第七次作业

作业

name:姚逸飞
student number：2013301020096

Homework

• $\Bbb homework\; L1:$ 2.19
• $\Bbb homework\; L1:$ use vpython to visual the trajectory of cannon or baseball.

Abstract

This passage is mainly discuss about the trajectory of a batted baseball. As we all know ,baseball is an extremely popular sport in some conutries(it's regretful that it's not so popular in China).
This passage is going to discuss about the trajectory of baseball. To a certain extent,baseball is likecannon. But baseball is in a high rotating velocity. So the trajectory of baseball should be pictured in a 3-D coordinate system.

Passage

Here we will discuss about the high velocity roatating baseball. From reference, the baseball is driven by Magnus force :

$$F_M=S_0\vec{v}\times\vec{\omega}$$ Here $S_0$ can be seen as an constant in this passage.
Of course , the gravity and friction force should also be considered. As all the above ,we can get the equation of the baseball: $$\frac{{\rm d^2}\vec{r}}{{\rm d}t}=\vec{g}+\frac{B}{m}v^2\hat{v}+\frac{S_0}{m}\vec{v}\times\vec{\omega}$$ it's easy to give the solution :
$$\begin{cases} \frac{{\rm d}x}{{\rm d}t}=v_x\\ \frac{{\rm d}y}{{\rm d}t}=v_y\\ \frac{{\rm d}z}{{\rm d}t}=v_z \end{cases}$$
$$\begin{cases} \frac{{\rm d}v_x}{{\rm d}t}=-g-\frac{B_2}{m}v_x^2\hat{v}_x+\frac{S_0}{m}(\omega_yv_z-\omega_zv_y) \\ \frac{{\rm d}v_y}{{\rm d}t}=-g-\frac{B_2}{m}v_y^2\hat{v}_y+\frac{S_0}{m}(\omega_zv_x-\omega_xv_z)\\ \frac{{\rm d}v_z}{{\rm d}t}=-g-\frac{B_2}{m}v_z^2\hat{v}_z+\frac{S_0}{m}(\omega_xv_y-\omega_yv_x) \end{cases}$$

• usually and firstly, we try to use Euler method to find the influence of diffrent steps on the trajectory of baseball.

• Then we try to find the influence of the drag forece:

• From the picture ,we can see the drag force has deep effect on the trajectory of baseball owing to the light mass. So we can see that's different from the cannon.
  Then we can picture the trajectory of the baseball in the 3-D coordinate system:
  
  
  

Conclusion

Because from now on , I can not use the Vpython sufficiently, it's regret to not use it to picture the trajectory of baseball vividly. And I haven't caculate the error yet. If time allows, I'll try my best to complete this passage.

Reference

• Thanks for the Cmd Markdown: https://www.zybuluo.com/codeep/note/163962#9