Home run data from Major League Baseball's Statcast can be described by adding a lift force to the equations of projectile motion commonly used in undergraduate computational physics courses. We discuss how the Statcast data can be implemented in the classroom.

## I. INTRODUCTION

Projectile motion in sports is an interesting, realistic example that can bring enthusiasm to a physics classroom. In particular, the flight of a baseball has drawn the attention of many physics educators and scientists as evidenced by the numerous articles in this and other journals (see Refs. 1–8 to mention a few). Recently, Major League Baseball added a “Statcast” feature to their website that lists data on the flight of home runs, as well as other situations. We point out here how one can use this home run data in a computational physics course.

The data we examine are from Ref. 9, which are the longest 50 home runs from the 2015 postseason. The site lists the data in a convenient format that can be copied and pasted into a text file with four columns to be read and analyzed by a computer code. The data consist of the initial speed *v*_{0}, the launch angle *θ*_{0}, the maximum height *h*_{max}, and the range *R* of the home run. The data span a fair spread of values with $395\u2009ft<R<459\u2009ft,\u2009100.1\u2009mph<v0<112.5\u2009mph,\u200917.9\xb0<\theta 0<35.9\xb0$, and $51.6\u2009ft<hmax<136.0\u2009ft$. The *v*_{0}, *θ*_{0}, and *h*_{max} data are actually measured, and are given to the tenth's place in their respective units. The range *R* data are given to the nearest foot and represent the projected distance since the ball usually lands in the stands.^{10}

## II. MODELING THE TRAJECTORY

The home run data can be modeled using ideas from Refs. 1–8. The force that the air exerts on a flying baseball can be separated into a component opposite to the direction of the velocity $v\u2192$, and one perpendicular to the velocity. The component opposite to $v\u2192$ is referred to as drag. It is common to include this air frictional drag force in numerical exercises, and to take its magnitude proportional to the speed squared (*v*^{2}). The component of force perpendicular to $v\u2192$ is due to the Bernouli effect and is referred to as the Magnus force; its direction is perpendicular to $v\u2192$ and $\omega \u2192$, where $\omega \u2192$ is the angular velocity vector of the ball. For a classroom treatment, we use the most basic equations that best describe the motion. Since we are trying to determine two data points, *R* and *h*_{max}, we should have at most two free parameters in our equations. For these considerations, we use the following ansatz:

where *a _{x}* and

*a*are the acceleration components, and

_{y}*v*and

_{x}*v*the components of the velocity in the

_{y}*x*and

*y*directions. (The

*y*direction is vertical and the

*x*direction is horizontal, pointing in the direction the ball is traveling.) The acceleration

*g*due to gravity enters only in the equation for

*a*. The parameters

_{y}*v*and

_{T}*l*take into account air friction and the Magnus force as follows.

_{T}The force of air resistance, or drag, is $|F\u2192d|=Fd=12CD\rho Av2$, where *ρ* is the density of air, *A* is the cross section of the baseball, and *C _{D}* is the drag coefficient. The drag force is in the $\u2212v\u2192$ direction and can be expressed in terms of the terminal speed

*v*. At the speed

_{T}*v*, the drag force equals the object's weight: $mg=(1/2)CD\rho AvT2$. The resulting expression for the magnitude of the drag force is $Fd=mgv2/vT2$; the force itself is therefore

_{T}The magnitude of the Magnus force is $|F\u2192M|=FM=(1/2)CL\rho Av2$, where *C _{L}* is the lift coefficient. The constant

*C*depends directly on the spin factor

_{L}*S*≡

*rω*/

*v*, where

*r*is the radius of the ball. Consequently, to a good approximation

*F*is proportional to

_{M}*ωv*. The vector $F\u2192M$ can be decomposed into a component that is horizontal, $FH\u2192$, and one that lies in the vertical plane, $Fl\u2192$. The component $FH\u2192$ can make the baseball curve left or right, while $Fl\u2192$ can cause the ball to rise or sink. It is the $Fl\u2192$ component that will affect

*h*

_{max}and

*R*the most, and we only include this component in the equations of motion. We also assume that $\omega \u2192$ is constant, or equal to its average value, throughout the flight. With these assumptions, the lifting component of the Magnus force will be proportional to

*v*and can be included using only a single parameter, which we will call

*l*. It is convenient to use the terminal speed in the parameterization by writing $|Fl\u2192|=lTmgv/vT$. As such, the parameter

_{T}*l*is the ratio of

_{T}*F*to the ball's weight when the ball is traveling at

_{l}*v*. The vector $F\u2192l$ lies in the

_{T}*xy*-plane and will be perpendicular to $v\u2192$; it can therefore be written

## III. NUMERICAL METHODS AND RESULTS

Equations (1) and (2) can be solved numerically using finite difference methods, and without the *a*_{lift} terms are classic examples. We solved these equations using the Euler method with a time step of 0.01 s. The initial position of the ball was taken to be *x*_{0} = 0 and *y*_{0} = 1 m, assuming the batter hits the ball on average when it is around a meter above home plate. The initial velocity in the *x*-direction is $vx0=v0\u2009cos\u2009\theta 0$ and in the *y*-direction $vy0=v0\u2009sin\u2009\theta 0$. We determined the range *R* as the value of *x* when *y* becomes negative. For more accuracy, one can interpolate linearly between the positive and negative values of *y* to find where *y* = 0. The range could differ be as much as one foot, since the distance covered in 0.01 s by an object traveling at 100 mph is around 0.4 m. Typical values for the parameters of the home runs we analyzed are listed in Table I.

v (mph)
. _{T} | l
. _{T} | R (ft)
. | h_{max} (ft)
. |
---|---|---|---|

78 | 0.5 | 396 | 87 |

83 | 0.5 | 418 | 89 |

78 | 0.7 | 417 | 102 |

v (mph)
. _{T} | l
. _{T} | R (ft)
. | h_{max} (ft)
. |
---|---|---|---|

78 | 0.5 | 396 | 87 |

83 | 0.5 | 418 | 89 |

78 | 0.7 | 417 | 102 |

There are two parameters that determine the fate of the ball: *v _{T}* and

*l*. One can match the range

_{T}*R*using only

*v*and setting

_{T}*l*= 0. However, for every home run, the predicted

_{T}*h*

_{max}is well below the data. In fact, for 47 of the 50 home runs and all home runs where

*θ*

_{0}< 31.5°, the maximum height is larger than $vy02/2g$, the value obtained neglecting the effect of the air. The trajectory of these 50 home runs cannot be accounted for without some lift. So, one has two parameters,

*v*and

_{T}*l*, to fit two data points,

_{T}*h*

_{max}and

*R*. The students can determine

*v*and

_{T}*l*for each home run and examine if the values are consistent and reasonable. For some home runs, Statcast lists an estimated hang time. For these cases, the students can check their predictions, although the data are only approximate.

_{T}Values for *v _{T}* and

*l*that “best fit” each home run can be carried out by varying the parameters in expected ranges to minimize a

_{T}*χ*

^{2}function. Using nested loops, we varied

*v*from 70–100 mph in increments of 1 mph, and varied

_{T}*l*from 0.2–1.0 in increments of 0.01. Our

_{T}*χ*

^{2}function was $\chi 2=(Rcalc\u2212Rdata)2+(hmax,calc\u2212hmax,data)2$. The grid sizes of ±1 mph for

*v*and ±0.01 for

_{T}*l*were small enough to match, within one foot, the range and maximum height data. One could search a smaller grid for better predictions, however, the model is too crude to justify more accuracy.

_{T}The values we obtained for the longest 50 home runs for the 2015 postseason were as follows. The terminal speeds *v _{T}* were all in the interval 75 mph <

*v*< 88 mph, with an average of 81.5 mph and a standard deviation of

_{T}*σ*= 3.3 mph. The terminal speed will depend upon air density and vary from park to park, though we note that none of the home runs in the 2015 post season were hit at high elevation. The relation $mg=(1/2)CD\rho AvT2$ can be used to obtain

*C*from the terminal speed. A value of

_{D}*v*= 81.5 mph yields a value for

_{T}*C*of 0.41. The values for

_{D}*l*were all in the interval 0.42 <

_{T}*l*< 0.71, with an average of 0.56 and a standard deviation of

_{T}*σ*= 0.08. Since $CL=2lTmg/\rho AvT2=lTCD$, the data have an average value for

*C*of approximately 0.23. This value for

_{L}*C*results in a spin factor

_{L}^{1,4}of around 0.25 or a rotation rate of ≈2400 rpm at

*v*. Thus, both

_{T}*C*and

_{D}*C*are “in the ball park” of accepted values.

_{L}^{11}

The results can lead to interesting classroom discussions. How accurate are the equations? How much could *ω* change during the flight? What considerations were not included? How much might wind affect the range? What variation could *v _{T}* have at sea level? What is

*v*for home runs hit in Denver? What rotation rates would give values of

_{T}*l*between 0.4 and 0.7, and how reasonable are these rates? The students can speculate about how far the home run ball would travel without the Magnus force, and so on. Answers to some of these questions can be found in the references. Every year there will be new home runs for the next class to analyze, and perhaps Statcast will include accurate estimates of hang time. We hope the Statcast home run data are a hit with the students, and lift their interest in the Bernoulli effect in baseball as well as other sports.

_{T}## References

To determine the projected HR distance, the MBL Statcast website “[c]alculates the distance of projected landing point at ground level on over-the-fence home runs.”

The coefficient *C _{D}* for a baseball does depend on the speed and rotation rate of the ball. From Ref. 4, most measurements of

*C*lie between 0.35 and 0.5. References 1 and 4 show plots relating a value for

_{D}*C*of 0.23 to a spin factor of approximately 0.25. The average rotation rate for major league pitchers is around 2240 rpm.

_{L}