Applications in Calculus: Basketball

Research/Problem: Over the past few weeks, I have been looking into the applications

of calculus in basketball. I began this process of research a bit about the general

applications of calculus in sports and then focused specifically on basketball. In order to

accurately display an example of the application, I decided to use one of the NBA’s best,

Steph Curry, to generate the numbers I would need. I found information about his three

point shots and general information about him that would be important in generating an

accurate situation to write a parametric problem from. Using his height of 6 feet 3

inches, I determine that the ball is released at roughly 8 feet three inches. This

estimation was found through an average 9 inch jump on a three point shot and

considered the distance of the ball over his head. Steph Curry's average release for a

three point shot is at a 50 degree angle so I planned to use this in my problem. I also

found the distance from the NBA three point line to the hoop to be 23.75 feet and the

height of the hoop is well known by most people. The hoop is 10 feet from the ground.

This was needed when creating the diagram on a coordinate plane.

To start, I used a feet to meters conversion calculator. I was able to convert these

distances and values to meters to be included in my solution that measures in meters

per second. Using these measurements, I created my diagram. I added a picture of

Steph Curry taking a three point shot into notability. Over the picture, I first added an

x-axis and a y-axis. I traced the trajectory of the ball and added lines for the height hoop

(3.048 meters), distance of the hoop from the three point line (7.239 meters), Steph

Curry’s height (1.905 meters) and the shot release height (2.4384 meters). I also

marked the peak of the shot.

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 After these initial markings were made on the original picture, I added them to a

coordinate plane as it was easier to see. Using Desmos, I matched the trajectory of the

ball to a quadratic equation. This equation is f(t) = -0.28t^2 + 1.6t + 2.718 where t is

measured in seconds. Using this equation, I wrote a problem that I would go on to

solve. This problem can be seen below.

“Steph Curry attempts a three point shot in a game against the Detroit Pistons. This

equation can be modeled by the function f(t) = -0.28t^2 + 1.6t + 2.718 and is measured

in meters per second. Using this equation, find the arc length of the shot.”

Solution: To solve this problem, I first gathered and recorded all important information I

would have needed. I took notes on parametric functions and noted the formula I would

use to find the length of a parametric curve. I then organized my variables. These can

be seen on the top left of my solutions page. Using my notes and these variables, I

found the vertical and horizontal positions of the ball. I used these values to find the

position vector of the particle. The positions vectors can be defined as < x(t), y(t)>.

These values were taken to the arc length equation. I took the derivative of x(t) and y(t).

This process can be seen on the solutions pages, directly under the arc length equation.

Using dx/dt and dy/dt I was able to fill in the information of my arc length equation. I

determined the bounds of the integral by timing a three point shot made by Steph Curry.

I determined a to be equal to 0 seconds and b to be equal to 2 seconds. I simplified

under the square root. This simplification can be seen on the solutions pages where the

arc length formula is boxed. Following this step, numbers specific to this situation

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involving Steph Curry are substituted to get an arc length of 21.02 meters per second.

The steps from the substitution to the answer can be followed and begin after the boxed

formula.

Importance: By showing my solution and the steps it took to generate an answer, this

real life application of calculus can be used as an example for others to reference.

Finding the arc length of the trajectory of the ball can prove to be very important to

players because a different arc creates a different angle when the ball is coming at the

hoop. This angle determines whether the shot is made or missed, so in order to

increase a player's three point shot percentage, the angle must be accurate.

Since Steph Curry releases his three point shot at the ideal angle of 50 degrees,

we know that the ideal arc length for a NBA three point shot is roughly 21.02 meters in

distance.

If a player wanted to improve their percentage of three point shots made, they

could record the important data from a shot. This would include their height, the height

at which the ball was released, and the release angle. They could calculate the arc

length of the shot and compare it to 21.02. By comparing these numbers they could

make adjustments in order to increase or decrease the length of the arc. The player

may need to jump higher to get a higher release point or maybe the player needs to

increase the angle at which the ball is released in order to get the perfect length. Once

they determine the problem or problems in their current shot, they can work on fixing it

to ultimately increase their overall accuracy and therefore three point shot percentage.