_{1}

^{*}

We consider the impact of drag force and the Magnus effect on the motion of a baseball. Quantitatively we show how the speed-dependent drag coefficient alters the trajectory of the ball. For the Magnus effect we envision a scenario where the rotation of the ball confines the Magnus force to the vertical plane; gravity, drag force and the Magnus force make a trio-planar system. We investigate the interplay of these forces on the trajectories.

In introductory physics, engineering and math courses, undergraduate students are traditionally introduced to the concept of projectile motion such that a projectile is thrown at an angle in a vertical plane in a vacuum. The motion in the vacuum is analyzed because in the absence of air, utilizing the Newton’s second law yields to a set of two independent second-order linear ODEs. The trivial solution of these equations provides information about the kinematics of the projectile, such as the trajectory, range, time of flight, etc. [

The motivation of this article stems from the former question, namely “At what angle above the horizontal a baseball should be batted making the range a maximum?” To answer this question, we analyze the problem systematically. In the course of our analysis, we stumbled upon a few interesting related issues. To the author’s amazement, the analysis led also to uncharted territories unveiling features of a flying baseball that to date have not been reported in scientific literature. In this article we confirm the previously reported quantitative results and extend the investigation for the “what-if” scenarios.

We consider a ball projected in the air at an arbitrary initial angle above the horizontal. In addition to gravity the ball encounters air resistance. Irrespective of its speed, the drag force (the air resistance) acts in the opposite direction of the motion retarding its movement. In _{D}, respectively. In practice a batted ball also spins; it may spin backward (backspin) or forwards (topspin). We quantify the spin by its angular velocity ω. We consider a spinning ball with angular velocity vector perpendicular to the vertical plane. Hence, a back-spin ball orients its ω parallel to the ground and outward to the vertical plane. Conversely, a top-spun ball orients its ω parallel and inward to the ground. _{D} and F_{M} a trio coplanar. For the given scenario the ball stays in the vertical plane, and therefore, the analysis of the problem becomes two-dimensional.

Here we consider the characteristics of the last two forces. For speeds relevant to the MLB (Main League

Baseball) the airflow around a baseball is considered as laminar [

cording to

posite direction of motion. The parameters in this equation are C_{D}, ρ, A, the drag coefficient, the density of the media (air) and the cross sectional area of the ball i.e. πR^{2}, where R is the radius of the ball. Utilizing a Computer Algebra System (CAS) recently this equation has been applied to a “real-life” situation. However, assuming a speed independent coefficient the impact of the drag force has been trivialized leading to inaccurate conclusions [

ter. In the course of our search we encounter

both in m/s units. Noting in SI, _{D}; this yields

ing, for slow speeds, C_{D} approaches ~0.52 i.e.

a baseball in SI units, namely _{D} varies vs. the practical range of speed.

According to _{D}’s for the first and the last range are constants and are as high as 0.52 and as low as 0.22, respectively. The middle range is the transition range where the C_{D} varies from the high value of 0.52 to the low value of 0.22. Consequently, a baseball batted at a typical MLB speed ~88 MPH (=37 m/s) encounters a weak drag force at the beginning, and while traversing along its trajectory continuously slows down encountering a relatively strong drag force. In short, the impact of the drag force is not constant. One intuitively expects the impact of the drag force at the beginning to be less than at the end of the flight. In Section 3, quantitatively we validate our intuition.

The other force is the Magnus force, for a baseball it is parametrized as

the ball is standing still and the air passes it by with the velocity v from right to left; see

depicted in ^{−}^{4}

[

As we explained in the previous section a batted back-spun baseball with its initial angular velocity vector parallel to the horizontal flies and stays in a vertical plane. A vertical plane is the one that contains the initial velocity vector. In other words, the motion of the ball occurs in a 2D space. One such plane with a Cartesian xz coordinate is shown in

Note the last term of Equations (1), (2) are only operative for a spinning baseball. In other words, the equations associated with a non-spinning baseball are simpler than those given by Equations (1), (2). For a back-spun ball the horizontal components of the drag and the Magnus force are constructively additive, while along the vertical direction they are destructive. Equations (1), (2) describe the motion of the baseball. These are highly coupled non-linear ODEs. The solution of these equations are conducive to a set of explicit time dependent coordinates, namely

where ^{−}^{1} while ^{−}^{1}; all four parameters are constant. It is obvious that a set of such equations may not have symbolic solutions! Despite of our insistence we were unable to convert these equations to solvable entities. We then apply Mathematica symbolic DSolve command, but it also fails to produce analytic solutions. As a last resource we attempt solving them numerically. We introduce a set of meaningful, practical initial conditions and utilize Mathematica’s numeric routine, namely, NDSolve. This simple seven letter word successfully produces an output. Utilizing

The initial condition is comprised of the specifications of the initial velocity; on the vertical xz-plane this is

The initial angles are discrete, i.e. they are 10˚ apart and so are their corresponding ranges. Utilizing Interpolation command we fill in the gap and generate the needed continuous coordinates. With these continuous coordinates on hand we display the discrete solutions along with the interpolated continuous spectrum. These are shown in

With these plots it is easy to draw a few conclusions. For instance, the first and the last plots depict the two extreme scenarios. The first graph corresponds to the vacuum, and the last graph shows the impact of the combined drag and the Magnus effects. The difference between these two scenarios is quite pronounced. Their maximum range not only occurs at two different angles, but their numeric values are almost off by a factor of two. Similarly, one may also compare the second dot to the third plot. Both figures describe the impact of the media; the second plot is with the pure Magnus effect, while the third plot shows the impact of the pure drag force. Although these scenarios do not represent real-life situations they illustrate the impact of the individual forces.

Furthermore, with the interpolated continuous function at hand we search for the initial angle that maximizes the range. The output of the search is tabulated in

The first column of

Utilizing the corresponding time of fight in

These figures are self-explanatory. The headings are the same as in

Finally, utilizing the solutions of Equations (1), (2), namely

"max range, m-->" | 139.694 | 162.579 | 78.2832 | 84.095 |

θ^{ο} "-->" | θ → 45 | θ → 38.3973 | θ → 39.4280 | θ → 35.1726 |

Trajectories depicted in

We evaluate the maximum height, the results are shown pictorially in

We continue exploring some more properties of a batted baseball, namely the arc length of a projectile. The length of the projectile comes about from the integration of the arc length element along the trajectory;

Having these plots on hand, one may now ask at least two questions. The first questions is “At what angle a ball should be batted to gain the maximum arc length?”. The results are tabulated in

According to

The second question stems from a careful analysis of the last plot of

Accordingly, this trio angles are {32.51˚, 82.35˚, 87.12˚}. One needs to keep in mind that these angles are specific to the initial speed 37 m/s. In other words, these angles are speed-dependent.

The author would like to comment that although since Newton’s era the projectile motion has been visited numerous times no such observation has been reported in scientific literature!

Scientific literature is flooded with information about the physics of the baseball. A thorough review of these resources reveals that there is no similarity between the outputs of the presented investigation vs. the historic body of knowledge. In this article we analyzed the impact of the drag force and the Magnus effect on the character of a projectile. Specifically we considered a baseball with a MLB character and included the impact of the speed-dependent drag coefficient. Depending on the choice of factors, four different scenarios were considered. Because of the complexity of the cases we heavily relied on the numeric solutions of the ODEs describing the motion. Results were interpreted and conclusions were reported for individual scenarios. For a comprehensive understanding, the results of each section are depicted graphically. In the last section we reported a new discovered result, namely a set of trio-angles conducive to an equal trajectory arc length.

"max length, m-->" | 167.548 | 184.428 | 100.646 | 103.567 |

θ^{ο} "-->" | θ → 56.4556 | θ → 45.744 | θ → 62.166 | θ → 53.8664 |

A version of this investigation was presented at the 11th International Conference on Computer Science and its Applications, Santander/Spain, 2011.