Calculate the trajectory of our cannon shell including both air drag and the reduced air density at high altitudes so that you can reproduce the results in the figure.Perform your calculation for different firing angles and determine the value of the angle that gives the maximum range.

Projectile motion

• Newton's second law in two spatial dimensions

  
  
  
  
  
  

(1)

Euler method for second-order ODE

• write each of these second-order equations as two firest-order differential equations

  
  
  
  
  
  
• finite difference form
  
  

Resistance

• the magnitude of the drag force is given by

  
  
  
  
  
  which
  

• the componets of the drag force

  
  
  
  
  
  

(2)

• Euler method with resistance
  
  
  
  

How to decide the point of fall?

• the last point above the ground(n) and the first point below the ground(n+1) is given by

  
  
  
  
  
  

Code(python)

• without air resistance

import numpy as np
import math
import matplotlib.pyplot as pl
class cannon_shell:
    def __init__(self,time_step=0.1,total_time=30,gravity=9.8,initial_dis=0):
        print("enter the angle of the cannon shell ->")
        self.rad=float(input())
        self.angle=(self.rad/180)*math.pi
        print("enter the speed of the cannon shell ->")
        self.velocity=input()
        self.g=gravity
        self.dt=time_step
        self.v_x=[self.velocity*(math.cos(self.angle))]
        self.v_y=[self.velocity*(math.sin(self.angle))]
        self.x=[initial_dis]
        self.y=[initial_dis]
    def run(self):
        while(self.y[-1]>=0):
            self.x.append(self.x[-1]+self.v_x[-1]*self.dt)
            self.y.append(self.y[-1]+self.v_y[-1]*self.dt)
            self.v_x.append(self.v_x[-1])
            self.v_y.append(self.v_y[-1]-self.g*self.dt)
    def show_result(self):
        pl.plot(self.x,self.y)
        pl.title('connon shell without air resistance')
        pl.xlabel('x/R')
        pl.ylabel('y/H')
        pl.ylim(0.0)
        pl.show()
a=cannon_shell()
a.run()
a.show_result()

• consider the resistance of air

import numpy as np
import math
import matplotlib.pyplot as pl
class cannon_shell:
    def __init__(self,time_step=0.1,resis_coeff=4e-5,total_time=30,gravity=9.8,initial_dis=0):
        print("enter the angle of the cannon shell ->")
        self.rad=float(input())
        self.angle=(self.rad/180)*math.pi
        print("enter the speed of the cannon shell ->")
        self.v=input()
        self.B2_m=resis_coeff
        self.g=gravity
        self.dt=time_step
        self.v_x=[self.v*(math.cos(self.angle))]
        self.v_y=[self.v*(math.sin(self.angle))]
        self.x=[initial_dis]
        self.y=[initial_dis]
    def run(self):
        while(self.y[-1]>=0):
            self.x.append(self.x[-1]+self.v_x[-1]*self.dt)
            self.y.append(self.y[-1]+self.v_y[-1]*self.dt)
            self.v_x.append(self.v_x[-1]-self.B2_m*self.v*self.v_x[-1]*self.dt)
            self.v_y.append(self.v_y[-1]-self.g*self.dt-\
            self.B2_m*self.v*self.v_y[-1]*self.dt)
    def show_result(self):
        pl.plot(self.x,self.y)
        pl.title('connon shell without air resistance')
        pl.xlabel('x/R')
        pl.ylabel('y/H')
        pl.ylim(0.0)
        pl.show()
a=cannon_shell()
a.run()
a.show_result()

(3)

Effects of altitude

• isothermal (constant temperature) ideal gas

• adiabatic approximation
  
  
  
  which
  

• the drag force due to air resistance is proportional to the density

  
  
  

consider the reduce of air density

import numpy as np
import math
import matplotlib.pyplot as pl
class cannon_shell:
    def __init__(self,time_step=0.1,resis_coeff=4e-5,total_time=30,gravity=9.8,initial_dis=0):
        print("enter the angle of the cannon shell ->")
        self.rad=float(input())
        self.angle=(self.rad/180)*math.pi
        self.v=700
        self.B2_m=resis_coeff
        self.g=gravity
        self.dt=time_step
        self.v_x=[self.v*(math.cos(self.angle))]
        self.v_y=[self.v*(math.sin(self.angle))]
        self.x=[initial_dis]
        self.y=[initial_dis]
    def run(self):
        while(self.y[-1]>=0):
            self.rho=(1-(2.257e-5)*self.y[-1])**2.5
            self.x.append(self.x[-1]+self.v_x[-1]*self.dt)
            self.y.append(self.y[-1]+self.v_y[-1]*self.dt)
            self.v_x.append(self.v_x[-1]-self.rho*self.B2_m*self.v*self.v_x[-1]*self.dt)
            self.v_y.append(self.v_y[-1]-self.g*self.dt-\
            self.rho*self.B2_m*self.v*self.v_y[-1]*self.dt)
        labeltext=str(self.rad)
        pl.plot(self.x,self.y,label=labeltext)
def show_result():
    pl.title('connon shell without air resistance')
    pl.xlabel('x/m')
    pl.ylabel('y/m')
    pl.ylim(0.0)
    pl.legend(loc='upper left')
    pl.show()
a=cannon_shell()
b=cannon_shell()
c=cannon_shell()
d=cannon_shell()
e=cannon_shell()
f=cannon_shell()
a.run()
b.run()
c.run()
d.run()
e.run()
f.run()
show_result()

show the result

(4)