Introduction
Pitcher and batter combat is the most exciting moment during the baseball game. As a result, pitcher can make good use of breaking ball to confuse batter and weaken batter threat. Each kind of breaking ball has its distinctive trajectory, which is mainly caused by the airflow during flight. After analyzing the motion of the baseball, we find that drag force, gravity and Magnus force affect the ball’s trajectory dramatically. Using classical mechanics, we can derive plenty of differential equations to describe the motion of a flying baseball. Using MATLAB we can solve those differential equations numerically by using 4th Runge-Kutta method.
Chien-Ming Wang, A famous pitcher in MLB from Taiwan. (Wikipedia)
Finally, applying different initial conditions (velocity, spin rate, rotation axis, etc.) we simulate various breaking balls successfully, such as slider, sinker, and curveball. Moreover, we also simulate the most dazzling breaking ball, the knuckleball, or known as butterfly ball. Knuckleball trajectory is a chaotic phenomenon, and our simulation shows a bizarre trajectory that isn’t observed in a normal breaking ball.
A great video for those who are not familiar with Knuckleballs
The following discussion is mainly based on the analyzes done by professor Richard Fitzpatrick [1]. Matlab code and numerical analysis are done personally, including some discussion. The breaking ball’s trajectory is displayed by Origin. Moreover, the fascinated animation was done by my friend using Unity.
Motion Analysis
In order to analyze the trajectory of the baseball, we consider the total force acting on the baseball between the journey from the pitcher to the catcher. We can decompose the total force into gravity, air drag, Magnus force, and lateral random force. The ratio of lateral random force is too small so that it only has an effect in low spin rate ball (low spin rate ball have extremely small Magnus force).
After the above consideration, we find air drag, Magnus force and gravity play an important role in baseball motion. Following we going to introduce air drag, Magnus force, and lateral random force respectively.
Air Drag
The direction of air drag is parallel to the direction of the ball and the amplitude depends on the velocity. Air drag can be described by the drag equation
Where C_d is the drag coefficient, and usually depends on the geometry of the object, and can be measured by experiment.
In order to describe the air drag acting on the baseball, we can fit the experimental data in Fig. 2 and derive to describe the amplitude of air drag (eq. 3).
Therefore the drag force in x,y,z component is
Magnus Force
The Magnus effect is an observable phenomenon that is commonly associated with a spinning object moving through the air or another fluid. The force can be described as
Random Lateral Force
During a low spin rate (Knuckleball), Magnus force can be neglect and the system is dominated by lateral force. Lateral force act on the baseball is caused by the asymmetry of the baseball since the direction of the stream is distinct in different angle. The lateral force act on the baseball was measured in the wind tunnel (Fig. 3).
Experiment data shows a periodic relationship between the lateral force and the angle.
Fitting the data, lateral force can be approximate as Eq. 2.
Finally, we can combine each component together. And derive the equation of motion of the breaking ball including the initial condition.
Different types of breaking balls with different initial conditions listed in Table 1.
Those ODE equations can be solved by MATLAB ODE45 function numerically.
(Use both files together in MATLAB, to display the baseball trajectory. Different initial conditions can be changed as well.)
Modeling results
- 3-Dimensional with the strike zone and different types of breaking ball
- Pitcher v.s Batter
Using the parameter in Table 1 and MATLAB to solve the ODE equation numerically, we can obtain (x, y, z, t) data. Plotting the data in 3D space, we can simulate the trajectory of different breaking balls.
Besides, observing the trajectory at the x-z plane, we can find vertical shift isn’t obvious initially between fastball and changeup ( Δt~0.1). Therefore, the batter can be easily confused by the breaking ball and cause swing and the miss.
- Different types of breaking balls are shown beneath with the catcher view.
Each chart includes the parameter used in the simulation.
- The trajectory of the knuckleball is abnormal, which seems like a flying butterfly.
Animation demo using Unity in batter vision with real-time!
Try if you can distinguish a strike or ball 😆, therefore MLB player is really a great batter!
Fastball, Changeup, Curveball, Cutter, Sinker, Slider, Knuckleball
Reference
1. http://farside.ph.utexas.edu/teaching/329/lectures/lectures.html , Richard Fitzpatrick
The Magnus force, Simulations of baseball pitches, The knuckleball
2. Aerodynamics of a knuckleball, Robert G. Watts and Eric Sawyer, American Journal of Physics 43, 960 (1975)
3.The physics of baseball, R.K. Adair (Harper & Row, New York NY, 1990).