MathSource Reviews 9(3)

Swatting Flies: Airborne Baseball Trajectories

Edited by Matthew M. Thomas

There are, to be sure, more important applications for the tools of physics than the sport of baseball. But scientists who are baseball fans often become baseball fans before they become scientists, and baseball becomes an avocational pursuit of these scientists. That such is so is seen in sessions of, say, the American Association for the Advancement of Science (AAAS): The February 2000 AAAS Annual Meeting featured a "Science of Baseball" symposium, whose presenters included Harvard's Stephen Jay Gould ("Can Anyone Hit 71 Home Runs?"), Illinois' Alan M. Nathan ("How Would a Physicist Design a Bat?"), Yale's Robert K. Adair ("Batting and Thinking"), and Arizona's A. Terry Bahill ("The Rising Fastball and Ideal Bat Weights"). Indeed, as of this November 2000 writing, Prof. Nathan's paper on the dynamics of the baseball-bat collision [Nathan 2000] was just published in the American Journal of Physics -- arguably the preferred refereed journal for examining the physics of baseball. Even Vannevar Bush, director of many very important applications for the tools of physics, once took time to correspond with stroboscopic photography pioneer Harold "Doc" Edgerton on ... a baseball struck by a bat [Bush 1965]. This MathSource Review addresses trajectories of airborne batted baseballs. If you deem this Review topic frivolous, let this paragraph convince you that frivolity loves company.

Earlier in the year of this writing, Major League Baseball staged its first all-New York City championship round ("World Series") since New York's two previous National Baseball League teams moved to Los Angeles and San Francisco. But it is the first post-move World Series between the New York-anchored American Baseball League champion "Yankees" and a transplanted New York-originated National League champion (the San Francisco "Giants") that provides a seemly setting for introducing this Review. The year is 1962; the scene is World Series decisive Game Seven in San Francisco's Candlestick Park, with the visiting Yankees leading 1-0 in the bottom of the ninth inning. Ralph Terry is the Yankee pitcher; the Giant baserunners are Matty Alou at third base and Willie Mays at second; two are out; the Giant batter is Willie McCovey. Note that two years before -- in the bottom of the ninth of Game Seven of the 1960 World Series between the Yankees and Pittsburgh -- Ralph Terry yielded the dramatic home run by Bill Mazeroski that won the game and the Series for Pittsburgh. Note that Terry yielded more home runs (40) than any American League pitcher during the 1962 regular season. Note that Willie McCovey would become a Major League Baseball Hall-of-Fame member whose 521 career home runs remain the most by any lefthanded batter in the National League, and whose 18 career grand-slam home runs remain the most by any batter -- lefthanded or righthanded -- in the National League. And note that McCovey had taken Terry's measure: McCovey homered off Terry in Game Two of that 1962 World Series, hit a 410-foot triple into a 35 mile-per-hour (MPH) wind in from center field off Terry in the seventh inning of Game Seven, and, on that first pitch of this current at-bat, hit a home run-length drive down the right field line that curved foul. Terry's next pitch to McCovey was a ball. The climactic 1-1 pitch from Terry was a fastball that McCovey lined ... right at Yankee second baseman Bobby Richardson in short right field for the final out, giving the Yankees their twentieth World Series victory and Ralph Terry 1962 World Series Most Valuable Player honors. A teacher at a school in the Giants' new neighborhood -- Professor Paul Kirkpatrick of Stanford -- asserted that had McCovey's bat collided with that fateful Terry pitch one millimeter closer to the ground, the baseball would have flown past Richardson for the Series-winning hit [Kirkpatrick 1963, p. 606].

As time goes by, of course, that McCovey line drive becomes a harder and harder hit baseball, thus causing the Richardson putout to become a better and better fielding play. So goes baseball lore. There are reports, however, that McCovey's line drive put not backspin but topspin on the baseball, and that Richardson as such made a relatively easy catch at chest level [see the References for details]. Question: What is required to simulate the fly-ball or line-drive trajectory of a batted baseball? Response: As of this writing, such simulations were available via the World-Wide Web, at URLs www.exploratorium.edu/baseball/scientificslugger.html, www.netgworld.com/demos/baseball.htm, and www.scri.fsu.edu/~jac/Java/baseball.html. Follow-up: Can Mathematica play a role in such simulation? Obvious answer: Yes, and the example is MathSource item WhatsNew/0211-318/Baseball1.nb by Robert Lee of Millersville University. Let us now, in order, review the relevant physics, review the code in Baseball1.nb, put the program through its paces, and deliver an overall assessment.

BACKGROUND

If you are interested in the physics of airborne baseballs, there are two books that you ought to read: The Physics of Baseball by the aforementioned Robert K. Adair, and Keep Your Eye On The Ball: Curve Balls, Knuckleballs, and Fallacies of Baseball by Robert G. Watts and the aforementioned A. Terry Bahill. Adair, Sterling Professor of Physics at Yale, served as "Physicist to the National League" (1987-1989) under National League President A. Bartlett Giamatti (formerly President of Yale); his well-written 142-page volume contains a wealth of graphs but a paucity of equations. Watts, a former semiprofessional pitcher who teaches in the Tulane Mechanical Engineering department, and Bahill, Professor of Systems Engineering at Arizona, offer more of a baseball-physics textbook, appealing more to the technologist and less (by just a bit) to the technology-adverse. Together, these two books provide an excellent starting point for research into the physics of airborne baseballs. And if some evil force were to constrain you to the archives of just one journal for such research, your best choice would be the American Journal of Physics: From the brand-new work by Alan M. Nathan to the earliest seminal works, much physics-in-baseball research has found a home in its pages.

Within the topic of the physics of baseball, there are three prominent subtopics: pitched baseball curvature; baseball-bat collision; batted baseball trajectory. Lyman J. Briggs of the National Bureau of Standards was among the first [Briggs 1959] to inspect the lateral movement of a pitched baseball out of its initial vertical plane. He attributed that movement to the "Magnus effect" named for German engineer G. Magnus, whose examination of projectile spin and its effects was published in Berlin in 1852 (long after Benjamin Robins' 1742 work New Principles of Gunnery ostensibly offered a similar examination) [see also Adair 1994, pp. 12-15 and Watts and Bahill 2000, Chapter Three]. Watts and Ferrer [1987] credit Briggs with correcting the Robins/Magnus explanations for their observations, but challenge Briggs' contention that the magnitude of baseball movement is proportional to the rate of spin and to the square of velocity: They assert that this magnitude is in proportion to rate of spin and to velocity (rather than to its square). The previously cited papers by Kirkpatrick [1963] and Nathan [2000] address baseball-bat collisions, with Kirkpatrick pinpointing thirteen variables (time and three coordinates each of position, angular orientation, and linear and angular momentum) that define the bat at the point of contact, and with Nathan showing how internal bat vibrations influence the post-collision velocity of the baseball. See also Adair [1994, pp. 71-79] and Watts and Bahill [2000, Chapter Five] for more on baseball-bat collisions.

It is the last of these subtopics -- batted ball trajectory -- that is the most pertinent to this Review. Forces at work on an airborne baseball include gravity, drag, and spin. The Magnus effect is inherent in spin [de Mestre 1990]. Newton's Second Law of Motion demands that the net effect of these forces equal the time rate of change of the baseball's momentum. Motivated by what's to come, let us adopt a right-handed Cartesian coordinate system in which the apex of home plate is the origin, the initial vertical plane of the batted baseball extends from the origin into "fair territory" along the positive x-axis, and the positive z-axis extends from home plate to the heavens at a right angle to that xy-plane which is the baseball field. Gravity can then be said to act in the -z direction on the baseball, while drag acts in the opposite direction of baseball velocity vector (Overscript[v, ⇀]) (though still in the xz-plane). If the spin is pure backspin, angular velocity vector (Overscript[ω, ⇀]) is in the -y direction, and (Overscript[ω, ⇀])×(Overscript[v, ⇀]) is normal to drag force direction -(Overscript[v, ⇀]) though still in the xz-plane -- in this case, the force from the spin is "lift". Pure topspin leads to (Overscript[ω, ⇀]) in the +y direction and an "anti-lift" effect. Pure clockwise sidespin (as seen from above) has (Overscript[ω, ⇀]) in the direction of the pure anti-lift effect in the xz-plane, and has (Overscript[ω, ⇀])×(Overscript[v, ⇀]) in the -y direction; pure counterclockwise sidespin has (Overscript[ω, ⇀]) in the pure lift direction (xz-plane) and has (Overscript[ω, ⇀])×(Overscript[v, ⇀]) in the +y direction. Thus, clockwise sidespin often pulls fly balls to right field toward right-field foul territory (à la the McCovey first strike in the 1962 World Series Game Seven bottom of the ninth), while counterclockwise sidespin often pulls same to left field toward left-field foul territory. Of course, rare is the baseball whose post-collision spin is purely topspin, backspin, or sidespin, or which for that matter is not also affected by wind.

If you neglect all three forces, then the baseball will fly off the bat at a constant speed in the initial (and unchanging) direction of (Overscript[v, ⇀]). This case is obviously the trivial case. Consider only gravity, and you get those trajectory equations seen in the early chapters of textbooks from first courses in physics: time of projectile flight = (2(Subscript[v, 0])sin θ)/g; flight range = (((Subscript[v, 0])^2)sin(2θ))/g; trajectory apex = (((Subscript[v, 0])^2)(sin^2)θ)/2g (where (Subscript[v, 0]) is the initial speed of the batted baseball, g is the acceleration due to gravity, and θ is the angle that (Overscript[v, ⇀]) initially makes with the +x axis). Add drag to the trajectory model [Frohlich 1984], and you simultaneously render it more realistic and more difficult to solve. Baseballs facing gravity and drag forces while free from lift, sidespin, or similar (Overscript[ω, ⇀])×(Overscript[v, ⇀]) forces are baseballs with no spin -- on the relatively rare occasions when batters put such balls in play, the baseballs tend to go to center field, where center fielders tend to find these "batted knuckleballs" virtually uncatchable [Adair 1994, p. 84]. Add backspin to the batted baseball, and lift enters the trajectory model [Watts and Baroni 1989; Watts and Bahill 2000]. To a point, backspin increases the flight range of an airborne batted baseball, while topspin decreases it. If the modeler stops with pure backspin/topspin and the forces generated thereof, he constrains his problem to the xz-plane; maintain the baseball rotation axis parallel to the y-axis but let it lose its perpendicularity to the xz-plane, and a 3-D problem is born.

Adding to the subtlety of airborne baseball trajectory determination are the 108 stitches (or 216, depending on how you actually define the stitch) holding together the two cowhide strips encasing each baseball. The effect of the resulting seam on baseball drag is not fully understood. Frohlich [1984, pp. 326-327] had found that the seam height is approximately 0.5 mm, and has deemed the ratio of seam height to baseball diameter large enough to affect drag (in comparison to surface roughnesses from other spheres in an experimental study by Achenbach [1974]). But when Erlichson drew contrasts between the non-uniform seam-based baseball surface roughness and the uniform surface roughness of Achenbach's spheres, Frohlich responded that his results were more qualitative than quantitative, given a lack of data for the velocity dependence of baseball drag coefficients at that time [Erlichson 1985]. Frohlich also emphasized a sharp (if not rigidly quantified) reduction in baseball drag coefficient magnitude for a given regime of higher baseball velocities. But by contrast, subsequent work by Watts and Ferrer [1987] found drag coefficient to be only a weak function of Reynolds number (and thus velocity), and follow-up work by Watts and Baroni [1989, p. 43] uses a constant drag coefficient for baseball trajectory calculations to simplify those calculations. All involved concur that accurate quantitative baseball drag coefficient measurement remains a goal to be attained, with Adair [1994, p. 10] providing perhaps the most succinct assessment of the situation. Suffice it to say that just as we have assumed pure backspin/topspin on the batted baseball, so shall we assume a constant baseball drag coefficient during flight.

Driven by the considerations laid down above, and following the lead of Watts and Baroni [1989, pp. 42-43] and Watts and Bahill [2000, pp. 159-166], let's write Newton's Second Law in both the x- and z-directions:

x: m(((d^2) x)/(dt^2))= - (Subscript[F, D]) cos θ - (Subscript[F, L]) sin θ
z: m(((d^2) z)/(dt^2))= - (Subscript[F, D]) sin θ + (Subscript[F, L]) cos θ - mg

where m is the baseball mass, (Subscript[F, D]) and (Subscript[F, L]) are the drag force and lift force (respectively) on the baseball, and θ and g are as defined above. Using standard definitions of drag and lift forces in terms of their respective coefficients, we write (Subscript[F, D]) = 0.5 ρ(Subscript[C, D])A(v^2) and (Subscript[F, L]) = 0.5 ρ(Subscript[C, L])A(v^2), where (Subscript[C, D]) and (Subscript[C, L]) are unitless drag and lift coefficients (respectively), ρ is the density of air, A is the cross-sectional area (π (r^2)) of the baseball, and v is the magnitude of the baseball velocity, defined as v = (Sqrt[(((dx/dt))^2) + (((dz/dt))^2)]). Lift coefficient data obtained in laboratory settings from baseballs allow us to express (Subscript[C, L]) as a function of ω: (Subscript[C, L]) = rω/v, where r is the baseball radius [Watts and Ferrer 1987, Figure 5; Watts and Baroni 1989, Equation 17]. In addition, exploiting the trigonometry of the force directions leads to cos θ = (1/v)((dx/dt)) and sin θ = (1/v)((dz/dt)). Applying these definitions to our Second Law expressions, and subsequent simplification, give us

x: (((d^2) x)/(dt^2))= -(1/2)((ρ/m))(Subscript[C, D])π(r^2)(Sqrt[((((dx/dt)))^2) + ((((dz/dt)))^2)])((dx/dt)) - (1/2)((ρ/m))π(r^3)ω((dz/dt))
z: (((d^2) z)/(dt^2))= -(1/2)((ρ/m))(Subscript[C, D])π(r^2)(Sqrt[((((dx/dt)))^2) + ((((dz/dt)))^2)])((dz/dt)) + (1/2)((ρ/m))π(r^3)ω((dx/dt)) - g

Note that when ω = 0, there is no spin on the batted baseball, the lift terms above disappear, and these equations represent the case in which gravity and drag are the only forces on the baseball [see Frohlich 1984, p. 327]. We can now use Mathematica to examine the theoretical trajectories of batted baseballs in the xz-plane, given post-collision baseball speed (Subscript[v, 0]) and launch angle θ. Air density at 80°F is 0.0735 lbm/(ft^3); we follow Watts and Baroni [1989, p. 43] in using "constant" drag coefficient (Subscript[C, D]) = 0.5; baseball rotation rate ω is in radians per second. Major League Baseball Rule 1.09 calls for a baseball to weigh not less than five nor more than 5.25 ounces avoirdupois, and measure not less than nine nor more than 9.25 inches in circumference. The mid-range of these tolerances gives us m = 5.125 oz. avd. = 0.320 lbm, and r = 9.125 in./2π = 1.47 in. = 0.121 feet. We take g to be 32.174 ft/(sec^2), and make use of NDSolve[]: The inputs to trajectory[] are v0 (as defined above, in MPH), θ (also defined above, in degrees), and rpm (baseball rotation rate, in revolutions per minute).

\(trajectory[v0_, \ θ_, \ rpm_] := 
      Module[{ω, \ vx0, \ vz0, \ ρ, \ cd, \ r, \ m, \ sol, x, z, 
          tmax}, 
ω\  = \ rpm\ Pi/30 // N; \  (*\ 
          converting\ revolutions/minute\ to\ radians/
              second\ *) 
vx0\  = \ 
          v0\ Cos[θ\ Pi/180]\ \((5280/3600)\);  (*\ 
          x - component\ of\ post - collision\ velocity; \ 
          converting\ MPH\ to\ ft/sec\ *) 
vz0\  = \ 
          v0\ Sin[θ\ Pi/180]\ \((5280/3600)\);  (*\ 
          z - component\ of\ post - collision\ velocity; \ 
          converting\ MPH\ to\ ft/
              sec\ *) 
ρ\  = \  .0735; \ \ \  (*\ 
          air\ density\  .0735\ lbm/ft^3\ for\ 80  
            degF\ *) 
cd\  = \  .5; \  (*\ 
          coefficient\ of\ drag, \ 
          used\ by\ Watts/
              Baroni\ in\ 1989\ *) 
r\  = \ \((\((9.125/
                    12)\)/Pi)\)/2; \  (*\ 
          baseball\ radius\  = \ 0.121\ feet\ *) 
m\  = \ 
          5.125/16; \  (*\ 
          baseball\ mass\  = \ 0.320\ lbm\ *) 
\ \ \ \ \ sol = 
          NDSolve[{\(\(x'\)'\)[
                  t] == \(-0.5\)\ \((ρ/
                        m)\)\ cd\ Pi\ r^2\ \ Sqrt[\(x'\)[t]^2 + \(z'\)[
                            t]^2]\ \(x'\)[
                      t] - 
0.5\ \((ρ/
                        m)\)\ Pi\ \((r^3)\)\ ω\ \(z'\)[t], \  (*\ 
                x - component\ of\ 2  
                    nd\ Law\ \(\(Eqn\)\(.\)\)*) 
	\ \ \(z''\)[
                  t] == \(-0.5\)\ \((ρ/
                        m)\)\ cd\ Pi\ r^2\ \ Sqrt[\(x'\)[t]^2 + \(z'\)[
                            t]^2]\ \(z'\)[t] + \ 
                  0.5\ \((ρ/m)\)\ Pi\ \((r^3)\)\ ω\ \(x'\)[
                      t]\  - \ 32.174, \  (*\ 
                z - component\ of\ 2  
                    nd\ Law\ \(\(Eqn\)\(.\)\)*) 
	\ \ \ \ \ \ \ x[
                  0] == 1, 
              z[0] == 3, \  (*\ 
                initial\ position\ \(conditions : \ 
                    ball\ struck\ 1\ ft\ in\ front\ of\ plate\ and\ 3\ ft\ above\ ground\)\ *) 
		\(x'\)[0] == vx0, \(z'\)[0] == vz0\  (*\ 
                initial\ velocity\ components\ *) }, 
				\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {x, z}, {t, 0, 10}]; 
		x[t_] = 
          x[t] /. sol[\([1]\)]; \ 
        z[t_] = z[t] /. sol[\([1]\)]; \ \ \ \ 
		tmax = 
          t /. \ FindRoot[z[t] == 0, {t, 10}]; 
\ 
        ParametricPlot[\ {x[t]\ , \ z[t]\ }, {t, 0, tmax}, \  (*\ 
            plot\ the\ result\ *) \ {\ 
            AxesLabel -> {range \((ft)\), \ 
                height \((ft)\)}, 
PlotRange\  -> \ {{0, 600}, {0, 200}}}]]\)

With a tool now at our disposal, let's examine a batted baseball launched at 110 MPH and thirty degrees above the horizon, with no spin on the baseball:

trajectory[110, 30, 0]

Given this baseline, let us now examine the benefit of 3700 revolutions per minute worth of pure backspin on the batted baseball, at the same post-collision baseball speed and launch angle:

trajectory[110, 30, 3700]

Lift is good: The backspin extends the range of the batted baseball by 80 feet (as well as roughly doubling the trajectory apex). But what if the spin were pure topspin in lieu of pure backspin under these conditions?

trajectory[110, 30, -3700]

Obviously, topspin lowers the trajectory apex, and, more importantly, decreases the batted baseball range by roughly 100 feet in comparison to the baseline case. Finally, let's consider a massive (albeit theoretical) amount of pure backspin, as perhaps generated by a blatantly illegal bat with ample amounts of carved grooves and pine tar well above the handle:

trajectory[110, 30, 7000]

Lift may be good, but too much of anything has its drawbacks: Note that lift force taken to the extreme produces not only a huge trajectory apex, but also (and more importantly) a theoretical batted baseball range virtually identical to that of the baseline case.

The vigorous reader who uses trajectory[] to generate plots for comparison with Watts and Bahill [2000, Figures 54-57] will see that, given analogous inputs and physical properties, the plots correspond in shape and magnitude. Figures 55-57 of Watts and Bahill [2000] are apparently the English-unit equivalents of Figures 5-7 of Watts and Baroni [1989]; both the latter work and Frohlich [1984, p. 325] cite 1.29 kg/(m^3) (0.0805 lbm/(ft^3)) as the density of air for their calculations. But that density value applies at 0°C -- why so cold? Given that more Major League Baseball games are played at 80°F than at 32°F, the use of ρ(air, 80°F) = 0.0735 lbm/(ft^3) seems more appropriate. A close examination of Figures 54-57 from Watts and Bahill [2000, pp. 162-166] reveals that a batted baseball launched at a 30-degree angle with the horizon at 110 MPH and 6000 revolutions per minute travels 315 feet in Figure 54, 340-360 feet in Figure 55, 360 feet in Figure 56, and 330 feet in Figure 57 -- for four graphs based on the same phenomenon, there is some imprecision from graph to graph. Given the physical properties used by Watts and Baroni/Bahill, trajectory[] generates output that falls within their precision limits, allows users to examine lift effects on baseballs, and provides a tool for checking MathSource contributions such as WhatsNew/0211-318/Baseball1.nb by Robert Lee.

MILLERSVILLE CONTRIBUTION

The formal title of said MathSource entry is "Baseball Trajectory With Graphics" and, as the five-line explanatory introduction makes note, it had its original use in fulfilling the requirements of an undergraduate physics class at Millersville University (roughly 120 km west of Philadelphia in Pennsylvania, U.S.A.). The introduction states that the user can "input the pitch and bat speed", and that the program then finds and illustrates the batted baseball trajectory at "random angles" in the xy-plane -- no effort is made to tackle the admittedly arduous task of determining whether the theoretical batter "pulls the ball" or "goes to the opposite field". The introduction also states that air drag is considered in the trajectory calculation, thereby noting that the program goes beyond the gravity-only approach to projectile trajectory calculation found in first-year physics textbooks, without saying anything about spin effects.

WhatsNew/0211-318/Baseball1.nb is a 14-kB notebook file, but its only contents are the Mathematica code for the baseBall[] package. The Module[] function plays the lead role in this package, integrating elements of graphics database creation, simultaneous ordinary differential equation (ODE) solution, and graphics generation from this creation and solution. Variables g1 through g9 "contain" the graphics database, which Robert Lee creates through extensive use of the RGBColor[] and Polygon[] functions. In-line (* comments *) describe which graphics elements correspond to what parts of a baseball park (field, grandstands, side of grandstands, the baseball diamond, a billboard, flags, and even birds flying above the park). NDSolve[] is used for simultaneous ODE solution. The Show[] operation within a Do[] loop creates an animation to illustrate the baseball trajectory, with ParametricPlot3D[] plotting the NDSolve[] output, and with Graphics3D[] illustrating the baseball on its trajectory (and its shadow on the field), stitching together elements in the graphics database, and displaying the text that tells how far and to which field the baseball was hit.

In terms of baseball trajectory calculation, the immediate prelude to and execution of NDSolve[] is at the core of Baseball1.nb. Robert Lee sets the angle with the horizon at which the baseball leaves the bat this way: If the pitch speed is less than 70 ft/sec, the angle is a randomly set value between 45 and 60 degrees; if the pitch is faster than or equal to 70 ft/ sec, the angle is a randomly set value between 30 and 45 degrees. The initial values of dx/dt and dz/dt are then determined from that angle, from the pre-collision pitch and bat speeds, and from conservation of momentum between baseball and bat. Those initial values serve as initial velocity conditions in NDSolve[]; initial position conditions are (Subscript[x, 0]) = 3 and (Subscript[z, 0]) = 4 (i.e., the batter hits the baseball three feet in front of the plate and four feet above the ground). The simultaneous ODEs actually solved in Lee's use of NDSolve[] are

(((d^2) x)/(dt^2))= - 0.00165 (Sqrt[((((dx/dt)))^2) + ((((dz/dt)))^2)])((dx/dt))
and
(((d^2) z)/(dt^2))= - 0.00165 (Sqrt[((((dx/dt)))^2) + ((((dz/dt)))^2)])((dz/dt)) - 32

Lee imposes the right-hand Cartesian coordinate system upon the baseball field such that the apex of home plate is the origin, the positive x-axis extends from the origin to dead center field, and the positive z-axis extends from home plate to the heavens. For each x[t] solution generated by NDSolve[], a y[t] is found from y[t_]=x[t] Random[Real,{-1,1}] (where positive y[t] values lie on the third-base side of the field, and their negative counterparts lie on the first-base side). Thus the magnitude of y[t] can equal but never exceed that of x[t], so these definitions apply: y[t]=x[t], baseball hit down the third-base foul line; y[t]=0, baseball hit to dead center field; y[t]=-x[t], baseball hit down the first-base foul line. The time span allowed for the numerical ODE solution is 0-10 seconds, with time of projectile flight tmax found from FindRoot[z[t]==0,{t,10}]; flight range x1max=N[x[tmax],4], and "random angle" parameter y2max=N[y[tmax],4] to determine where in the xy-plane (between the left- and right-field foul lines) the baseball was hit. The following seven-condition Which[] execution determines a somewhat (but not fully) arbitrary category of batted baseball:
If x1max>400 and -100<=y2max<=100, Home-Run in Center;
If x1max<100, Out;
If x1max<200, Single;
If x1max<300, Double;
If x1max>(-y2max+2100)/5 and y2max>100, Home-Run in Left Field;
If x1max>(y2max+2100)/5 and y2max<-100, Home-Run in Right Field;
If x1max<400, Triple.

BATTING PRACTICE

Having seen how Baseball1.nb is laid out, let us now figuratively step into the batter's box and hit some baseballs. To match the first example set by Lee, let us call for a slow pitch (80 ft/sec) and swing hard (100 ft/sec) at it:

baseBall[80, 100]

Our result: A 375-foot triple into the right-center field "power alley". If the pitch were also 100 ft/sec just before collision, then that 100 ft/sec swing of the bat adds 20 feet to the batted baseball range -- enough for a shot to right field to clear the fence for a home run:

baseBall[100, 100]

If not only the pitch but also the bat moves at only 80 ft/sec, the range of the batted baseball falls to under 300 feet, earning the "double" classification (assuming that the center fielder does not catch this fly ball -- Baseball1.nb defines type of hit as a function of range, without considering how "playable" a batted baseball is by the fielders):

baseBall[80, 80]

Finally, when we swing softly at a soft pitch, the result is a pop-up to either the third baseman or the shortstop:

baseBall[40, 40]

Having spent our dollar at a quarter per swing, let us comment on what we've accomplished. First, note that each application of baseBall[] results in an animation -- space constraints require the showing of just one frame from each animation. Note also that, since the interior pages of this journal do not accommodate color, the preceding four frames have been modified to appear in shades of gray. Running baseBall[] with a color monitor produces a green field (a realistic choice, given the color of grass and artificial turf) and a purple grandstand (a bit outré for Major League Baseball, perhaps). These issues, however, are issues of aesthetics. There are three other, more serious issues in need of addressing. All three could have been avoided through good documentation ... or, for that matter, any documentation.

1) Units. Seeing "32" in the ODE for the second derivative of z[t] with respect to time leads the technically inclined to assume that English units are at work, since the acceleration due to gravity is roughly 32 feet per second per second. Seeing "400" in the Which[] conditions for a home run to dead center field leads the baseball fan also to assume that English units are at work, since 400 feet is a reasonable distance for the center field fence to be from home plate in a Major League Baseball park. But apart from these two rather subtle clues in the Mathematica code, the user has no idea that s/he is supposed to enter the initial pitch speed and initial bat speed for baseBall[] in feet per second. The first choice of the baseball fan would be MPH: Some Major League Baseball parks feature radar guns that report pitch speed on the scoreboard, invariably in MPH, and when the ESPN cable television network did a study of Major League Baseball bat speeds a few years ago, it reported its results in MPH. By contrast, a Mathematica user from outside of North America might reasonably opt for meters per second as the speed units for baseBall[] inputs. Given no guidance, the casual baseBall[] user knows not whether to enter pitch and bat speeds in feet per second, meters per second, miles per hour, or furlongs per fortnight.

2) Choice of drag coefficient. Lee's set-up of x- and z-direction Second Law equations is correct, mirroring that of Frohlich [1984] in considering drag and neglecting spin. In the absence of documentation, though, we only know that -0.5 (ρ/m)(Subscript[C, D])π(r^2) has been set equal to -0.00165: one equation, four variables -- three degrees of freedom. From before, let us take the density of air to be ρ = 0.0735 lbm/(ft^3), the mass of a baseball to be m = 0.320 lbm, and the baseball radius to be r = 0.121 feet; solving for (Subscript[C, D]) thus produces (Subscript[C, D]) = 0.312. This drag coefficient value is on a par with those reported by Watts and Bahill [2000, Figure 52], albeit on the low side (in comparison with the 0.5 value used in trajectory[]). Anecdotally, the top Major League Baseball pre-collision pitch and bat speeds are taken to be 100 MPH (146.67 ft/sec); the reader can verify that baseBall[146.67,146.67] produces a mammoth batted baseball range of 600 feet -- a range far greater than what is expected in Major League Baseball play [Adair 1994, pp. 100-106], especially in the absence of batted baseball spin. This excessively large output from Lee's model can be traced in part to too little drag force, in proportion to its smaller value for (Subscript[C, D]). On the other hand, if we retain the (Subscript[C, D]) = 0.5 value (and baseball mass and radius) from before, solving for the density of air produces ρ = 0.0459 lbm/(ft^3) (0.74 kg/(m^3)). Standard atmosphere property tables reveal 0.74 kg/(m^3) to be the density of air at an altitude of 5000 m and a corresponding temperature of -18°C. Thus, under these circumstances, the baseBall[146.67,146.67] output in the absence of batted baseball spin takes on a greater degree of reality ... for any Major League Baseball game played atop Mount Ararat in Turkey, that is. There is a certain amount of exploratory fun to be had with such parameter tweaking, but an explanation as to how -0.5 (ρ/m)(Subscript[C, D])π(r^2) became -0.00165 would have obviated the need for such tweaking.

3) Conservation of momentum. Though fundamentally at odds with each other, the pitcher and batter conspire (through speeds of the pitch and bat, respectively) to fix the initial velocity of the batted baseball. To determine initial velocity, Lee uses (Subscript[(Overscript[v, ⇀]), 0]) = ((1/4)vPitch + (5/4)vBat) cos θ \(x\&^\) + ((1/4)vPitch + (5/4)vBat) sin θ \(z\&^\). The equation satisfies (Subscript[(Overscript[v, ⇀]), 0]) = f(vPitch, vBat, θ), but what is the basis for coefficients 1/4 and 5/4? If we examine conservation of momentum, using m = mass, v = velocity, 1 = baseball, 2 = bat, B = before collision, and A = after collision, we can write (Subscript[m, 1])(Subscript[v, 1  B]) + (Subscript[m, 2])(Subscript[v, 2  B]) = (Subscript[m, 1])(Subscript[v, 1  A]) + (Subscript[m, 2])(Subscript[v, 2  A]), where (Subscript[v, 1  B]) = -vPitch and (Subscript[v, 2  B]) = vBat. We also define the coefficient of restitution "e" as the ratio of relative speed of the bat and baseball after impact to that before impact; that definition and a simplifying assumption of impact at the center of bat mass lead to e = -((Subscript[v, 1  A]) - (Subscript[v, 2  A]))/((Subscript[v, 1  B]) - (Subscript[v, 2  B])) [Watts and Baroni 1989, pp. 40-41; Watts and Bahill 2000, pp. 139-140]. Solving these two equations simultaneously for (Subscript[v, 1  A]) yields

(Subscript[v, 1  A]) = (((Subscript[m, 1]) -  (Subscript[m, 2]) e)/((Subscript[m, 1])  +  (Subscript[m, 2])))(Subscript[v, 1  B]) + (((Subscript[m, 2]) +  (Subscript[m, 2]) e)/((Subscript[m, 1])  +  (Subscript[m, 2])))(Subscript[v, 2  B])

But recall that (Subscript[v, 1  B]) = -vPitch (the negation results from the direction change of the baseball upon impact), and (Subscript[v, 2  B]) = vBat. Also note that (Subscript[v, 1  A]) is the initial baseball speed after impact, or (Subscript[v, 0]). We thus have

(Subscript[v, 0]) = (((Subscript[m, 2]) e  -  (Subscript[m, 1]))/((Subscript[m, 1])  +  (Subscript[m, 2]))) vPitch + (((Subscript[m, 2]) +  (Subscript[m, 2]) e)/((Subscript[m, 1])  +  (Subscript[m, 2]))) vBat

From before, (Subscript[m, 1]) = 0.320 lbm. In Major League Baseball, two ranges for the mass of a wood baseball bat are 1.938 lbm ≤ (Subscript[m, 2]) ≤ 2.250 lbm [Adair 1994, p. 109] and 1.938 lbm ≤ (Subscript[m, 2]) ≤ 2.063 lbm [Watts and Bahill 2000, p. 125]. Though Rule 1.09 doesn't require as much, the coefficient of restitution of a baseball is taken as 0.55 [Adair 1994, p. 75; Watts and Bahill 2000, p. 102], with a range of 0.514 - 0.578 deemed acceptable [Adair 1994, p. 86]. Comparing this last equation to the magnitude-only version of Lee's yields ((Subscript[m, 2])e - (Subscript[m, 1]))/((Subscript[m, 1]) + (Subscript[m, 2])) = 1/4 and ((Subscript[m, 2]) + (Subscript[m, 2])e)/((Subscript[m, 1]) + (Subscript[m, 2])) = 5/4; thus, for our "unknowns" (Subscript[m, 2]) and e, we now have two equations. Alas, the equations are not independent: Adding one to each side of the first yields the second. Given the resulting degree of freedom, we let e = 0.55 and solve for (Subscript[m, 2]), getting (Subscript[m, 2]) = 1.3 lbm -- ideal for a Little Leaguer [Watts and Bahill 2000, p. 122]; woefully light for a Major Leaguer. As an alternate approach, if we set (Subscript[m, 2]) to 2 lbm (32 ounces) and solve for e, we get e = 0.45 -- below the allowable range of e presented earlier. Using (Subscript[m, 2]) = 2 lbm, e = 0.55, and the equation for (Subscript[v, 0]), Major League Baseball would re-write Lee's equation as (Subscript[(Overscript[v, ⇀]), 0]) = ((1/3)vPitch + (4/3)vBat) cos θ \(x\&^\) + ((1/3)vPitch + (4/3)vBat) sin θ \(z\&^\). Mr. Lee may well have had a good reason for using coefficients less by one-twelfth than those for Major League Baseball, but in the absence of documentation, we'll never know that reason.

ASSESSMENT

Your reviewer hopes that Robert Lee received high marks for MathSource item WhatsNew/0211-318/Baseball1.nb in his Millersville University undergraduate physics class. His graphical rendition of a baseball park may not be worthy of the latest video game technology, but it leaves no question that what one sees is indeed a baseball park, and it represents a creative use of the Polygon[] function. The animations depicting the flight of the baseball (as well as the shadow of the baseball on the field) are well-constructed. Use of the y-axis to indicate the field to which the baseball is hit is a clever strategy, in conjunction with the seven-condition Which[] function, even though such use implies a level of physics application (vis-à-vis sidespin forces) that doesn't exist in Lee's approach. Most significantly, consideration for both gravity and drag in the simultaneous ODE solution is given in a physically correct manner. Twenty years or so ago, when your reviewer was an undergraduate student, introductory physics courses tiptoed around problems requiring ODE solutions; after all, most of the students in these courses were concurrently taking their first courses in calculus. Kudos to the Millersville physics instructors if indeed they use NDSolve[] to introduce otherwise untouchable problems to their new students.

Unlike other recent Mathematica packages and notebooks reviewed in this space, Baseball1.nb worked fine "as is"; no debugging of the code was necessary. Unfortunately, Baseball1.nb is sorely in need of the support that documentation provides, and it is but one of many MathSource items plagued by this need. It may well be that, in submitting Baseball1.nb to meet his physics class requirements, Robert Lee concurrently submitted a thorough explanation of the equations (and especially their numerical coefficients) used in the baseball trajectory simulation. But with no such explanation available to users of this MathSource item, the users are left to derive these equations  (and their numerical coefficients) through their own means. If one sought to analyze the line drive Willie McCovey hit to Bobby Richardson thirty-eight years ago, s/he could not do so with the unaltered version of Baseball1.nb: The required units for the input speeds are not defined; an explanation for the included physical properties is absent; choices for "baseball game properties" such as bat mass and baseball coefficient of restitution are not only unexplained but also outside allowable Major League Baseball ranges. In fairness not only to Lee but also to the MathSource caretakers, the aforementioned non-Mathematica Web-based simulators were likewise free from equation explanation; still, two or more wrongs fail to make a right. As a repository and a showcase for Mathematica applications, MathSource has reached the point where mandatory minimal documentation for its items is in order, lest MathSource degenerate into a junkyard whose items need repair before they can be of use.

ABOUT THE AUTHOR

Born and reared several miles north of Busch Stadium in St. Louis, MO, where oppressive August heat and humidity continue unabated past sundown, Matthew M. Thomas attended his first and only baseball game in San Francisco's Candlestick Park on an August 1988 evening. Attired in a short-sleeved shirt, he realized he might be in trouble when he saw his fellow spectators arriving in jackets with blankets in tow. He endured the entire 13-inning game, thus earning a Croix de Candlestick; his arms have yet to thaw.

thomas@wuche2.wustl.edu

REFERENCES

ACHENBACH, ELMAR. "The Effects of Surface Roughness and Tunnel Blockage on the Flow Past Spheres", Journal of Fluid Mechanics, 65(1), 113-125 (1974).

ADAIR, ROBERT KEMP. The Physics of Baseball, 2nd ed. HarperPerennial, New York (1994).

BRIGGS, LYMAN J. "Effect of Spin and Speed on the Lateral Deflection (Curve) of a Baseball; and the Magnus Effect for Smooth Spheres", American Journal of Physics, 27(8), 589-596 (1959).

BUSH, VANNEVAR. 01 August 1965 letter to Harold E. Edgerton. Institute Archives and Special Collections, Hayden Library, Massachusetts Institute of Technology, Cambridge, MA (http://libraries.mit.edu/archives/exhibit-bush/bush.html as of November 2000).

DE MESTRE, NEVILLE. The Mathematics of Projectiles in Sport (Number 6 in the Australian Math. Society Lecture Series). Cambridge University Press, Cambridge (1990).

ERLICHSON, HERMAN. "Is a Baseball a Sand-Roughened Sphere?" American Journal of Physics, 53(6), 582-583 (1985) [contains response by Cliff Frohlich].

FROHLICH, CLIFF. "Aerodynamic Drag Crisis and its Possible Effect on the Flight of Baseballs", American Journal of Physics, 52(4), 325-334 (1984).

KIRKPATRICK, PAUL. "Batting the Ball", American Journal of Physics, 31(8, Part 1), 606-613 (1963).

NATHAN, ALAN M. "Dynamics of the Baseball-Bat Collision", American Journal of Physics, 68(11), 979-990 (2000).

NEW YORK TIMES, Vol. CXII (No. 38,252), 17 October 1962, pp. 1, 42-43: Articles by John Drebinger, Bill Becker, and Arthur Daley provide day-after accounts of 1962 World Series Game 7. Drebinger uses the adjective "tremendous" to describe both McCovey's seventh-inning triple ("into the teeth of the gale") and his ninth-inning foul ball down the right-field line. Daley writes that McCovey "drilled a massive drive to deepest center" for his triple. Becker reports that McCovey's series-ending line out came off a Ralph Terry fastball (implying at least a proportionally fast line drive). Drebinger states that McCovey "blasted" his final line drive, and Daley described it as a "wicked liner". These accounts can be contrasted with a World-Wide Web sketch of McCovey, available as of November 2000 at URL www.celebrate2000-ang.com/sportsfigures.asp?Sports=Willie_McCovey. In this latter account, which ranks McCovey 23rd in the Top 50 Bay Area Sports Figures of the Century, author Jeff Faraudo quotes Bobby Richardson as claiming that McCovey's final liner had "overspin -- [Mickey] Mantle used to hit balls like that", and reports that Richardson caught the ball chest-high (implying that perhaps the one-millimeter-lower point of contact suggested by Kirkpatrick [1963] might not have sufficed to score Matty Alou and Willie Mays).

WATTS, ROBERT G. AND BAHILL, A. TERRY. Keep Your Eye On The Ball: Curve Balls, Knuckleballs, and Fallacies of Baseball. W.H. Freeman and Company, New York (2000) [Sign errors plague the drag force components in the x-direction Newton's Second Law equations on pp. 160-161 of the first printing].

WATTS, ROBERT G. AND BARONI, STEVEN. "Baseball-Bat Collisions and the Resulting Trajectories of Spinning Balls", American Journal of Physics, 57(1), 40-45 (1989).

WATTS, ROBERT G. AND FERRER, RICARDO. "The Lateral Force on a Spinning Sphere: Aerodynamics of a Curveball", American Journal of Physics, 55(1), 40-44 (1987).