The motion of a baseball

程熹 2013301020038 物基一班
Homework

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Abstract

This article investigate how the motion of a baseball is influenced by air drag, spin, its texture and the initial velocity and the initial angle. By changing different initial velocity, we can draw different trajectory of the baseball's motion.

Key Word

Baseball motion

Introduction

The baseball features a rubber or cork center, wrapped in yarn, and covered, in the words of the Official Baseball Rules "with two strips of white horsehide or cowhide, tightly stitched together." Some are wrapped in a plastic-like covering.
A significant characteristic of the baseball is the stitching that holds together the covering of the ball. After a ball has been pitched, these raised stitches act like wings on a plane, catching the wind and causing the ball to swerve slightly on its way to the catcher. Whether the ball swerves to the right, to the left, downward, or a combination thereof, and whether it swerves sharply or gradually, depends on which direction, and how fast, the stitches have been made to spin by the pitcher. See, for example, curveball, slider, two-seam fastball, four-seam fastball, sinker, cutte

Main Features

In this article, we use the Euler method to translate the differential equation of the motion of baseball into equations suitable to design a program.

$$\begin{cases} \frac{dv_x}{dt}=-Dvv_x+\frac{S_0}{m}(\omega_yv_z-\omega_zv_y)\\ \frac{dv_y}{dt}=-g-Dvv_y+\frac{S_0}{m}(\omega_zv_x-\omega_xv_z)\\ \frac{dv_z}{dt}=-Dvv_z+\frac{S_0}{m}(\omega_xv_y-\omega_yv_x)\\ \frac{dx}{dt}=v_x\\ \frac{dy}{dt}=v_y\\ \frac{dz}{dt}=v_z\\ \end{cases}$$

Result

1.Different texture balls in the initial velocity=30.56m/s,initial angle=45 and at the height of 1m

2.Different drag coefficient of different textures of balls

3. Normal ball with the initial velocity=30.56 and the initial angle=45

Conclusion

1. From the diagram 2 we can find that the three types endure the same air drag force at a low speed, and then,with the increasement of the speed, the rough ball have the least air drag force among the three types of balls. However,when the speed of the ball continue to enlarge, the air drag that all three types of ball endure don't grow, with the rooth ball enduce the least air drag force.
2. At he initial speed of 30.56m/s, a rough ball can travel the longest among these three types of balls.

Acknowledgement

I have used the origin code of 陈天懿，and run his program to get some of the upper result. So thanks to him here.

Reference

Computational Physics(second edition) Nicholas J.Giordano and Hisao Nakanishi
Wikipedia Baseball
https://en.wikipedia.org/wiki/Baseball_(ball)