are ENDLESS in the field of sports.

Athletes, trainers, and coaches use

**calculus to achieve**

**O**

**N**

**O**

**A**

**I**

**M**

**I**

**T**

**P**

**I**

Calculus is the mathematical study of change.

the human body’s physiological capacity for running can be measured with set of simple differential equations

In track and field,runners

must decide how to pace themselves to achieve their quickest times.

the theory explains...

2

Applied Calculus

in Sports

**Z**

**T**

Players are constantly researching the calculus behind their respective sports to gain a competitive advantage over their colleagues.

Calculus can be used in basketball to find the arc length of a shot from the shooter's release to the net.

The arc is the path the basketball flies from the time it leaves the shooter's hand until it arrives at the basket.

After the shot, given the strength of relase and the angle of release, a mathematician can accurately predict the path and arc length because gravity is the only force affecting the ball.

Photo 1 shows the parabolic path that the ball takes from the shooter's hand to the basket.

Photo 2 shows the arc formed over time with the red line.

Figure 1: Extreme Entry Points: 90° Arc enters directly from above, 0° Arc enters directly from the side

Figure 2: (Red) 30˚, (Green) 45˚ and (Blue) 60˚ Arcs

KEY TERMS

Hoop Size

is 18 inches, which is the diameter of the hoop ring.

Because a basketball size is smaller than the hoop, there is a constant hoop margin.

Hoop Margin

is the amount of open space left after the ball enter the 18 inch ring of the hoop. If a ball enters the hoop directly from above, since a men's ball is 9.7 inches in diameter, the amount of open space left in the hoop is 18 – 9.7 = 8.3 inches.

Jump shots, free throws, and three pointers enter at an, which makes the hoop more of an oval entrance. This changes the Hoop Margin.

Apparent Hoop Size

is the apparent opening of the hoop to the ball. The flatter the arc, the smaller the ellipse.

Because there are various ways a shot can be taken and many angles the ball can fly the Apparent Hoop Margin also varies.

Apparent Hoop Margin

is the Apparent Hoop Size minus the diameter of the ball. The greater the margin, the greater the error of margin that a shooter can take.a

Analysis

Although the greatest AH Margin and AH Size is obtained when the ball travels a 55 degree Arc, the actual optimal shot is for it to travel at a 45 degree Arc. This is because at 55 degrees, it is proven significantly harder for the shooter to aim and release at that angle constantly and consistently without maintaining release strength. The distance control of a higher shot suffers greatly. Nonetheless, professional athletes in the NBA have practiced and mastered their release to an angle closer to 55 degrees to fully gain an advantage of the AH Margin and AH Size.

One of these great shooters have forever changed the game and is currently redefining basketball history.

Stephen Curry

During the 2012–13 season, Curry set the NBA record for three-pointers made in a regular season with 272. He surpassed that record in 2015 with 286, and again in 2016 with 402. During the 2013–14 season, he and teammate Klay Thompson were nicknamed the "Splash Brothers" en route to setting the NBA record for combined three-pointers in a season with 484, a record they broke the following season (525) and again in the 2015–16 season (678).

Velocity

Let r(t) be a differentiable vector valued function representing the position of a particle. Then the speed of the particle is the magnitude of the velocity vector. In one variable calculus, we defined the acceleration of a particle as the second derivative of the position function. Jerk is the third derivative. In our case, the basketball is the particle.

Arc

Finding the arc length needed to make a 3 point shot:

The derivatives of x(t) and y(t) in respect to t are:

Using the average velocity of a basketball shot, 2.24 m/s, the shot angle that would produce maximum efficiency, 45 degrees, in the time it would take the ball to travel from the free throw line, about 2 sec, the arc length can be calculated.

Therefore

Basketball

Equations

Velocity of a 3 Point Shot

Equation at which player must shoot a basketball in order for it to land perfectly in the basket.

Formula for solving for the range of the basketball

Ideal Angle: 45 degrees

Graph of the Range Function of How Hard the Player Must Shoot the Ball

Solving for Velocity

Substitute 25 ft in for the range and solve.

In order for to make the shot, the player must shoot the ball at approximately 28 feet/sec

**Calculus in**

Running

Running

"Running a world-record track and field race depends on a lot of factors: psychological, physiological, biological, and physical. Perhaps not surprisingly, underlying much of it, is the mathematical."

—Alex Miller

His model predicts that for all races under 291 meters, the optimum strategy is to sprint at 100% acceleration or at V02 max for the entirety of the race. Races longer than 291 meters require a more complicated strategy to optimize performance.

Joseph Keller’s

“A Theory of Competitive Running”

OPTIMIZATION

VO2 max refers to the maximum amount of oxygen that an individual can utilize during intense or maximal exercise. It is measured as "milliliters of oxygen used in one minute per kilogram of body weight."

JESSE OWENS

Owens was an American track and field athlete and four-time Olympic gold medalist. Owens specialized in the sprints and the long jump and was recognized in his lifetime as "perhaps the greatest and most famous athlete in track and field history". A sprinter in the 100m, Owens also specialized in the high jumps. He set three world records and tied another in less than an hour at the 1935 Big Ten track meet in Ann Arbor, Michigan. It was deemed"the greatest 45 minutes ever in sport". Owens was ranked as the sixth greatest North American athlete of the twentieth century and the highest-ranked in his sport.

Calculus in Baseball

Calculus can also be used in baseball to optimize a pitcher's throwing mechanic to maximize efficiency. In addition, calculus can be used to calculate the projectile motion of a baseball trajectory. Lastly, related rates is important to check if runners can make the next base on time given their speed and the speed of a hit ball.

Bob Bowman, president of MLB Business and Media, explains new baseball statistics and how they might be displayed during a broadcast at a news conference. This new system will also allow coaches to use calculus to improve their players in training.

Links

https://d284gedng9vuu0.cloudfront.net/article_media/2016/04/sports-metrics-7ba7cba7.jpg.885x497_q90_box-0,297,4571,2864_crop_detail.jpg

http://jwilson.coe.uga.edu/EMAT6680/Huffman/Mathematics%20in%20Sports/MathematicsSports.html

https://packbackbooks.com/answers/1932911/How-is-calculus-used-in-the-world-of-sports

http://www.ams.org/samplings/feature-column/fcarc-sports

http://realworldmath.tumblr.com/post/63694329609/mathematics-and-running-pace-strategy

http://www.noahbasketball.com/make-shots

https://prezi.com/x9ak7iq393zg/calculus-in-sports/

http://www.stewartcalculus.com/data/ESSENTIAL%20CALCULUS%202e/upfiles/instructor/eclt_wp_0706_inst.pdf