第六次作业

作业

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name:姚逸飞
student number：2013301020096

homework

• $\Bbb homework\; L1:$ 2.9

• $\Bbb homework\; L2:$ developed 2.10（wind resistance is considered————“precision\ strike\ system”

• $\Bbb homework\;L3:$ develop“super precision strike system”

abstract

These problems will be solved in this article.
I will give the trajectory of cannon with Euler method in different step. And then consider the drag force ,picture the trajectory of the cannon,and compare the two diffrent situation.

passage

This article is mainly discuss about the trajectory of cannon.And for convenience,we can presume the cannon as a mass point.

• air resistance is ignored

we can get the equation:

$$\begin{cases} \frac{{\rm d}v_x}{{\rm d}t}=0 \\[2ex] \frac{{\rm d}v_y}{{\rm d}t}=-g \end{cases}$$

This equation is very simple to solve：

$$\begin{cases} v_{x}=v_{x0} \\[2ex] v_{y}=v_{y0}-gt \end{cases}$$
Then we caculate the answer with Euler method in different step ,we can achievepicture as follows:

Here we can see the difference bwtween various steps.When the step<0.1,we can find its hard to tell apart from the ture solution ,so when we sove the follow problems ,all of steps below 0.1.Thus we can get the accurate answers.

• air resistance is considerated[1] : $${\scr F}=-(1-\frac{ay}{T_0})^{\alpha}B_2v^2$$ Here ${\scr F}$ is the fraction force ;$a,T_0,\alpha,B_0$ are all coefficents:$\alpha=2.5;\;a=6.5\times10^3 K/m;\;T_0=300K;\;B_2/m=4\times10^{-5}m^{-1}$
  To solve this problem ,its convenient to use the class of the cannon.
  
  Here we can purely observe the cannon trajectory with diffrent initial angle $\theta$ ,the horizantal distance is first increase and then decrease as the angle increase.

Reference

Thanks for Chen yangyao's splendid homework.It helps me a lot.
From that ,I begin to know the class which is very useful to me .