Why Does a Fly Ball Carry Better to Centerfield?
Alan Nathana-nathan@illinois.edu, @pobguy
University of Illinois at Urbana-Champaign(Dated: June 25, 2020)

I.

INTRODUCTION

Have you ever noticed that baseballs hit to centerfield carry farther than those hit
to left or right field? I’ll show some direct evidence for the phenomenon shortly. For now,
simply take my word for it so I can pose the question in the title: Why is this so? My
gut feeling has always been that it has something to do with spin on the ball, particularly
the spin axis. Because of the dynamics of the ball-bat collision, balls hit to hit to left or
right field have considerably more sidespin than balls hit to center. The primary effect of
the sidespin is a phenomenon easily observed when watching a game: It causes the ball tofollow a curved path, breaking toward the foul line.
But aside from this instinct, I have to admit that I never gave the matter much thought
until recently. What stimulated my recent interest was a journal article1 that claimed thereason for the reduced carry is the curved path followed by balls hit with sidespin as opposedto the straight path followed by balls hit with no sidespin. The horizontal distance of thelanding point from home plate is reduced if the ball follows along a curved rather thana straight path, assuming otherwise identical launch conditions. The argument certainly
makes sense qualitatively. But does it make sense quantitatively? And if not, then what are
the primary reasons for the phenomenon? These are the questions I have asked myself andwill address in this article.

II.

THE EVIDENCE

The evidence for the ball carrying better to centerfield and the connection with sidespin
is shown in Figs. 1-3. Each of these figures shows Statcast fly ball data for games played in
Tropicana Field, which was chosen for this study since the atmospheric effects (temperature,2etc.) are constant and wind is absent. The data include exit velocity, launch angle, sprayangle, spin rate, spin axis, and distance. Only fly balls were chosen for which the batted ball
was tracked to at least 80% of its eventual distance, assuring good accuracy in the distancemeasurement. The combination of spin rate and spin axis allows a separation of the total
spin rate ω into backspin and sidespin rates, ωb and ωs , respectively.2 Note that the signof ωs is positive for balls breaking toward the left field foul line and negative toward theright field foul line. All the data shown have an exit velocity in a range bracketing 100 mph,launch angles in the range 25◦ -30◦ , and ωb in the range 2000-3000 rpm.
Fig. 1 shows the relationship between distance and ωs . Fig. 2 shows the relationshipbetween ωs and spray angle for left-handed and right-handed batters separately. Fig. 3shows the relationship between distance and spray angle, color coded by the magnitude ofωs . In the latter two figures, the spray angle has been adjusted by batter handedness so that
it is negative for the pull field and positive for the opposite field.
EV=100 mph; LA=25−30; backspin=2000−3000
425Distance (ft)

400

375

350

325
−2000

−1500

−1000

−500

0500

1000

1500

2000

Sidespin (rpm)FIG. 1: Statcast data showing that fly ball distance decreases with sidespin ωs , with all other launchvariables (exit velocity, launch angle, and backspin) in narrow ranges. The curve is a parabolic fit
to the data.

There is wealth of information contained in these three graphs, which is now summarized.
• Fly ball distance depends on ωs (Fig. 1); ωs depends on the spray angle (Fig. 2);
therefore, fly ball distance depends on spray angle (Fig. 3). This phenomenon was theprimary motivation for the study and will be discussed in Sec. III.
• Fly ball distance is largest when ωs =0 and is roughly symmetric about ωs =0. Indeed,
as indicated in the caption to Fig. 1, the distance-ωs data are well described by an3
EV=100 mph; LA=25−30
2000−3000 backspin
2500
2000Sidespin (rpm)

1500
1000
500

Batter Hand0L
R

−500
−1000
−1500
−2000
−2500
−30

−20

−100

102030Adjusted Spray Angle (deg)FIG. 2: Statcast data showing the dependence of sidespin on the spray angle for both left-handed
and right-handed hitters. The spray angle has been adjusted for batter handedness so that negative and positive values correspond to balls hit to the pull and opposite fields, respectively. Thesidespin is zero at a spray angle of approximately -10◦ (i.e., slightly to the pull side of straightawaycenterfield).

EV=100 mph; LA=25−30
2000−3000 backspin
420Distance (ft)400sidespin
38001000
2000
3000

360
340
320
−30

−20−100

102030Adjusted Spray Angle (deg)FIG. 3: Statcast data showing the dependence of distance on spray angle (adjusted as in Fig. 2),
color coded by the magnitude of sidespin (rounded to the nearest 1000). These data show thatthe distance depends on spray angle, with a maximum at -10◦ (i.e., slightly to the pull side ofstraightaway centerfield). The data further show that sidespin reduces the carry of the ball (seealso Fig. 1).

inverted parabola symmetric about 0.
• ωs =0 (and therefore fly ball distance is maximum) at a spray angle slightly to the pullside of straightaway centerfield (≈10◦ ). This is true for both left- and right-handedbatters. This was a secondary motivation for the study and will be discussed in Sec. IV.

4• As a consequence of the previous bullet point, balls hit at a given spray angle to thepull side (where the magnitude of ωs is smaller) will carry farther than balls hit at thesame spray angle on the opposite side (where the magnitude of ωs is greater).
• As a further consequence, balls hit to the pull side of 10◦ break toward the right-fieldfoul line (i.e. hook). Balls hit to the opposite side of 10◦ break toward the left-fieldfoul line (i.e., slide). In particular, balls hit to straightaway centerfield (zero sprayangle) slice toward the opposite field. For a given spray angle, the amount of slice foropposite field fly balls is greater than the amount of hook for the same spray angle on
the pull side.

III.

WHY DOES DISTANCE DEPEND ON SIDESPIN?

To attack this question, I utilized my Trajectory Calculator, a spreadsheet-based tool
I created over the years to calculate the trajectories of batted balls. The tool uses a modelfor the aerodynamic forces on the ball, specifically the so-called drag and lift coefficients,
and the parameters of the model were determined by fitting to actual trajectory data.3 The
advantage of using this tool is that certain features of the model can be switched on or off
or parameters altered to investigate how these changes affect the trajectory, particularly thefinal distance.
While results for a variety of different initial conditions were investigated, those summarized below were for initial conditions similar to those of the data in the Figs. 1-3: exit
velocity=100 mph, launch angle=27.5◦ , ωb =2500 rpm, and ωs either 0 or 1500 rpm. The
latter resulted in a reduction of the fly ball distance by 12 ft. Here is what I have learned
about the reasons for this reduction.
1. Approximately 40% of the reduction comes from the feature of the drag coefficient√that it increases with the total spin ω = ωb 2 + ωs 2 . Therefore when increasing ωsfrom 0 to 1500 rpm, with ωb fixed at 2500 rpm, ω increases from 2500 to 2915 rpm,
resulting in a 3% increase in the drag coefficient, from 0.387 to 0.399, and a reduction
in distance.
2. Approximately 15% of the reduction comes from the curved trajectory, the fact thatthe horizontal distance of the landing point from home plate is smaller when the5trajectory is curved than when it is straight. This small fraction of the total effect isat variance with the claim in the literature1 that got me started on this project in thefirst place.
3. The remaining 45% of the reduction comes from three different factors–some of them
subtle–associated with the Magnus force, the force responsible for the movement ofa spinning baseball. These three factors all act to reduce the lift on the baseball,resulting in a reduction of the distance. It is not straightforward to quantify theindividual contribution of these factors, since they are coupled together. However, it
is easy to describe them.
(a) When the ball leaves the bat, the backspin axis is horizontal and perpendicular
to the horizontal direction of the baseball. As the baseball changes its horizontal
direction due to the sidespin, that axis is no longer perpendicular to the direction;in effect, some of the backspin gets converted into gyrospin, thereby reducing thelift and decreasing the distance.
(b) When the ball leaves the bat, the sidespin axis is primarily in the upward direction. However, since the ball starts out at a non-zero launch angle, a componentof the sidespin points in the backward direction. As the ball changes direction in
the horizontal plane, that backward component results in a downward force on
the ball (i.e, “negative lift”), thereby reducing the total lift and the distance.
(c) The relationship between the Magnus force and spin rate is nonlinear. For example, increasing ω by 16.6% (from 2500 to 2915 rpm) increases the Magnus force by
only 8.3%. As a result, increasing the spin rate by adding sidespin while leaving
the backspin unchanged actually reduces the lift due to the backspin.

IV.

WHY IS SIDESPIN MINIMIZED ON PULL SIDE OF CENTERFIELD?

This topic has been previously discussed by Kagan4 and Nathan,5 so I won’t repeat all
the arguments in detail here. In a nutshell, here is their explanation. The primary reason forsidespin on a baseball is the horizontal angle of the bat with respect to the incoming pitch.
When the batter is “out in front”, the angle is such that the ball is hit predominately to thepull field, with hooking sidespin. When the batter is behind, the ball is hit predominantely

6to the opposite field, with slicing sidespin. If the bat is perpendicular to the incoming
pitch, the ball is hit to centerfield, with essentially zero sidespin. With this mechanism, thedependence of ωs on the spray angle would look similar to that in Fig. 2, except shifted sothat ωs =0 at zero spray angle.
The reason for the shift of the zero point to the pull side is that the bat is not normally horizontal at impact but is usually tilted, with the barrel of the bat below the knob.
Consider what happens when the bat is horizontal and perpendicular to the pitch. If thedirection of the bat at contact is under the center of the ball, the result will be a fly ballto straightaway centerfield with pure backspin. However, when the bat is tilted downward
but still perpendicular to the pitch, the spin axis gets rotated, reducing the backspin andresulting in a slicing sidespin.
So there are two different mechanisms for generating sidespin: the horizontal angle of thebat and the tilt angle of the bat. The combined effect of these two mechanisms is to shift
the zero of ωs a little bit to the pull side, exactly as observed in the data. Detailed models of
the ball-bat collision are in agreement with the data.6 The precise amount of shift depends
on the tilt angle, and the data in Fig. 2 represent an average over many fly balls.

V.

WHY IS BACKSPIN MAXIMIZED FOR OPPOSITE FIELD?

Having satisfied myself that I understand the primary reasons why balls carry betterto centerfield than to the pull or opposite field, I have to admit that there is something in
the Statcast data that I initially did not understand, namely, the fact that ωb is largest for
opposite-field hits and smallest for pull-field hits (see Fig. 4). This feature was something
totally unexpected by me. Thanks to a suggestion from David Kagan and others, I think I
now have at least a qualitative understanding of the phenomenon.
The suggestion is that the attack angle of the bat is not constant but changes throughout
the swing, starting out negative during the downswing, passing through zero, and ending
up positive. To model that behavior requires an assumption about the actual path of thebat during the swing. I assumed that the barrel of the bat follows a parabolic path, which
implies an attack angle that varies linearly with the angle of the bat in the horizontal plane;
it is negative (i.e., downward) when the bat axis is angled backward toward the catcher, zero
when the axis is perpendicular to the incoming ball, and positive (i.e., upward) when the7Backspin, v0=95−105051015202530

354045

6000
3000

Backspin Rate (rpm)0−30006000
3000
0−3000−25025−25025−25025−25025−25025Adjusted Spray Angle (deg)FIG. 4: Statcast data showing the dependence of backspin rate on spray angle (adjusted as in Fig.
2), for exit velocities in the range 95-105 mph and for different 5◦ buckets of launch angle. Thesolid curve is a smooth fit to the data while the dashed line indicates 0 rpm. These data show thatthe backspin rate is largest for opposite-field hits and smallest for pulled-field hits.

05101520

253035

6000
4000
2000

Backspin Rate (rpm)

06000
4000
2000
0
−20

020

6000
4000
2000
0−20

020

−20020Adjusted Spray Angle (deg)FIG. 5: Calculation of backspin vs. spray angle for different 5◦ buckets of launch angle, where thebarrel of the bat was assumed to follow a parabolic path. These results are in qualitative agreementwith the data of Fig. 4.

axis is angled forward toward the pitcher. With that assumption, it was a straightforward
exercise to inject it into my ball-bat collision model6 to calculate the parameters of thebatted ball, particular the dependence of backspin on spray angle. The results, shown inFig. 5, are in qualitative agreement with the Statcast data and give confidence that theessential physics is understood.8VI.

SUMMARY

In this article, I have shown evidence from Statcast data that fly balls carry better tocenterfield than to right or left field. I have also shown that the reason for reduced carry
to left and right is sidespin. The reduced carry due to sidespin comes from two primarycauses: the increased drag as the total spin increases and subtle changes to the Magnus force
that reduce the lift. Finally I have shown that the curved path of a ball hit with sidespin
contributes very little to the reduction of distance. The puzzle regarding the dependence ofbackspin on spray angle is now at least qualitatively understood.

Acknowledgments

Thanks to my friend and colleague Dr. David Kagan (@DrBaseballPhD) for some helpfulcomments on this article. Thanks also to David and several of my Twitter followers for thesuggestion about the attack angle.

References
1

H. Nakashima, G. Horiuchi, and S. Sakurai, “Three-dimensional batted ball in baseball: effectof ball spin on the flight distance,” Sports Engineering 21, 493-499 (2018).

2It should be noted that while the Trackman radar system, an integral part of Statcast, measuresthe total spin ω directly, it only infers the spin axis (and therefore ωb and ωs ) from the trajectory,
using a proprietary algorithm.

3

A description of the model along with links for downloading the tool can be found at the author’s
web site: http://baseball.physics.illinois.edu/trajectory-calculator-new3D.html.

4DavidKagan,

Hardball“TheTimesPhysics(Mayof22,aFoul2014),

Down

theLine,”David

Kagan,

The

https://tht.fangraphs.com/tht-live/

the-physics-of-a-foul-down-the-line/.
5

Alan Nathan, “Spinning Out Of Control,” FanGraphs (October 26, 2016), http://www.
fangraphs.com/blogs/spinning-out-of-control/.

6Alan Nathan, unpublished. The details of the 3D ball-bat collision model will be described in9a future publication. The 2D model is described here: http://baseball.physics.illinois.
edu/ppt/ModelingBallBatCollision.pptx.