The Deflection of a
Curveball in Baseball and Fast pitch Softball
Anna Tapio
Physics Department
Introduction
Baseball is not simply a sporting event – it is physics, especially the curve ball. People curiously studied the physics of a pitch for years – whether balls actually curved during flight. The work of Daniel Bernoulli, Bernoulli’s Principle, and Heinrich Magnus, the Magnus Effect, holds most importance when examining the physics of a curve ball.
Consider a spherical baseball as fluid in motion through air. According to Bernoulli’s principle, a pressure difference surrounds the baseball along its boundary layer. The air pressure at the front of the baseball is greater than the pressure at the top since as the ball accelerates, its kinetic energy increases (Bahill 42). As air travels beyond the top of the baseball, its speed decreases, thus its pressure decreases. In the region behind the baseball, the fluid flow experiences a disruption since the viscosity of air is not zero, called the wake region (Bahill 46). Within the wake region, a swirling flow occurs due to the spinning of the baseball, similar to the swirling behind boats. This wake affects the curve of a curve ball.
The Magnus Effect suggests that if the air resistance is greater on one side of the ball, the ball will curve in the opposite direction (Adair 26). For instance, the wake region behind a baseball creates a drag force. Since the wake is uneven, the created force redirects the path of the pitch as well as decreasing its velocity. The spin of the baseball will shift the wake region, causing the ball to curve left or right. Clockwise spin curves the ball to the right, whereas counterclockwise spin curves it to the left (Bahill 48). The force from the spin and velocity of a pitch has an equation of
F = 0.50πρωvR^{3} (1) (Bahill 76)
or F = (2mdv^{2})/ l^{2} (2) (Bahill 26)
where ρ is the density of the fluid, ω is the rotation rate, or spin, v is the velocity of the pitch, R is the radius of the ball, d is the deflection, and l is the length from the pitcher’s mound to home plate. Applying this force to simple laws of motion yields an equation for the deflection of the curve ball of
d = (πρωl^{2}R^{3})/(4mv) (3) (Bahill 77)
Equation (3) shows that the deflection of the curve ball has a linear relationship with the rotation rate. Conversely, the deflection has an inverse relationship with the pitch velocity. Thus, many combinations of pitch velocity and spin produce differing curve balls and deflections.
The Experiment/Program
Fast pitch softball follows the same physics as baseball. Pitchers in both games throw similar pitches, including the curve ball. In order to compare the curve balls in baseball and fast pitch softball, I created a program in Fortran 90. My first goal for this program was to compare the deflection of a curve ball in baseball versus softball. The second goal of my program was to determine the relationship between force and shape. Throughout all calculations, the softball and baseball undergo the same pitch velocities, as seen in tables 2 and 3 below. Table 1, as seen below, shows the constants used throughout all calculations in the program.
Table 1 Values
used in calculations of deflection and force

Radius
(m) 
Spin (rad/s) 
Mass (kg) 
Distance
to Home Plate (m) 
Baseball 
0.038 
199 
0.1488 
16.76 
Softball 
0.0486 
156 
0.1878 
0.1878 
Using equations (1) and (3), the program first calculated the deflections and forces for the baseball and softball. The program then graphed deflection versus velocity, in order to compare the differing deflections of the baseball and softball. I expected that the softball would have smaller deflections than the baseball, since the softball is larger in volume.
My program then calculated the forces on a baseball and softball for curve balls with similar deflections and velocities. By comparing the ratio of the forces to the ratio of the volumes, I could determine the relationship between force and volume. I performed the comparison with three deflections and three velocities, as seen in tables 2 and 3 below. I expected to find that the ratio of forces would approximately equal the ratio of volumes since deflection depends on shape, as seen in equation (3).
Results
My program provided three tables and two figures from the calculated data. Tables 2 and 3, as seen below, show the different deflections and forces calculated with constant rotation rates and similar pitch velocities for the softball and baseball, as well as with similar velocities and deflections.
Table 2 Baseball Data Collected in Fortran 90 Program
Baseball Data at spin = 199 rad/s
Velocity (m/s) Deflection (m) Force (N)
===================================
22.352 0.905
0.4792
23.246 0.871 0.4984
24.140 0.838
0.5176
25.034 0.808 0.5367
25.928 0.781
0.5559
26.822 0.754
0.5751
27.716 0.730
0.5943
28.611 0.707
0.6134
29.505 0.686
0.6326
30.399 0.666
0.6518
Baseball Data at Similar Deflection and
Velocity
Velocity (m/s) Deflection (m) Force (N)
====================================
22.352 0.100
0.0529
23.246 0.300
0.1718
24.140
0.500
0.3087
Table 3 Softball Data Collected in Fortran 90 Program
Softball Data at spin = 156 rad/s
Velocity (m/s) Deflection (m) Force (N)
===================================
22.352 0.821 0.7839
23.246 0.790
0.8152
24.140 0.760
0.8466
25.034 0.733
0.8780
25.928 0.708
0.9093
26.822 0.684
0.9407
27.716 0.662
0.9720
28.611 0.641
1.0034
29.505 0.622 1.0347
30.399 0.604
1.0661
Softball Data at Similar Deflection and
Velocity
Velocity (m/s) Deflection (m) Force (N)
===================================
22.352 0.100
0.0955
23.246 0.300
0.3098
24.140 0.500
0.5568
Figure 1, as seen below, is a graph of deflection versus velocity where the deflection is not similar.
Figure 1:
Graph of deflection versus velocity for the baseball and softball.
My program also calculated the ratio of the forces and ratio of the volumes under similar deflections, printing these values to the screen. The following figure depicts an example of this outputted information.
amtapio@lin23.cs.csbsju.edu
30% curve
The spin of the softball
is 155.5967
Ratio of forces is 1.803618 1.803618 1.803618
Ratio of volumes is 2.091983
Figure 2 Output
from program printed to the screen
These tables and figure allowed for the comparison of deflections of the softball versus the baseball, permitting me to interpret the relationship between the force on the curve ball and shape of the ball used.
Discussion
From the data my
program calculated, I saw that my expectations were correct. In calculating deflections under constant
rotation rates and similar velocities, the deflections of the softball were
smaller than the deflections of the baseball.
Furthermore, I saw that the force on the curve ball relates to the shape
of the ball.
As seen in tables 2 and 3, when the deflections vary, the softball has deflections approximately 0.1 meter smaller than the deflections of the baseball. Figure 1 also confirms that at similar velocities with varying deflections, the deflections of the softball are smaller. The deflections values themselves, however, are erroneous. The values suggest that the baseball and softball deflect at maximum values of roughly 0.9 and 0.8 meters, or almost 3 feet. Even the minimum values suggest deflections of almost 2 feet. Yet the width of home plate is only 17 inches or approximately 1.5 feet. The values thus claim that the curveballs deflected between the width of home plate and twice of that width. Personal observations of baseball and softball games, however, depict curve balls with significantly smaller deflections. These results suggest that errors occurred in the program.
One possible error is the values for the rotation rates. The rotation rates of the softball and baseball are 156 and 199 radians per second, respectively (Bahill 68). These values may not be the true values of the rotation rates since research came exclusively from books and personal experience for this program. Finding more accurate rotation rates may produce reasonable deflection values. Another possible error results from the assumption that the two shapes are spherical. For instance, the program does not account for the effect from the laces. Furthermore, the radii are average values, not an exact value. Regulation radii of softballs and baseballs must fall within a range of values, giving uncertainty to the radii used in this program. Lastly, the velocity values of 22 to 31 meters per second, or 50 to 70 miles per hour, are estimates based on pitch speeds taken from personal observations of softball and baseball games.
Tables 2 and 3, as well as figure 2, shows that the force relates closely to the shape of the ball. Figure 2 illustrates that the ratio of the forces is approximately two; therefore, a fast pitch softball pitcher must exert twice the force than a baseball pitcher in order to achieve the same deflection. This increase in force relates to the shape of the ball. Equation (2) shows that the deflection of the curve ball has a linear relationship with the shape of the ball used. The volume of the softball is approximately twice the volume of a baseball. Since the force exerted on the ball relates linearly to the deflection, the shape of the ball directly affects the force. Future research regarding the path of the curve ball, more accurate rotation rates, and the actual speed range of this pitch would provide more insight to the curve ball, as well as providing logical values of its deflection.
References
Adair, Robert K. The Physics of Baseball.
Bahill, Terry A.
and Robert G. Watts. Keep Your Eye on the Ball – The Science and
Folklore of Baseball.
Chapman, Stephen J. Introduction to Fortran
90/95.
Acknowledgements
The author would like to acknowledge Professor Jim Crumley for his help in creating her program.