Prof. Basir Shafiq
June 8, 2026
DYNAMICS (INGE 3032): FINAL PROJECT
The Physics of Baseball Breaking Pitches:
Fastball Pitch
Fabián A. Díaz Monge – Sec. 096
Romulo Corrada – Sec. 096
Ignacio Bautista – Sec. 036
Isaac Morales Del Valle – Sec. 086
Introduction
Baseball pitchers routinely throw fastballs, curveballs, and sliders, each consistently
following its own characteristic trajectory. These flight paths do not occur by chance; rather, they
are governed by the fundamental principles of Dynamics.
The objective of this project is to analyze and explain the trajectory of a fastball using these
identical principles, thereby describing how linear momentum, angular momentum, impulse, and
aerodynamic forces influence the flight path of the ball. Furthermore, a comprehensive numerical
simulation will be developed to model this trajectory, allowing for a direct comparison with realworld slow-motion video analysis to evaluate the similarity of the results and derive the
corresponding engineering conclusions.
Theory and Governing Equations
To perform the proper analysis, the corresponding equations for the mass moment of
inertia, as well as linear momentum, angular momentum, and their respective impulses are
required. Additionally, critical fluid dynamic and aerodynamic forces that govern the ball's flight
path will be integrated into the framework. Within each section, the physical meaning, notation,
and standard units for every mathematical variable will be explicitly defined.
a) Mass Moment of Inertia of a Solid Sphere:
Inertia, the fundamental property of a body with mass to resist any
change in its state of motion, depends on the distribution of mass within the
object. The shape of this object is assumed to be a uniform solid sphere.
Therefore, the following equation was used:
𝐼𝐺 =
2
𝑚𝑟 2
5
Where:
𝐼𝐺 : Mass moment of inertia about the center of mass (𝑘𝑔 ∙ 𝑚2 ), m: Mass of the baseball
(𝑘𝑔), r: Radius of the baseball (𝑚)
b) Linear Impulse and Momentum Principle:
The pitcher, by applying a net force over a specific time interval,
produces a linear impulse. In this context, this vector impulse is equal to
the change in linear momentum.
𝑝 = 𝑚𝑣
𝐼 = 𝐹∆𝑡 = 𝑚𝑣𝑓 − 𝑚𝑣𝑖
𝑚𝑣𝑖 + ∫ 𝐹𝑑𝑡 = 𝑚𝑣𝑓
Where:
𝑝: Linear Momentum vector (𝑘𝑔 ∙ 𝑚/𝑠), 𝑣𝑖 , 𝑣𝑓 : Ball Velocity (𝑚/𝑠) or (𝑚𝑝ℎ), t: Time (𝑠),
𝐹: 𝑁𝑒𝑡 𝐹𝑜𝑟𝑐𝑒 (𝑁)
c) Angular Impulse, Momentum Principle and Rotational Kinetic Energy:
With this property, the spin of the ball upon being released is
explained. The fingertips, by exerting a tangential force, produce a torque
over a specific time interval, thereby causing an angular impulse and
creating a change in angular momentum. In addition, this creates rotational
kinetic energy.
𝐿 = 𝐼𝐺 𝜔
𝐼𝐺 𝜔𝑖 + ∫ 𝑀𝑑𝑡 = 𝐼𝐺 𝜔
𝑅𝐾𝐸 =
1
𝐼 𝜔2
2 𝐺
Where:
𝐿: Angular Momentum (𝑘𝑔 ∙ 𝑚2 /𝑠), 𝜔: Ball Angular Velocity (𝑟𝑎𝑑/𝑠), 𝑀: Moment (N∙
𝑚)
d) Fluid and Aerodynamic Forces:
These are the aerodynamic forces that interact with the ball while it
is in the air; of this type, only two forces are present. The drag force is the
resistance force that slows down the velocity of the ball due to air friction.
There is also the force generated by the backspin, the Magnus force. This is
caused by Bernoulli's principle, since, by generating low pressure on top of
the ball, it creates a vertical upward force.
𝐴 = 𝜋𝑟 2
𝐹𝐷 =
1
𝐶 𝜌𝐴𝑣 2
2 𝐷
1
𝐹𝑀 = 𝐶𝐿 𝜌𝐴𝑣 2
2
Where:
𝐹𝐷 : Drag Force (𝑁), 𝐹𝑀 : Magnus Force (𝑁), 𝐶𝐷 : Drag Coefficient, 𝐶𝐿 : Lift
Coefficient, 𝜌: Air fluid density (𝑘𝑔/𝑚3 ), 𝐴: Cross- sectional area of a sphere
(𝑚2 )
Calculations
Part I – Baseball Properties
The parameters utilized for baseball were based on the official regulations of one of the
most important professional baseball leagues, the MLB. The ball features an average mass of 145g,
an average circumference of 232mm, and an average diameter of 74mm [10]. In Table 1, these
important values can be observed, along with their respective conversions to ensure compatibility
with the dynamic equations.
Although the ball doesn’t behave as a solid sphere, and it deforms throughout its trajectory,
for the sake of this analysis it shall be treated as a sphere through its trajectory.
2
2
𝐼𝐺 = 𝑚𝑟 2 → (0.145𝑘𝑔)(0.037𝑚)2
5
5
𝑰𝑮 = 𝟕. 𝟗𝟒 × 𝟏𝟎−𝟓 𝒌𝒈 ∙ 𝒎𝟐
Table 1. Baseball Properties
𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦
𝑉𝑎𝑙𝑢𝑒
𝑀𝑎𝑠𝑠 (𝑚)
0.145𝑘𝑔
𝐷𝑖𝑎𝑚𝑒𝑡𝑒𝑟 (𝑑)
0.074𝑚
𝑅𝑎𝑑𝑖𝑢𝑠 (𝑟)
0.037𝑚
𝑀𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝐼𝑛𝑒𝑟𝑡𝑖𝑎 (𝐼𝐺 )
7.94 × 10−5 𝑘𝑔 ∙ 𝑚2
Part II – Linear Momentum Analysis
In this part, the linear momentum analysis will be analyzed when the pitcher begins the
pitch. It was considered that the velocity of the ball at release was zero. Furthermore, since the
pitch is a fastball, according to “Average Fastball Speed MLB: Trends and Impact on the Game”,
written by James Anderson, the average fastball in the MLB is around 94.2 mph [11]. The final
velocity we decided to utilize, when the pitcher releases the ball to throw it, is 95mph (42.47m/s).
With these parameters in mind, the impulse required to accelerate the ball was successfully found.
In Table 2, the necessary properties can be found.
𝑚 = 0.145𝑘𝑔
𝑚𝑣𝑖 + 𝐹∆𝑡 = 𝑚𝑣𝑓 → 0 + 𝐼 = 𝑚𝑣𝑓
𝐼 = 0.145𝑘𝑔 ∙ 42.47
𝑚
𝑠
𝑰 = 𝒎𝒗𝒇 = 𝟔. 𝟏𝟓𝟖𝑵 ∙ 𝒔
Table 2. Baseball Translational Motion Data.
𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦
𝑉𝑎𝑙𝑢𝑒
𝑅𝑒𝑎𝑙𝑒𝑠𝑒 𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 (𝑣𝑖 )
0 𝑚/𝑠
𝐹𝑖𝑛𝑎𝑙 𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 (𝑣𝑓 )
42.47 𝑚/𝑠
𝐼𝑚𝑝𝑢𝑙𝑠𝑒 (𝐼)
6.158 𝑁 ∙ 𝑠
FBD:
Figure 1. Impulse-Momentum Free-Body Diagram
Part III – Angular Momentum Analysis
To analyze the angular momentum of a fastball once it is airborne, a spin rate of 2,400 rpm
was utilized. According to Baseballtips.com, the average revolutions per minute for a fastball range
from 2,000 to 2,400 rpm [12]. This information is necessary to calculate the angular momentum
and rotational kinetic energy, which are presented in Table 3. The fastball spin rate utilized was
2,400 rpm.
a) Fastball spin rate in rad/s:
𝜔 = 2400𝑟𝑝𝑚 × 2𝜋 ×
𝝎 = 𝟐𝟓𝟏. 𝟑𝟑
1𝑚𝑖𝑛
60𝑠
𝒓𝒂𝒅
𝒔
b) Angular Momentum (𝐿):
𝐿 = (7.94 × 10−5 𝑘𝑔 ∙ 𝑚2 ) (251.33
𝑟𝑎𝑑
)
𝑠
𝑳 = 𝟎. 𝟎𝟏𝟗𝟗𝟔𝒌𝒈 ⋅ 𝒎𝟐 /𝒔
c) Rotational Kinetic Energy (𝐾𝐸):
1
𝑟𝑎𝑑 2
)
𝐾𝐸 = (7.94 × 10−5 𝑘𝑔 ∙ 𝑚2 ) (251.33
2
𝑠
𝑲𝑬 = 𝟐. 𝟓𝟎𝟕𝟕 𝑱
Table 3. Baseball Rotational Motion Data.
𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦
𝑉𝑎𝑙𝑢𝑒
𝐴𝑛𝑔𝑢𝑙𝑎𝑟 𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 (𝜔)
251.33 𝑟𝑎𝑑/𝑠
𝐴𝑛𝑔𝑢𝑙𝑎𝑟 𝑀𝑜𝑚𝑒𝑛𝑡𝑢𝑚 (𝐿)
0.01996 𝑘𝑔 ⋅ 𝑚2 /𝑠
𝑅𝑜𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝐾𝑖𝑛𝑒𝑡𝑖𝑐 𝐸𝑛𝑒𝑟𝑔𝑦 (𝐾𝐸)
2.5077 𝐽
The spin of the ball is highly important in determining its overall trajectory. In this case,
since it is a fastball, the spin rate ranges from 2,000 to 2,400 rpm [12]. This backspin, driven by
the Magnus force, is necessary to produce the characteristic straight-line effect and extend its flight
time in the air. Similarly, depending on the specific velocity and rpm combinations, other distinct
pitches can be created, such as a curveball or a slider.
Part IV – Angular Impulse-Momentum Analysis
For the Angular Impulse-Momentum Analysis, different assumptions were made regarding
the pitcher’s fingers contact time with the ball, the initial velocity of the ball, where we know that
the ball is initially at rest (vi=0, 𝜔𝑖 = 0) since it’s a pitch. The pitcher’s fingers contact time was
considered to be 0.02s, for the sake of ease in analysis and calculation. Also, the final angular
velocity of the ball (𝜔) was assumed to be 251.33 rad/s.
𝜔𝑖 = 0 , ∆𝑡 = 0.02𝑠 , 𝜔 = 251.33
𝑟𝑎𝑑
𝑠
𝐼𝐺 = 7.94 × 10−5 𝑘𝑔 ∙ 𝑚2
↓
𝐼𝐺 𝜔𝑖 + 𝑀∆𝑡 = 𝐼𝐺 𝜔
0 + 𝑀(0.02𝑠) = (7.94 × 10−5 𝑘𝑔 ∙ 𝑚2 ) (251.33
𝑀=
𝑟𝑎𝑑
)
𝑠
0.01996𝑁 ∙ 𝑚 ∙ 𝑠
→ 𝑴 = 𝟎. 𝟗𝟗𝟕𝟖𝑵 ∙ 𝒎
0.02𝑠
This result appears reasonable because the average force on a finger is about 90N, this force
is mostly used to thrust the ball forward, so a small percentage of this is used to generate spin.
Also, we must keep in mind that, since the time is assumed to be very short, the torque might be a
different number than the values researched [2].
Part V – Flight Dynamics
When moving into this section, the final velocity of the pitcher from part 4 will be used as
the initial velocity in this case. This happens because before it was the motion of the pitcher from
a velocity at rest until his final velocity, which now in this part 5 is the initial velocity, which is
when the ball is in the air. In few words, when the ball no longer has contact with the pitcher. It is
clear to add that angular acceleration does not exist since the angular velocity is constant.
Figure 2. Flight Dynamics Free-Body diagram
We assume that the trajectory of the ball is fully horizontal, and the drag force (FD) acts
against the velocity of the baseball. In addition, the direction of the weight is assumed to be
opposite to the Magnus force (FM.) Both forces require the air density (𝜌), which is taken as
𝑘𝑔
1.225 𝑚3 , as well as the cross-sectional area (A) calculated as follows:
𝐴 = 𝜋(0.037𝑚)2 = 0.0043𝑚2
a) Force Drag:
The first force to calculate is the Drag Force (FD). For this, the coefficient of drag (𝐶𝐷 ) can
be found in two ways, one is with Reynolds number, being the measure of ratios of inertial forces,
and the other is with a spin factor. Both equations are shown below:
𝑟𝑎𝑑
𝑟𝜔 0.037𝑚 × 251.33 𝑠
𝑆𝑝𝑖𝑛 𝑓𝑎𝑐𝑡𝑜𝑟 (𝑆) =
=
= 0.219
𝑚
𝑣
42.47 𝑠
Viewing figure 3, a spin factor of 0.219 gives the drag coefficient of approximately 0.34 [8].
Figure 3. Drag coefficient as a function of spin factor. Adapted from Lyu et al. (2020).
The Reynolds number for a baseball in flight has been calculated to be approximately
220,000 by NASA Glenn Research Center [9]. Now looking at Figure 4, the drag coefficient at a
Reynolds number of 220,000 is approximately 0.35 [8].
Figure 4. Drag coefficient as a function of Reynolds number. Adapted from Lyu et al. (2020).
We can take the average value of the drag coefficient value to be 0.345, therefore the drag force
would be:
1
𝐹𝐷 = 𝐶𝐷 𝜌𝐴𝑣 2
2
1
𝑘𝑔
𝑚
𝐹𝐷 = (0.345)(1.225 3 )(0.0043𝑚2 )(42.47 )2
2
𝑚
𝑠
𝑭𝑫 = 𝟏. 𝟔𝟑𝟗𝑵
When applying Newton’s Second Law for the x-direction we get the following value for
acceleration:
∑ 𝐹𝑥 = −𝐹𝐷 = 𝑚𝑎𝑥
𝑎𝑥 = −
𝐹𝐷
1.639𝑁
𝒎
=−
= −𝟏𝟏. 𝟑𝟎 𝟐
𝑚
0.145𝑘𝑔
𝒔
The negative value for acceleration is correct because it acts in the same direction as the drag
force. This value for acceleration is understandable since the baseball reduces its velocity during
the trajectory from the pitcher to home base. When calculating the final velocity of the baseball
with this acceleration, we get that the final velocity is 37.55m/s. This means that the fastball is
thrown at 95 mph and reaches the batter at 84 mph. This corresponds to a loss of 11 mph during
its flight, which is somewhat greater than the 8-mph reduction typically observed for MLB
fastballs [13]. The discrepancy may result from assumptions made in the drag coefficient, spin, or
from simplifications in the flight model.
b) Magnus Force:
The angular momentum vector 𝐿 = 𝐼𝐺 𝜔 defines the spin axis of the baseball, and it is this
spin axis that determines the direction of the Magnus force through the cross-product ω × v. For
a fastball with backspin, L points horizontally along the -y axis, and when crossed with the
forward velocity v in the +x direction, produces a Magnus force in the +z direction (upward),
which is what reduces the ball's gravitational drop on its way to the plate.
To find the Magnus Force (FM), a specific dimensionless coefficient is used; in this case, it
is the lift coefficient (CL).
From a study done by Alan M. Nathan at the University of Illinois, he found a comparison
of lift coefficients on baseballs at different velocities and graphed his results. This graph is
calculated when using a spin factor between 0.15-0.25, which falls between our spin factor of
0.219. From that graph we can estimate a coefficient of lift of approximately 0.23 [1].
Figure 5. Coefficient of lift with respect to velocity. Adopted from Nathan, A. M. (2008).
Therefore, the Magnus Force is:
𝐹𝑀 =
1
𝑘𝑔
𝑚
(0.23)(1.225 3 )(0.0043𝑚2 )(42.47 )2
2
𝑚
𝑠
𝑭𝑴 = 𝟏. 𝟎𝟗𝟑𝑵
As we can see when we throw a fastball, the ball does not follow a parabolic path but
instead follows a linear trajectory. This happens because, among the vertical forces, specifically
the Magnus force FM induced by backspin, counteracts the force of gravity, thereby preventing the
ball from dropping significantly.
This result states that the acceleration of the Magnus force is less than the acceleration due to
gravity (FM < mg). Proving that the ball still falls but drastically slower due to Magnus Force (may = FM - mg).
∑ 𝐹𝑦 = −𝑚𝑔 + 𝐹𝑀 = 𝑚𝑎𝑦
To find the resulting acceleration, we can cancel out the masses, because it’s the ball’s mass, and
we get:
∑ 𝑎𝑦 = −𝑔 + 𝑎𝑀 = 𝑎𝑦
𝑚
𝑚
𝑎𝑦 = −9.81 2 + 7.54 2
𝑠
𝑠
𝒎
𝒂𝒚 = −𝟐. 𝟐𝟕 𝟐
𝒔
This final acceleration, of -2.27 m/s2 is the rate at which the ball is falling, considering the Magnus
force. It is negative because the downward direction is negative. This is why the fastball seems to
“rise”, it falls slower than how we perceive objects to act towards gravity.
Part VI – Numerical Simulation
The trajectory data used for the simulator was obtained using the Trajectory Calculator
spreadsheet by Dr. Alan M. Nathan (University of Illinois), publicly available at The Physics of
Baseball (baseball.physics.illinois.edu) [3]. This excel uses all equations of aerodynamic motion
incorporating aerodynamic drag and Magnus lift using empirically validated coefficients from
different studies (TrajectoryCalculator-new-3D-May2021.xlsx. Accessed 2 June 2026.) [4].
The spreadsheet uses a drag coefficient of the form 𝐶𝐷 = 𝐶𝐷0 + 𝐶𝐷 𝑠𝑝𝑖𝑛 × (spin/1000), with
𝐶𝐷0 = 0.3008 and 𝐶𝐷 𝑠𝑝𝑖𝑛 = 0.0292, yielding 𝐶𝐷 = 0.3709 at 2400 RPM. For the lift coefficient, the
spreadsheet uses 𝐶𝐿 = 𝐶𝐿2 ·S/(𝐶𝐿0 + 𝐶𝐿1 ·S) with 𝐶𝐿0 = 0.583, 𝐶𝐿1 = 2.333, and 𝐶𝐿2 = 1.120, resulting
to 𝐶𝐿 = 0.224. The drag coefficient is slightly higher than our estimated drag coefficient 𝐶𝐷 = 0.345
used in our analytical model, while the lift coefficient is very similar to our analytical value of
0.23. Both coefficients are assumed to be constant throughout the trajectory of the baseball.
In the excel spreadsheet, we placed some initial conditions and assumptions to compare
the simulation to our theoretical values. The release height was set to z₀ = 6.0 ft, within the realistic
range of 5.0–6.5 ft for professional pitchers and with an initial velocity of 95 MPH. The release
angle was set to θ = 0° (fully horizontal), only backspin was applied at 2400 RPM, with no sidespin
and the spin rate was held constant throughout the trajectory. The simulation was run at sea level
(ρ = 1.225 kg/m³) with no wind and measured over 60.5 ft of distance from the pitcher to home
plate.
Figure 6. Spreadsheet for simulation
In this figure we see all the parameters that the excel spreadsheet uses to calculate the
trajectory of the baseball. The values highlighted in orange and blue are the values we inserted,
and the rest are values already assigned by the analysis.
Position vs. Time Graph
70
POSITION (FT)
60
50
40
30
20
10
0
0.00
0.10
0.20
0.30
TIME (S)
Figure 7. Position vs. time graph
0.40
0.50
In this graph, the position is modeled with respect to time, forming a near-linear function.
The time it takes for the ball to reach the batter is around 0.46s. When comparing this to online
data we find that our time of 0.46s is longer than the average time of 95mph fastball to reach the
batter with a time of 0.435s [7]. The time difference is very small meaning that the simulation is
accurate at determining the time the ball reaches home plate at 60.5ft which is the measurement
from the MLB from the mound to home plate.
Height vs. Position Graph
7
HIEGHT (FT)
6
5
4
3
2
1
0
0
10
20
30
40
50
60
70
POSITION (FT)
Figure 8. Height vs. Position graph
From this graph, we can see that when a fastball is thrown perfectly horizontally, the ball
drops only one foot throughout its trajectory. This indicates that the Magnus force greatly affects
the baseball, preventing it from falling as much as it would under the influence of gravity alone.
When using the simulator, we found that at a release angle of -1°, the ball drops two feet instead
of one, and at a release angle of -2°, the ball's height at home plate decreases by three feet. This
linear relationship suggests that for every 1° decrease in the release angle, the ball's vertical
position at home plate decreases by approximately 1 ft.
Using a second simulator, Baseball Pitch Trajectory Simulator, which uses the same
aerodynamic model developed by Alan M. Nathan while incorporating MLB pitcher statistics, we
found that pitchers typically throw a fastball at an average release angle between -1.5° and -2.5°
[5]. This suggests that most fastballs drop a vertical distance of approximately 2 to 3.5 ft before
reaching the batter.
Part VII – Video Analysis
For the video analysis, we used a YouTube video posted by PlayStation called MLB The
Show 19 - Slow Mo Guys With Ken Griffey Jr | PS4 [6]. In this video, there are many pitches and
their respective slow-motion videos. We used the pitch thrown at 2:04 (time stamp) with its slow-
motion video. Using an editing software (Imovie), we grabbed the video segment of the pitch
and saw that the ball was in the air between the frames of 0.4s and 0.5s. We used the intermediate
time of 0.45s for our analysis. Then, we found the speed of the pitch, the distance d between the
mound and the home plate was 60.5ft, then we used the following formula:
To relate the original video to the slow-motion version, we took the time for the slow-motion
pitch to reach the batter and used ratios to find out how much it was slowed down. The pitch time
in slow motion was 12.2 seconds, creating a slowdown factor of 27.1.
𝑆𝑙𝑜𝑤𝑑𝑜𝑤𝑛 𝐹𝑎𝑐𝑡𝑜𝑟 =
𝑠𝑙𝑜𝑤𝑒𝑑 𝑡𝑖𝑚𝑒
12.2𝑠
=
= 27.1
𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑡𝑖𝑚𝑒 0.45𝑠
a) Release velocity
For release velocity we estimated that in the first 0.8s of the pitch the ball traveled 4ft. Using the
same slowdown factor of 27.1, we get the following release velocity:
𝑡𝑟𝑒𝑎𝑙 =
𝑡𝑠𝑙𝑜𝑤
0.8𝑠
=
= 0.0295𝑠
𝑠𝑙𝑜𝑤𝑑𝑜𝑤𝑛 𝑓𝑎𝑐𝑡𝑜𝑟 27.1
𝑅𝑒𝑙𝑒𝑎𝑠𝑒 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 =
𝑑
4𝑓𝑡
1.2192𝑚
𝑚
=
=
= 41.33 = 92.5𝑚𝑝ℎ
𝑡 0.0295𝑠
0.0295𝑠
𝑠
This release velocity is slower than the average fastball release velocity of 95mph, but the
pitch was not thrown in a game, and the pitcher probably didn’t throw a full speed fastball to avoid
injury, therefore it can be concluded that the release velocity estimated is close to the average
velocity of a fastball.
b) Spin Rate:
To find the spin rate of the baseball in rpm we calculated how many times the ball
completed a full rotation in a time interval of the slow-motion video. We measured that the ball
made three full rotations in exactly two seconds of the slow-motion video meaning we could
calculate the rpm with the following formulas:
𝑆𝑙𝑜𝑤 𝑠𝑝𝑖𝑛 𝑟𝑎𝑡𝑒 =
3𝑟𝑒𝑣
𝑟𝑒𝑣
= 1.5
2𝑠
𝑠
𝑟𝑒𝑣
𝑟𝑒𝑣
× 27.1 = 40.65
𝑠
𝑠
𝑟𝑒𝑣
𝑆𝑝𝑖𝑛 𝑅𝑎𝑡𝑒 = 40.65
× 60 = 2439 𝑅𝑃𝑀
𝑠
𝑅𝑒𝑎𝑙 𝑠𝑝𝑖𝑛 𝑟𝑎𝑡𝑒 = 1.5
This value is very close to our researched spin rate of 2400RPM for a fastball, meaning that the
calculations and assumptions we took were correct.
c) Spin axis orientation:
To find the spin axis orientation we saw the slow-motion video and saw that the ball was almost
spinning fully counterclockwise, with a small amount of sidespin. This means that the ball has
backspin, backing up the theory of a fastball.
d) Average Velocity:
𝑣=
𝑑 60.5𝑓𝑡 18.44𝑚
𝑚
=
=
= 40.98 = 91.7𝑚𝑝ℎ
𝑡
0.45𝑠
0.45𝑠
𝑠
This value is similar to the average velocity of the simulation of 88.1mph only varying by
3.6mph.
Discussion
VIII – Engineering discussion
The purpose of this project was to apply the principles of Dynamics, specifically the
principles of Impulse-Momentum, to study the science behind a fastball pitch. Through numerical
and graphical analysis, it can be safely said that impulse generates linear momentum because an
object’s mass accelerates another object (in this case a resting one) through a period, thus
generating momentum. And, by the same principle, a torque, a force times perpendicular distance,
can generate angular momentum. Spin can influence the trajectory of the ball through the air flow
vectors, where a backspin traps these vectors creating a force that assists the ball to not fall as fast
as it is supposed to, making the range of the projectile longer. On one hand, for the fastball, the
backspin rate is close to being about the y-axis (side), this means that the fastball does not break
pitch, making it a straight path. On the other hand, for the other pitches, such as the curve ball and
sliders, as the names suggest, they have a distinct trajectory from the fastball, where their
trajectories break and they stray away from a natural, straight, path. These types of pitches serve
to deceive the batter, making the ball harder to hit, since the batter expects a straight, natural, path.
It can be said, with this data, and with engineering analysis, that the applications of Dynamics in
sports are helpful because they can help to analyze injuries, such as boxing injuries through impact
analysis, improve equipment through work analysis, such as a baseball bat hitting a very fast ball
and it breaking, and help with strategies for athletes in the field, such as a soccer player kicking a
ball that curves thanks to the wind.
Conclusion
Through careful numerical and graphical analysis, we can conclude that the principles of
dynamics play a big role in finding the pitching quality, speed, and type of pitch. The measured
values and the theoretical values coincided, proving once again why engineering analysis methods
are effective to predict whatever phenomenon is being studied. There were some numbers that
didn’t coincide with their theoretical values, such as the final pitching speeds and average pitch
velocity, where the values that were researched were faster than our calculated values and
simulation values. For example, in the simulation of Figure 7, the slope of the graph, being the
average velocity, was calculated to be 88.1mph, slower than the average velocity of the video of
91.7mph. These discrepancies have a multitude of possible errors, such as only taking into
consideration backspin, higher values of drag and lift coefficients, and the assumption of fully
horizontal velocity. Another thing that we found while making this analysis is that the Magnus
Force has a big effect on translating-rotating objects, this force greatly reduces the object’s vertical
acceleration, as seen in the simulation found in Figure 8, where the ball only drops 1ft after 60.5ft
of distance.
References
[1] Nathan, A. M. (2008). The effect of spin on the flight of a baseball. American Journal of
Physics, 76(2), 119–124. https://baseball.physics.illinois.edu/AJPFeb08.pdf
[2] Kinoshita, H., Obata, S., Nasu, D., Kikuta, J., Kawashima, N., & Furuya, S. (2017). Finger
forces in fastball baseball pitching. Human Movement Science, 54, 172–181.
https://doi.org/10.1016/j.humov.2017.04.007
[3] Nathan, A. M. (n.d.). Trajectory calculator. The Physics of Baseball.
https://baseball.physics.illinois.edu/trajectory-calculator.html
[4] Nathan, A. M. 3D Trajectory calculator TrajectoryCalculator-new-3D-May2021.xlsx
[5] Skill-Vis. (n.d.). Baseball pitch trajectory simulator. https://skillvis.github.io/MyTrajectorySimulator/
[6] PlayStation. “MLB the Show 19 - Slow Mo Guys With Ken Griffey Jr | PS4.” YouTube, 12
Apr. 2019, www.youtube.com/watch?v=2Q-gwKLVNYs.
[7] SportsRec. (n.d.). How to calculate fastball reaction time.
https://www.sportsrec.com/4232989/how-to-calculate-fastball-reaction-time
[8] Lyu, F., Nathan, A. M., & Smith, L. (2020). The dependence of baseball lift and drag on spin.
Retrieved from https://baseball.physics.illinois.edu/LyuDragLiftSpin.pdf
[9] NASA Glenn Research Center. (n.d.). Drag on a baseball. National Aeronautics and Space
Administration. https://www1.grc.nasa.gov/beginners-guide-to-aeronautics/drag-on-abaseball/
[10] Sports Law Blogger. “What Is the Mass of a Baseball?” Sports Law Blogger,
https://sportslawblogger.com/what-is-the-mass-of-a-baseball.html.
[11] Anderson, James, and James Anderson. “Average Fastball Speed MLB: Trends and Impact
on the Game.” Southern Pigskin |, 20 Aug.
2024, www.southernpigskin.com/recruiting/average-fastball-speed-mlb-trends.
[12] Coach John Peter. “The Complete Guide to Baseball Spin Rate: Master the Physics of
Pitching”, https://baseballtips.com/baseball-spinrate/#:~:text=For%20reference%2C%20MLB%20averages%20are,2%2C300%2D2%2C
500%20RPM%20for%20sliders.
[13] “What Is the Rate of Slowdown of a Varsity Baseball Pitch?” Sports Law Blogger,
https://sportslawblogger.com/what-is-the-rate-of-slowdown-of-a-varsity-baseballpitch.html.