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.

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 v0, the launch angle θ0, the maximum height hmax, and the range R of the home run. The data span a fair spread of values with $395 ft, and $51.6 ft. The v0, θ0, and hmax 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

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 $v→$, and one perpendicular to the velocity. The component opposite to $v→$ 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 (v2). The component of force perpendicular to $v→$ is due to the Bernouli effect and is referred to as the Magnus force; its direction is perpendicular to $v→$ and $ω→$, where $ω→$ 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 hmax, we should have at most two free parameters in our equations. For these considerations, we use the following ansatz:

$ax=−vxgvvT2−lTgvyvT,$
(1)
$ay=−g−vygvvT2+lTgvxvT,$
(2)

where ax and ay are the acceleration components, and vx and vy the components of the velocity in the 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 ay. The parameters vT and lT take into account air friction and the Magnus force as follows.

The force of air resistance, or drag, is $|F→d|=Fd=12CDρAv2$, where ρ is the density of air, A is the cross section of the baseball, and CD is the drag coefficient. The drag force is in the $−v→$ direction and can be expressed in terms of the terminal speed vT. At the speed vT, the drag force equals the object's weight: $mg=(1/2)CDρAvT2$. The resulting expression for the magnitude of the drag force is $Fd=mgv2/vT2$; the force itself is therefore

$F→d=−mgvvT2(vxî+vyĵ).$
(3)

The magnitude of the Magnus force is $|F→M|=FM=(1/2)CLρAv2$, where CL is the lift coefficient. The constant CL depends directly on the spin factor S ≡ /v, where r is the radius of the ball. Consequently, to a good approximation FM is proportional to ωv. The vector $F→M$ can be decomposed into a component that is horizontal, $FH→$, and one that lies in the vertical plane, $Fl→$. The component $FH→$ can make the baseball curve left or right, while $Fl→$ can cause the ball to rise or sink. It is the $Fl→$ component that will affect hmax and R the most, and we only include this component in the equations of motion. We also assume that $ω→$ 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 lT. It is convenient to use the terminal speed in the parameterization by writing $|Fl→|=lTmgv/vT$. As such, the parameter lT is the ratio of Fl to the ball's weight when the ball is traveling at vT. The vector $F→l$ lies in the xy-plane and will be perpendicular to $v→$; it can therefore be written

$F→l=mglTvT(−vyî+vxĵ).$
(4)

The acceleration alift caused by $Fl→$ is the last term in Eqs. (1) and (2). One can check that $|a→drag|=gv2/vT2, |a→lift|=lTg$ when v = vT, and that $a→lift·v→=0$ in these equations, as expected.

Equations (1) and (2) can be solved numerically using finite difference methods, and without the alift 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 x0 = 0 and y0 = 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 $vx0=v0 cos θ0$ and in the y-direction $vy0=v0 sin θ0$. 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.

Table I.

Results using typical values for the home runs we analyzed. In each case, v0 = 105 mph and θ0 = 28°. To assist readers in checking their code, the Euler method was used with a time step of 0.01 s and the range R is the value of x when y becomes negative.

vT (mph)lTR (ft)hmax (ft)
78 0.5 396 87
83 0.5 418 89
78 0.7 417 102
vT (mph)lTR (ft)hmax (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: vT and lT. One can match the range R using only vT and setting lT = 0. However, for every home run, the predicted hmax is well below the data. In fact, for 47 of the 50 home runs and all home runs where θ0 < 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, vT and lT, to fit two data points, hmax and R. The students can determine vT and lT 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.

Values for vT and lT that “best fit” each home run can be carried out by varying the parameters in expected ranges to minimize a χ2 function. Using nested loops, we varied vT from 70–100 mph in increments of 1 mph, and varied lT from 0.2–1.0 in increments of 0.01. Our χ2 function was $χ2=(Rcalc−Rdata)2+(hmax,calc−hmax,data)2$. The grid sizes of ±1 mph for vT and ±0.01 for lT 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.

The values we obtained for the longest 50 home runs for the 2015 postseason were as follows. The terminal speeds vT were all in the interval 75 mph < vT < 88 mph, with an average of 81.5 mph and a standard deviation of 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 $mg=(1/2)CDρAvT2$ can be used to obtain CD from the terminal speed. A value of vT = 81.5 mph yields a value for CD of 0.41. The values for lT were all in the interval 0.42 < lT < 0.71, with an average of 0.56 and a standard deviation of σ = 0.08. Since $CL=2lTmg/ρAvT2=lTCD$, the data have an average value for CL of approximately 0.23. This value for CL results in a spin factor1,4 of around 0.25 or a rotation rate of ≈2400 rpm at vT. Thus, both CD and CL are “in the ball park” of accepted values.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 vT have at sea level? What is vT for home runs hit in Denver? What rotation rates would give values of lT 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.

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9.
The URL for the home run data can be found at <http://m.mlb.com/statcast/leaderboard#hr-distance>. For this article, we used the 2015 post-season data.
10.

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.”

11.

The coefficient CD for a baseball does depend on the speed and rotation rate of the ball. From Ref. 4, most measurements of CD lie between 0.35 and 0.5. References 1 and 4 show plots relating a value for CL of 0.23 to a spin factor of approximately 0.25. The average rotation rate for major league pitchers is around 2240 rpm.