Effects of the Elbow Angle and Shoulder Rotation on a Free Throw
Jordan Tyler
Summer Ventures in Math and Science 2010
Visual and Image Processing
Dr. Rahman Tashakkori, Mr. Sina Tashakkori, Mr. Dale Henries
Appalachian State University
Abstract
The research shows how elbow and shoulder rotation will affect a basketball shooter’s free
throw percentage. This will be done using video editing and image processing software to find
data that will provide answers on how changing the degree of a shooter’s elbow and shoulder
rotation affect the shot their shot. Additional research also studies the trajectories and physics
behind the free throw line by analyzing angles from which a ball can be shot and the velocity
that is needed to make the basket. The research will show that the elbow and shoulder position
have a direct relationship with shot percentage.
1.0 Introduction
Basketball has been played for over a century. It was invented in 1891 by James Naismith in
a Y.M.C.A. training school in Springfield Massachusetts. The way that Naismith invented
the game was to dribble a soccer ball up and down a court and shoot the ball into a peach
basket. Basketball has come a long way in the past one hundred year by adding a standard
court size along with better basketballs, goals and rims. Basketball has become an American
tradition; millions of small children to old men play basketball. Basketball is a sport in
which fundamental skills are required. Basketball’s fundamental skills include dribbling,
passing, defense, and perhaps the most important shooting. Professional basketball players
practice shooting many hours daily, and even with all of that practice they do not make one
hundred percent of their shots. Most don’t even make fifty percent of their shots while being
guarded by another player. Basketball can become a very difficult sport if the player is being
guarded as that disrupts the players shot. This is because you have to create the room that
you need to shoot the ball, but why do so many players miss open shots such as free throws
in when they are guarded. Besides a layup a free throw should be the easiest shot in
basketball. This is because when players are at the free throw line you are not being
guarded, and they are at a set distance from the goal. A free throw should be exactly what it
sounds like; a “free” throw which should be made. So why do so many basketball players
miss their free throws? There are many factors that can affect a free throw shot, they are
mental, and fatigue factors, along with fundamental shot techniques. To be a good free
throw shooter the player must have a routine that he/she follow ever time. This routine
includes a stationary shot, which means that the player is not fading away from the basket.
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Fundamental shot techniques include balance, elbow, eyes, and follow through. Research
will look at how the angle of the elbow and shoulder rotation affects the free throw. The
shoulder and elbow were chosen to be studied because in order to “swish” the free throw the
first step is to be properly set up to the basket. The data will show what angles and shoulder
rotation provide the best results. The prediction for the experiment is that the optimal angle
for the elbow will be less than ninety degrees, and that the shooters’ shoulder should have
zero rotation which means their shoulder should be as close to their body as possible.
2.0 Method
A chart was created for different types of shots that would need to be taken to retrieve the
necessary data. Included were seventy-five shots at various elbow angle and shoulder
rotation. Video was taken as the shooter performed each shot. The shooter was filmed from
two different angles which were from the side of their shooting arm such that the angle of
their elbow could be found and from the front of the shooter such that the shoulder rotation
could be measured. For comparison the shooters’ free throw percentage was recorded
before the experiment. Each video was analyzed using video editing and image processing
software to look frame-by-frame to determine elbow angles and shoulder rotation. The data
was then placed into a spreadsheet where it was analyzed to determine how elbow and
shoulder rotation affected the shot. The types of shots that were taken from the elbow stand
point were, an acute angle shot, right angle shot, obtuse angle shot, straight arm shot and a
normal free throw for the person which was between the acute angle and the right angle
shot. Each of these was taken five times at different rotations of the shoulder.
3.0 Results
The optimal angle for the elbow during a free throw was determined to be from seventy to
ninety degrees. This was based on the number of free throws that were made and the angle
that they were shot from. As shown in Table 1 the total of the normal shots which ranged
from seventy to eighty degrees and the right angle shots which ranged from eighty-five
degrees to ninety-five degrees had the majority of the shots made.
As the shoulder rotation increased the amount of shots decreased. Along with
shoulder rotation the data suggests that as the angle of the elbow increases the number of
shots made decreased. The shooters original free throw percentage was eighty percent.
Table 1 shows the shots that were taken and the makes and misses.
It is clear in Table 1 that the shots taken from a normal shoulder position which is
located closest to the body. It is also clear that when the shooter took a normal shot the most
shots were made. The reason the right angle shot is almost as high as the normal shot is
because the normal shot for this shooter was from seventy to ninety degrees, which all of the
right angles were all plus or minus five degrees from ninety. Since there is not an effective
way to measure the angle of the elbow and the distance of the shoulder while the shooter
was shooting, image measurement software was used. The images were each looked at from
the side and the front such that the angles and distances could be measured. Figures 1 (a-c)
are the images from the normal shot, right angle shot and the straight arm shot.
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Table 1 – Data Collected on this Research
(a)
(b)
(c)
Figure 1- The Elbow Angles at a Normal, Right and Straight arm Shot
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Figure 2 (a-c) shows how the shoulder rotation was measured from the distance from
determining the right heel to the elbow during the same shot. The distance is the red line on
the inside of the blue line. Figure 2 (a-c) are all pictures that have no shoulder rotation.
(a)
(b)
(c)
Figure 2 – Shoulder Rotation with no Rotation
Figure 3 (a-c) are arranged from no shoulder rotation, to middle shoulder rotation, and full
shoulder rotation. Figure 3 (a) shows the distance from the blue line to be on right, Figure 3
(b-c) shows that the distance is on the left of the blue line. The data suggests that when the
distance on the left increases the shot percentage decreased. When the distance was inside of
the blue line (toward the shooter) the free throw percentage was much higher.
(a)
(b)
Figure 3 – Three Different Shoulder Rotations
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(c)
Shooting a free throw does not only have to do with shoulder rotation and elbow angle it has
to do with the angle at which the ball is shot from and the velocity that is needed to make
the shot. Figure 4 shows every angle that it is possible to make a free throw and the velocity
that is needed to make the shot assuming that the ball has no spin and goes straight through
the rim.
Figure 4- Relation between Free Throws and the Velocity of the Basketball
Using the mathematical model shown in (1) the chart in Figure 4 was formed to determine
initial velocity required to make the shot at any given angle. Figure 4 is set for a shooter
that would release the ball at eighty-two inches off the ground. This could be adjusted by
changing in the height in the mathematical model.
The actual shooting of the ball could become instinct with sufficient practiced
because the shooters’ body will naturally know at what angle they are and how hard they
need to shoot the ball, but the tricky part for most players they forget to set themselves up
correctly before the actual shot takes place which can make a significant effect on free throw
percentage.
𝑣0 =
𝐿
cos 𝜃
×√
−𝑔
2(𝐿×𝑇𝑎𝑛𝜃−ℎ)
5
(1)
4.0 Conclusion
This research could be very helpful to any basketball players who are looking to improve
their free throw percentage. As they can use the Mathematical model in (1) to find the
velocity and angle that they would need to shoot the ball on any position on the floor [1].
The actual shooting of the ball could become instinct with sufficient practice because the
shooters body would naturally know what angle their body is at and how much velocity is
needed to make the shot. The tricky part for most players is that they forget to set
themselves up correctly before the actual shot takes place. This can contribute profoundly
on the percentages of misses. This is why the player must keep their elbow and shoulder
rotation in mind because can drastically change their shot. This data helped find the optimal
elbow angle and shoulder position for the shooter, not every person shoots the same which
is why this data can not apply to everyone but the data does show that as the shoulder rotates
further away from the body the percentage drops, along with when the elbow angle
increases past a ninety-five degree angle the shot percentage also drops drastically.
Basketball is a fun sport it can be played anytime in your life at any recreation park or in a
drive way with a goal. Basketball is a simple game in which fundamental skills play a huge
roll such as making free throws. There are millions of games in the world that players or
teams have lost because they missed a free throw. A free throw is a simple shot, it is thirteen
feet and nine inches away from the goal, there is no one that is guarding you it is the ball,
the rim, and the shooter, and all that they have to do is make the shot. The error that could
have occurred in this lab is the shooters free throw percentages can throw the data off. The
image measurement software can be read wrong which can skew the results. The error was
only a few centimeters or degrees in the image measurement. The data was very precise and
shows clear results that the optimal elbow angle if from seventy to ninety-five degrees, and
that if the shoulder rotates away from the body the shot is least likely to go in.
5.0 References
[1]. Joerg M. Gablonsky, Andrew S. I. D. Lang, “Modeling Basketball Free Throws” Society for
Industrial and Applied Mathematics, Vol. 47, No 4, pp 775-798, 2005.
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