Exercise 6: The cannon 2

python homework

Abstruct

In this exercise, we will discuss the tajectories of cannon further.We also consider the drag force and altitude factor. The problem is described by ordinary differential equations in which initial values are given and can be solved using Euler method.

Content

L1:

• Problem 2.10

  Generalize the program developed for the previous problem so that it can deal with sistuations in which the target is at a defferent altitude than the cannon. Consider cases in which the target is higher and lower than the cannon. Also investigate how the minimum firing velocity required to hit the target varies as the altitude of target is varied.

• Numerial approach

  We know that the pressure p depends on altitude from thermology according to:
  

  $$p(y)=p(0)e^\frac{-mgy}{k_BT}\tag{1}$$
  m is the average mass of an air, y is altitude. For an ideal gas the pressure is propotional to the density, so this leads to
  $$\rho=\rho_0exp(-y/y_0)\tag{2}$$
  $y_0=k_B/mg≈1.0×10^4m$ $\rho_0$is density at sea level(y=0).
  Assume that the air is a poor conductor of heat, and that convection is very slow. We consider it as an adiabatic model. This approximation leads to dependence of the density on altitude:
  $$ρ=ρ_0(1-\frac{ay}{T_0})^\alpha\tag{3}$$
  So, the formula of the drag force due to air resistance can be rewriten as:
  $$F'_{drag}=\frac{\rho}{\rho_0}F_{drag}=-\frac{\rho}{\rho_0}B_2v^2\tag{2}$$
  Use Euler method, the ordinary differential equations can be writen as
  $$x_{i+1}=x_i+v_{x,i}△t$$
  $$v_{x,i+1}=v_{x,i}-\frac{\rho B_2vv_{x,i}}{m}△t$$
  $$y_{i+1}=y_i+v_{y,i}△t$$
  $$v_{y,i+1}=v_{y,i}-g△t-\frac{\rho B_2vv_{y,i}}{m}△t$$

• Realization by python
  There are many variations in problem 2.10. So, I choose a spercific firing angle. If the target's altitude is at the top of the cannon shell trajectory, the firing velocity of cannon is minimum. I draw the figure is the trajectory of cannon which firing velocity is at the range of 100~700m/s and the firing angle is 45°.
  Result：
  
  And here is the data:
  
  Codes
  Click "codes" to see the python code of this program.

L2

• Problem
  Design a program that make the cannon shell can hit the target presicley.
  Unfinished.