The Mathematical Modeling Behind a Perfect 3-Point Shot - Practice IA

1. INTRODUCTION
2. CONSIDERATIONS 
   • Observations and Assumptions
   • Method
   • Use of Technology
3. COLLECTED DATA 
   • Definition of Conditions 
   • Modelling and Scenarios 
   • Determining the Parabola for Each Condition 
4. EVALUATION 
   • Evaluation of Data
   • Strengths and Limitations 
   • Conclusion 

Michael Jordan, Aisha Sheppard, Stephen Curry, and Caitlin Clark are some of the best 3-point basketball shooters of all time, averaging between 32.7% to 43.3% of all 3 points attempted made (Google Search). But what makes or breaks their perfect throw? Beneath the heat and roar of the game, these flawless shots are determined by a complex array of mathematics and physics. When investigating the perfect shot, factors such as the height of a person, the height of the hoop, arc length, angle, velocity, force, and spin all contribute to the satisfying swish of a clean throw.

The presence of mathematics within basketball is personally fascinating, as the sport is a huge part of my life. Figuring out how to shoot a perfect 3-point would not only improve how I play basketball but would further my knowledge of the mathematical aspects involved and prove applicable to various other sports. The abundance of factors that go into determining the success of a basketball throw all contribute to the end result. Subsequently, a constancy within certain aspects is present, such as the distance from the hoop, height of the hoop, and the size of the hoop. This investigation specifically focuses on the quadratic equations and physics forces that go into a 3-point basketball shot, relevant to constant assumed control variables. The aim of this research is to calculate the appropriate parabola, degree angle, and force for a variety of heights, accordingly determining the perfect 3-point throw.

I will explore these factors in relation to their impacts on the probability of scoring a perfect 3-point shot, by manipulating the height variable to investigate three different scenarios. This investigation will explore the impact of height differences in players on the modelled curve of a shot and its ability to enter the net with a ‘clean swish.’

The assumption was made that constancy of individual body-related differences would be void within the investigation, such as arm length, proportions, hand size, and dexterity. This assumption ensured that these factors would not impact the calculations and or influence areas of uncertainty within the data. It was important to define the ‘perfect 3 point shot’ to evaluate the shot that would have the highest probability of going in. I had chosen to define a ‘perfect shot’ as the throw that went in without bouncing off the rim or the backboard, which in hand would result in the satisfying swish of the net. The perfect shot would be ‘achieved’ when the center of the basketball passed through the center of the hoop exactly at an angle that would not result in the ball hitting the rim.

Exploring this topic relied on mathematical concepts such as trigonometry, quadratics, linear regression, and moderate physics calculations. Therefore, preliminary research needed to be carried out to explore the standard variables that would be kept constant within the investigation; these were outlined in the Observations and Assumptions section above. To collect adequate data regarding the investigation, the following procedure was carried out:

•  Preliminary research regarding previous mathematical investigations of basketball was performed, investigating the varying heights, arc lengths, and angle degrees.
•  Data on each aspect was collected from the sources and used to construct a personal data collection method.
•  Specific heights were determined to act as the independent variable conditions.
•  All constant variables were substituted into the appropriate equations; then, the condition values were plugged into the equations to calculate the varying arc lengths, the force necessary, and angle degrees according to each height.
•  The data were modelled using the program Desmos to visualize the shot with all variables included.
•  The parabola of a ‘perfect shot’ for each condition was calculated.

The use of technology within the investigation was crucial for providing modelling for the conditions. A spreadsheet in Excel was used to prepare all the data, which were then transferred into the Desmos graphing program. All calculations were carried out primarily by hand and were then checked with online software and calculators. These programs allow for the accurate determination of angle degrees, arc lengths, quadratic equations, and physical forces present within each height condition.

A margin of entry angle for the ball to go through the center of the hoop was necessary to create a quadratic equation to model the arc of the ball across all conditions. I expected this angle to be around 50 degrees as if the ball was thrown at a 45-degree angle it would directly intersect the back rim. Due to this assumption, I created a  margin of angles based on the unit circle in the cartesian plane:

\(-90°<×<-50°\)​​​​​​​

Coordinate A: (0ft, 5.83ft)

Coordinate B: (24.5ft, 10ft)

Coordinate C: (12.25ft, 15.83ft)

Coordinate A: (0ft, 6.17ft)

Coordinate B: (24.5ft, 10ft)

Coordinate C: (12.25ft, 16.17ft)

These angles were plotted on a Desmos coordinate plane for each condition. Said coordinates were further used to determine a parabola function to fit each condition. The equation y=ax²+bx+c is the standard equation for a quadratic function and was used to determine the equation for each condition with the points outlined above.

Condition 1:

• Coordinate A: x=0 y=5.5
• y=ax²+bx+c
• 5.5=a(0)²+b(0)+c
• 5.5=c
• Coordinate B: x=24.5 y=10
• y=ax²+bx+c
• 10=a(24.5)²+b(24.5)+c
• 10=600.25a+24.5b+c
• Coordinate C: x=12.25 y=15.5
• y=ax²+bx+c
• 15.5=a(12.25)²+b(12.25)+c
• 15.5=150.06a+12.25b+c

This led to the developing of two quadratic equations specific to the two points in condition 1. To create a parabola to fit all three points, simultaneous equations were used. This solves the problem of having three undefined variables in the functions.

Equation 1: 5.5=c

Equation 2: 10=600.25a+24.5b+c

Equation 3: 15.5=150.06a=12.25b+c

As c was defined within the first equation to equal 5.5, it could be plugged into the two latter equations.

Equation 2: 10=600.25a+24.5b+5.5

Equation 3: 15.5=150.06a+12.25b+5.5

This resulted in two equations with two unknown variables that could be solved simultaneously.

• \(10=600.25a+24.5b+5.5\)
• \(4.5=600.25a+24.5b\)
• \(600.25a=4.5-24.5b\)
• \(a=4.5-24.5b/600.25\)
• \(a=0.0075-0.04lb\)​​​​​​​
  • \(15.5=150.06a+12.25b+5.5\)
  • \(15.5=150.06(0.0075-0.04lb)+12.25b+5.5\)
  • \(10=1.125-6.15b+12.25b\)
  • \(8.875=6.1b\)
  • \(b=1.455\)​​​​​​​
    • \(a=0.0075-0.04lb\)
    • \(a=0.0075-0.041(1.455)\)
    • \(a=0.052\)​​​​​​​
      • \(y=-0.052x^2+1.455x+5.5\)

The following calculations were repeated to determine the parabolas of the two other height conditions. Please see the table below:

Table 2 – Quadratic Equations for Each Condition

The data collected through this mathematical investigation provided insight into the effect of player height on the curve of a basketball shot. The investigation subsequently allowed the modelling of the perfect ‘clean throw’ in reference to an array of controlled variables. The equations were determined to suggest a low variance in the arc of a basketball when dependent on different heights. Furthermore, it implies that the taller the player is, the higher the vertex of the parabola would lie. In addition to the function findings, the ideal entry angle for the basketball accordingly provided insight into the mathematics behind a ‘clean throw,’ as there was only a narrow margin of angles that would result in the ball not intersecting the rim.

STRENGTHS AND LIMITATIONS

A strength of this investigation was that the equations were purposefully calculated to fit an array of points, leaving little room for variation. The investigation furthermore provided insight into the usefulness of quadratic modelling in sports due to the nature of the ‘bell curve’ shape of the arc. The evaluation of 3 different heights allowed for a comparison between the equations and parabolas to be made, in hand prompting conclusions derived from the differences between the functions. This could be further applied to a real-life scenario in basketball and could enhance your shot.

However, this investigation was limited by the control of variables influencing the equations. There were numerous variables that were set in order to create an applicable model that solely investigated the influence of height on a ‘perfect 3-point shot.’ Although this provided clarity within the findings, it limited the ability of the investigation to further explore the mathematics behind the parabola curve. One of the biggest controls was the implications of the set vertex of the parabola, which limited the possible shape the arc could take. Although this allowed for the equations to be calculated, in further investigation, I would explore a range of vertexes for each condition, allowing for a comparison between the ball's entry angles and the parabola's shape.

Furthermore, I feel an application of physics could have been applied to this research. This would have allowed the findings to be more applicable to a real scenario by exploring the effect of force on the curve of the basketball shot.

The application of mathematics in sports, although extremely prevalent, often goes overlooked by the players. Throughout this investigation, I have gained practical knowledge about the quadratics involved in my basketball shots, which will allow me to improve how I execute them in the future. These findings play a huge importance in my personal involvement with basketball, as it is a big part of my life, and discovering ways to improve my game is always necessary for becoming a better player. Not only is math involved in basketball, but in numerous other sports, which demonstrates its practicality within the world around us, subsequently influencing the way we perform in our favourite activities.