Chapter 2 Problem 19 Level 1 & Level 2

计算物理

---

Name： 陈锋
Student Number： 2013301020145
April 9, 2016

Abstract

This article discusses the trajectory of a batted baseball, considering topspin, no spin and backspin. Also, some other factors are considered, such as drag coefficient of various balls, the batting angle, the effect of wind and so on.

Introduction

As we all known, due to the spin, there will be a pressure difference at the edge of the ball, which called Magnus force. In the case of backspin, this force will offer a upward force, causing a longer flying distance than the no-spin situation. This increase the probability to get a home run.

Motion of Equations

The Magnus force is equal to the force differnce between the opposite edge of the ball:

$$\begin{equation} F_M \propto (v+r\omega)^2-(v-r\omega)^2 \sim vr\omega \end{equation}$$
where $\omega$ is the angular velocity of the ball, and $v$ is the flying velocity.
Considering the direction of Magnus force, the general vector form is:
$$\begin{equation} \overrightarrow{F}_M = S_0\overrightarrow{\omega}\times\overrightarrow{v}=S_0(\omega_yv_z-\omega_zv_y)\overrightarrow{i}+S_0(\omega_zv_x-\omega_xv_z)\overrightarrow{j}+S_0(\omega_xv_y-\omega_yv_x)\overrightarrow{k} \end{equation}$$
Therefore, the motion of equations become:
$$\begin{equation} \frac{dx}{dt}= v_x \\ \frac{dv_x}{dt} = -\frac{B_2}{m}vv_x + \frac{S_0}{m}(\omega_yv_z-\omega_zv_y) \\ \frac{dy}{dt}= v_y \\ \frac{dv_y}{dt} = -g -\frac{B_2}{m}vv_y + \frac{S_0}{m}(\omega_zv_x-\omega_xv_z) \\ \frac{dz}{dt}= v_z \\ \frac{dv_z}{dt} = -\frac{B_2}{m}vv_z + \frac{S_0}{m}(\omega_xv_y-\omega_yv_x) \end{equation}$$
where $B_2/m$ is given in the next section and $S_0/m = 0.00041$

Drag Coefficient of different kinds of ball

Textbook showed the drag coefficient curve of different balls at page 33 (Figure 2.6) and gave an approximate expression of drag coefficient of normal baseball (Equation 2.26). If we also want to consider the flying trajectroy of smooth ball and rough ball quantitatively, constructing their approximate expression is of necessity. Here, I give the expressions of them. Later, I will use these expressions to simulate the different situation of various kinds of ball.

Rough ball

$$\begin{equation} B_2/m=\left( 0.0013+\frac{0.0084}{1+e^{\frac{\left( x-21 \right)}{2.5}}} \right)\left( -\frac{3-e^{0.5\left( x-39 \right)}}{1+e^{0.5\left( x-39 \right)}}+4 \right) \end{equation}$$

Normal baseball

$$\begin{equation} B_2/m=0.0039+\frac{0.0058}{1+e^{\frac{\left( x-35 \right)}{5}}} \end{equation}$$

Smooth ball

$$\begin{equation} B_2/m=0.0013+\frac{0.0084}{1+e^{\frac{\left( x-65 \right)}{4}}} \end{equation}$$

Figure.1: Variation of $B_2/m$, which has a similar tendency of the Figure 2.6 in the textbook.

The tarjectories difference of various ball

Figure 2: Rough ball reach the longest distance and smooth one reach the shorest. This is the reason why making the baseball become rougher is forbidden.

Data Analysis

In general we choose the spinning axis parallelled to the z-direction, which means $\omega_z = \omega, \omega_x=\omega_y= 0$. Also, I assume that the angular velocity is a constant during this process.

Difference angular velocity

Initial velocity 100 mph; Angle: 40°

Figure 3: Calculated the trajectories of difference anguler velocity. Negative angular velocity means topspin and the positive means backspin. We can see that topspin makes the shorter height and shorter distance, while backspin makes an opposite reasult. When the angular velocity exceeds several hundrud rps, the trajectory will become very strange.

Difference batting angle

Initial velocity 100 mph; Anglar velocity: 40 rps

Figure 4: Calculated the trajectories of difference angles. In this case, batting the baseball at an angle about 35 degree can get the longest distance. Actually, this angle is related to the direction and magnitude of the angular velocity of spin.

Changing the spinning axis

Now, I consider the spinning axis at arbitary directions and I use $(\omega_x,\omega_y,\omega_z)$ to represent a angular velocity vector, where x is the horizontal direction and y is the vertical direction and z is the z direction.
Initial velocity 100 mph (${vz}_0$ = 0); Angle: 40°

Figure 5: Only via changing the direction of spin of the ball can we change the flying direction of the ball, which can cause difficulties for the opponents.

The influence of wind

When we consider the situation of wind, the expression of the drag force will be changed into:

$$\begin{equation} F_{drag,x} = -B_2|\overrightarrow{v}-\overrightarrow{v}_{wind}|(v_x - v_{wind,x}) \\ F_{drag,y} = -B_2|\overrightarrow{v}-\overrightarrow{v}_{wind}|(v_y - v_{wind,y}) \end{equation}$$
The results are as follows:
Initial velocity: 100mph; Angle: 40°; Angular velocity of spin: 40 rps; speed of wind: 10 mph.

Figure 6: Blue curves mean tailwind, green ones mean no wind and red ones mean headwind. As the figure showed above, the reasults are what we expect.

Argparse

Using argparse can increase the interaction of the programme, just like using a system application of Linux. Here are the results when adding argparse module into the previous programme.

Figure 7: I set four parameters for this programme, including the initial velocity, the batting angle, the angular velocity of spin($\omega_z$) and the height of batting position. There are also some optional arguments, six of which are about the transfermation of units, for diverse people is accustomed to different units. Also, '-s' is to saving all the datum into a .txt file and '-p' can help the users have a direct preview of their results.

Figure 8: Setting the initial velocity 30m/s, the initial angle 35 degree, the spin angular velocity 33 rps and height 1 meter. And '-p 2d' means plot the datum into a 2-dimensional space.

Level 2: Vpython

Setting the initial velocity: 100 mph; Angle: 40 degree; angular velocity: $(20,-20,17)$

Figure 9: Actually, the displacement of the baseball at the z-direction is not as large as the figure showed. It is much smaller than its horizontal distance, just as the results in Figure 5. However, in order to present its motion at the z-direction, I make a ten-fold increase in the velocity of that direction.
Changing the angular velocity of spin into $(20, -20, 300)$, we can get a strang motion trajectory.

Reference

1.Rod Cross, Physics of baseball & softball, Springer, 2011
2.Giodano, N.J., Nakanishi, H. Computational Physics. Tsinghua University Press, December 2007.
3.VPython Help, http://vpython.org/contents/docs/arrow.html

Programme Code

• Chapter2 Problem 19
• Chapter2 Visual baseball