The seventh assignment

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物基一班
陈天懿 2013301020146

Abstract

This article discusses the influences casued by different drag coefficient and spins, proves that the motion of the baseball will be hard to predict if included some extent of spin.

Background

This article mainly covers the motion of baseballs being hit, to study the movement of the ball, we must include two factors' impacts: one of them is the spin of baseball, it has the dicisive influence on the trajectory of the ball. The other factor is the texture of the ball, which decide whether it is smooth or rough and its influence on ball's drag coefficient. So called "knuckeball" is related to the second factor.

Main

Different drag coefficient of different textures of balls

The function of drag coefficient respect to the ball's velocity is drawn as follows. We used some approximation to make the question easier but still maintain the important part, such as the montonicity and the turning points.

Then we can construct the motion trajectories by the method used in 6th assignment. And we get the three kinds of ball's trajectories:

the initial conditions are $v_0=30.56m/s$, $\theta_0=45^\circ$ $h_0=1m$
We can't get too much information from the picture above, for the texture has a very complex relation with the velocity.

Motion trajectories after including different spin speeds

The spin of balls will make the air exerts a force on the balls, which is called magnus force in fluid mechanics. Theoretically we can derive the formula of magnus forece as below:

$$\vec{F_M}\propto (v+r\omega)^2-(v-r\omega)^2\sim vr\omega\\ \vec{F_M}=S_0\vec{\omega}\times\vec{v}$$
Decompose the equation to components:
$$\begin{cases} \frac{dv_x}{dt}=-Dvv_x+\frac{S_0}{m}(\omega_yv_z-\omega_zv_y)\\ \frac{dv_y}{dt}=-g-Dvv_y+\frac{S_0}{m}(\omega_zv_x-\omega_xv_z)\\ \frac{dv_z}{dt}=-Dvv_z+\frac{S_0}{m}(\omega_xv_y-\omega_yv_x)\\ \frac{dx}{dt}=v_x\\ \frac{dy}{dt}=v_y\\ \frac{dz}{dt}=v_z\\ \end{cases}$$
Where $D$ is the drag coefficient and $\frac{S_0}{m}$ is a constant.

The influence casued by spin

For simplicity, we consider a ball with normal texture, speed of wind is set to zero, the hitting angleis $\theta_0=45\circ$, and the initial speed is still $v_0=30.56m/s$.and the spin axe is set verticle to the ground, that is to say $\vec{omega_0}$ is verticle

And we draw another picture with $\vec{omega_0}$ is set horizontal

The resualt is quite clear that once we add the spin to the ball, its trajectory will become very chaos and unpredictive.
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Conclusion

The texture and spin both affect the trajectory of a baseball, and the spin will make it much more complicate and unpredictive.

Reference

Computational Physics-Nicholas J. Giordano, Hisao Nakanishi