l into (3.4) and obtain for our model l = v0 cos(θ0)T.(3.5) 780 JOERG M. GABLONSKY AND ANDREW S. I. D. LANG Similarly, the vertical equation of motion is given by y(t) = vt+ 1 2 gt2 = v0 sin(θ0)t+ 1 2 gt2,(3.6) where g = −32 ft s−2(−9.8m s−2) is the acceleration due to gravity. Substituting y(T ) = h, the vertical distance to the center of the basket, into the above equation, we obtain for our model h = v0 sin(θ0)T + 1 2 gT 2,(3.7) where h is the vertical distance to the center of the basket. Solving (3.5) for T , T = l cos(θ0)v0 ,(3.8) and substituting it into (3.7), we find the initial velocity v0 needed, for a given initial angle θ0, so that the basketball goes through the middle of the hoop: v0 = l cos θ0 √ −g 2 (l tan (θ0)− h) .(3.9) We note that this formula gives us sensible answers only for a limited range of θ0. Not only does it physically make sense to restrict initial angles to ones that result in forward motion, i.e., 0 < θ0 < 90◦, but also notice that the formula gives real values only for l tan(θ0)− h > 0 (remember g is negative). Physically this inequality corresponds to the ball having a sufficiently high initial release angle to reach the height of the rim. Thus we take the range of initial release angles to be tan−1 ( h l ) < θ0 < 90◦. In modeling, it is always good practice to note the range that your parameters can take. Otherwise you may unwittingly attain solutions which turn out to be nonphysical. We make special note of the physical range of θ0 here, because it is important for the numerical methods used to find solutions later in this paper. Furthermore, note that we assume that the ball will not hit the front of the rim on trajectories where the ball passes through the center of the hoop. We will show later that this might not be true, especially for shorter players. With the equations of motion modeled, we now derive the equations for the amount of error that can be made in the initial angle θ0 and still have the ball go directly into the basket. Keeping the initial velocity fixed, allowing the initial release angle to vary (this corresponds to our basketball player making an error in his release angle θoops0 ), and replacing l by x in (3.7) and (3.8), we work out the new horizontal position of the ball as it comes back down to the basket height, x = v0 cos (θ oops 0 ) −g ( v0 sin (θ oops 0 ) + √ v20 sin 2 (θoops0 ) + 2gh ) .(3.10) In the above equation θoops0 corresponds to a larger (or smaller) release angle (due to player error) than the ideal initial angle θ0 where the ball passes through the center of the hoop; see Figure 3.4. MODELING BASKETBALL FREE THROWS 781 Fig. 3.4 Comparison of the ideal trajectory that passes through the center on the hoop (v0, θ0) (red) and the trajectory with an error in the release angle (v0, θ oops 0 ) (blue). Fig. 3.5 The distance s between the front of the rim and the center of the ball. We now derive the following two criteria for the basketball to still go in the net: 1. To avoid contact with the front of the rim, the distance s between the rim and the center of the ball must remain greater than the radius of the ball throughout its trajectory, i.e., for all times t such that 0 < t < T ; see Fig- ure 3.5.2 Using for convenience the square distance, we have the following criterion for the ball not to hit the front of the rim: s2 = (x(t)− (l −Dr/2))2 + (y(t)− h)2 > (Db/2)2.(3.11) 2. From (3.10), we note that x+Db/2 is the horizontal distance to the rightmost part of the ball when the center of the ball is level with the basket. And since l + Dr/2 is the horizontal distance to the back of the rim, the criterion for having the ball hit the back of the rim as the center of the ball passes through the basket is x+Db/2 = l +Dr/2.(3.12) 2We note that this range of t, when the ball is near the rim, can be sharpened to l−Dr v0 cos(θ oops 0 ) ≤ t ≤ − 1 g (v0 sin(θ oops 0 ) + √ v20 sin 2(θoops0 ) + 2gh) −1. 782 JOERG M. GABLONSKY AND ANDREW S. I. D. LANG 3.5. Solving the Equations. To find the error allowed for a given initial angle θ0, we keep v0 fixed and solve numerically for the unique release angles θlow < θ0 and θhigh > θ0, which are, respectively, the solutions to the following equations: s2 − (Db/2)2 = 0,(3.13) the ball is released at an angle lower than θ0 and just misses the front of the rim as it goes in, and x− l + Db −Dr 2 = 0,(3.14) the ball is released at a angle higher than θ0 and hits the back of the rim and goes in. We note here that increasing the initial angle may increase xh, the distance to the center of the ball as it passes through the center of the hoop, but after a certain point xh will start to decrease. This can happen before the ball hits the back of the rim. So for certain trajectories, there is no solution to (3.14) and both θlow and θhigh are solutions to (3.13). This behavior will be made more apparent in the next few sections. After solving for θlow and θhigh, we find the minimum deviation from θ0, e (θ0) = min{θhigh − θ0, θ0 − θlow}.(3.15) The best release angle is the one that maximizes this function. In calculus, to find a maximum of a function, we usually differentiate it and then find the zero of the derivative. It can be shown that for a differentiable function, the derivative is zero at the maximum. This is a so-called necessary condition. To ensure a maximum, the second derivative must also be negative at that point. This additional condition is called a sufficient condition. So we are tempted to solve the equation e′ (θ0) = 0. Unfortunately if we tried to do this, we would run into trouble. To see why let’s examine Figure 3.6, a plot of the error function e (θ0) for various values of θ0. We created Figure 3.6 using linear interpolation (connecting the points with lines). The function is clearly not differentiable at the maximum; the left-hand slope is not equal to the right-hand slope. This can be easily explained by recognizing that (3.15) contains the min function, which can introduce nondifferentiability. Most standard optimization methods require at least first differentiability and therefore cannot find the maximum of this function. That is, we can’t take the derivative and set it equal to zero as we usually do to find the maximum because the function is not differentiable. So to find the maximum we use numerical methods that work for non- differentiable functions. You may have already seen in your calculus class numerical methods used to approximate function values (tangent line approximations) or to find roots of equations (Newton’s method). It is also possible to find maxima and minima numerically. The exact numerical method you use to find the maximum is not important, but the interested reader can find more information on optimization methods in standard texts; see, for example, [10]. Using a computer algebra system’s optimization routine,3 we get a best angle for our simplest model of θ∗center ≈ 48.18◦. To have the ball pass through the center of the hoop for this angle, Shaq would have to release the ball consistently at an initial velocity of 3We used MATLAB, but you could use Maple or Mathematica if you prefer. You can download our MATLAB code from http://epubs.siam.org/sam-bin/dbq/article/33955. MODELING BASKETBALL FREE THROWS 783 40 42 44 46 48 50 52 54 56 0 0.5 1 1.5 2 2.5 3 3.5 4 e (θ 0 ) θ0 Fig. 3.6 The error about θ0 for which the basketball still goes in. v∗center ≈ 6.62m/s (21.7 ft/s). Note that labeling an optimal solution with an ∗ is another common convention in mathematical modeling. It may seem silly to imagine Shaq stepping up to the line and thinking “48.18◦, 48.18◦, I must shoot at 48.18◦.”4 For some players though, it may have to start t
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