The Baseball model: Trace the ball

郭帅斐

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The summary:

Baseball model is just an upgraded version of cannon shell model, with Magnus Force be considered. We alomst can use the program in the cannon shell problem. That is the Magnus Force make the baseball more complicated but more interesting.

the psudocode:

We consider the condition that the ball move the the $x-y$ plane, and we have the differential equation:

$$\frac{dx}{dt}=v_x$$
$$\frac{dv_x}{dt}=-\frac{B_2}{m}vv_x$$
$$\frac{dy}{dt}=v_y$$
$$\frac{dv_y}{dt}=-g$$
$$\frac{dz}{dt}=v_z$$
$$\frac{dv_z}{dt}=-\frac{S_0v_x\omega}{m}$$
According to the equation, we can write a program, and calculate all the properties of the baseball, including the trajectories.

The result:

• First, a series of trajectories of the baseball in 2-dimentional were plotted:
  
  We can find that with different angular speed $w$ , the trajectory is very different! The magnus force is important to the motion of the baseball.
• Second, I plot a lot of 3_D figure with different condition and compare it with the ideal situation.
  When the $w$ is small, the two trajectory vary slightly:
  
  With a angular speed in $y$ direction, the position move a little along the $y$ axis.
  
  With the angular speed both in $y$ and $z$ axis, the trajectory is different from both.
  
  When the $w_z$ become very large, the trajectory becomes very different!
  
  When the angular speed have components in three directions,the trajectory is as follows:
  
  When the $w_y$ and $w_z$ become very big, the trajectory ia like this:
  

The conclusion:

In this program, a lot of 2D or 3D trajectories a plotted. We should remenber that the program can not be exactly consistent with the real life. But we can see the physics under the model.