Exercise_05 :Exercise 2.6 & 2.8 from Book

Cannon Shell Trajectories

冉峰 弘毅班 2014301020064

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Abstract

• There are so many different factors that may have effects on the movement of a cannon shell. So that I will try to calculate the different cannon shell trajectories by different factors.

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Background

From the Computational Physics we know the method of Euler.Now I want to take the Euler method into the projectile movement calculation:
Newton’s second law in two spatial dimensions :

$$\frac{dx^2}{dt^2}=0\qquad\frac{dy^2}{dt^2}=-g$$
Write each of these second-order equations as two firest-order differential equations:
$$\frac{dx}{dt}=v_x\qquad\frac{dv_x}{dt}=0$$
$$\frac{dx}{dt}=v_y\qquad\frac{dv_x}{dt}=-g$$
Finite difference form:
$$x_{i+1}=x_i+v_{x,i}\Delta t\qquad v_{x,i+1}=v_{x,i}$$
$$y_{i+1}=y_i+v_{y,i}\Delta t\qquad v_{y,i+1}=v_{y,i}-g\Delta t$$

There are some different situations

• Ideal situation
  $$x_{i+1}=x_i+v_{x,i}\Delta t\qquad v_{x,i+1}=v_{x,i}$$
  $$y_{i+1}=y_i+v_{y,i}\Delta t\qquad v_{y,i+1}=v_{y,i}-g\Delta t$$
• With air resistance
  $$x_{i+1}=x_i+v_{x,i}\Delta t\qquad v_{x,i+1}=v_{x,i}-\frac{B_2vv_{x,i}}{m}\Delta t$$
  $$y_{i+1}=y_i+v_{y,i}\Delta t\qquad v_{y,i+1}=v_{y,i}-g\Delta t-\frac{B_2vv_{y,i}}{m}\Delta t$$
• The g is not a constant
  $$g=\frac{GM_e}{(R_e+y)^2}$$
  where $GM_e=4.03\times10^{13}$, $R_e=6.37\times10^6$
  $$x_{i+1}=x_i+v_{x,i}\Delta t\qquad v_{x,i+1}=v_{x,i}-\frac{B_2vv_{x,i}}{m}\Delta t$$
  $$y_{i+1}=y_i+v_{y,i}\Delta t\qquad v_{y,i+1}=v_{y,i}-g\Delta t-\frac{B_2vv_{y,i}}{m}\Delta t$$

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The Main Body

Code for ideal situation

• code

import pylab as pl
import math
class trajectories:
    def __init__(self,mass=10, time_step=0.1,total_time=13,\
    initial_velocity_x=50,initial_velocity_y=50,
    initial_x=0,initial_y=0,B2=0):
        self.v_x = [initial_velocity_x]
        self.v_y = [initial_velocity_y]
        self.v=[math.sqrt(pow(initial_velocity_x,2)+pow(initial_velocity_y,2))]
        self.x=[initial_x]
        self.y=[initial_y]
        self.t = [0]
        self.m = mass
        self.dt = time_step
        self.time = total_time   
        self.B2=B2
        self.initial_velocity_x=initial_velocity_x
        self.initial_velocity_y=initial_velocity_y
    def run(self):
        _time = 0
        while(_time < self.time):
            self.v.append(math.sqrt(pow(self.v_x[-1],2)+pow(self.v_y[-1],2)))
            self.v_x.append(self.v_x[-1]-(self.B2/self.m)\
            *self.v[-1]*self.dt*self.v_x[-1])
            self.v_y.append(self.v_y[-1]-10*self.dt-\
            (self.B2/self.m)*self.v[-1]*self.dt*self.v_y[-1])
            self.x.append(self.v_x[-1]*self.dt+self.x[-1])
            self.y.append(self.v_y[-1]*self.dt+self.y[-1])
            self.t.append(_time)
            _time += self.dt    
    def show_results(self):
        font = {'family': 'serif',
                'color':  'darkred',
                'weight': 'normal',
                'size': 16,}
        pl.plot(self.x, self.y,label='angle is %f'%(math.atan(self.\
        initial_velocity_y/self.initial_velocity_x)*180/math.pi))
        pl.title('Cannon shell trajectories', fontdict = font)
        pl.xlabel('x ($km$)')
        pl.ylabel('y ($km$)')
        pl.legend()
        pl.show()
a = trajectories()
a.run()
a.show_results()

Result

• result for different angles
  

Code for situation with air resistance

• code

import pylab as pl
import math
class trajectories:
    def __init__(self,mass=10, time_step=0.1,total_time=13,\
    initial_velocity_x=50,initial_velocity_y=50,
    initial_x=0,initial_y=0,B2=0.01):
        self.v_x = [initial_velocity_x]
        self.v_y = [initial_velocity_y]
        self.v=[math.sqrt(pow(initial_velocity_x,2)+pow(initial_velocity_y,2))]
        self.x=[initial_x]
        self.y=[initial_y]
        self.t = [0]
        self.m = mass
        self.dt = time_step
        self.time = total_time   
        self.B2=B2
        self.initial_velocity_x=initial_velocity_x
        self.initial_velocity_y=initial_velocity_y
    def run(self):
        _time = 0
        while(_time < self.time):
            self.v.append(math.sqrt(pow(self.v_x[-1],2)+pow(self.v_y[-1],2)))
            self.v_x.append(self.v_x[-1]-(self.B2/self.m)\
            *self.v[-1]*self.dt*self.v_x[-1])
            self.v_y.append(self.v_y[-1]-10*self.dt-\
            (self.B2/self.m)*self.v[-1]*self.dt*self.v_y[-1])
            self.x.append(self.v_x[-1]*self.dt+self.x[-1])
            self.y.append(self.v_y[-1]*self.dt+self.y[-1])
            self.t.append(_time)
            _time += self.dt    
    def show_results(self):
        font = {'family': 'serif',
                'color':  'darkred',
                'weight': 'normal',
                'size': 16,}
        pl.plot(self.x, self.y,label='angle is %f'%(math.atan(self.\
        initial_velocity_y/self.initial_velocity_x)*180/math.pi))
        pl.title('Cannon shell trajectories', fontdict = font)
        pl.xlabel('x ($km$)')
        pl.ylabel('y ($km$)')
        pl.legend()
        pl.show()
a = trajectories()
a.run()
a.show_results()

Result

• result for different angles
• result compare with ideal situation
  
  

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Code for situation $g$ is not a constant

• code

import pylab as pl
import math
class trajectories:
    def __init__(self,mass=10, time_step=0.1,total_time=20,\
    initial_velocity_x=150,initial_velocity_y=150,
    initial_x=0,initial_y=0,B2=0.01):
        self.v_x = [initial_velocity_x]
        self.v_y = [initial_velocity_y]
        self.v=[math.sqrt(pow(initial_velocity_x,2)+pow(initial_velocity_y,2))]
        self.x=[initial_x]
        self.y=[initial_y]
        self.t = [0]
        self.m = mass
        self.dt = time_step
        self.time = total_time   
        self.B2=B2
        self.initial_velocity_x=initial_velocity_x
        self.initial_velocity_y=initial_velocity_y
    def run(self):
        _time = 0
        while(_time < self.time):
            self.v.append(math.sqrt(pow(self.v_x[-1],2)+pow(self.v_y[-1],2)))
            self.v_x.append(self.v_x[-1]-(self.B2/self.m)\
            *self.v[-1]*self.dt*self.v_x[-1])
            self.v_y.append(self.v_y[-1]-(4.03*pow(10,14)\
            /pow((self.y[-1]+6.371*pow(10,6)),2))*self.dt-\
            (self.B2/self.m)*self.v[-1]*self.dt*self.v_y[-1])
#use '(4.03*pow(10,14)\pow((self.y[-1]+6.371*pow(10,6)),2))' to change '10'
            self.x.append(self.v_x[-1]*self.dt+self.x[-1])
            self.y.append(self.v_y[-1]*self.dt+self.y[-1])
            self.t.append(_time)
            _time += self.dt    
    def show_results(self):
        font = {'family': 'serif',
                'color':  'darkred',
                'weight': 'normal',
                'size': 16,}
        pl.plot(self.x, self.y,label='angle is %f'%(math.atan(self.\
        initial_velocity_y/self.initial_velocity_x)*180/math.pi))
        pl.title('Cannon shell trajectories', fontdict = font)
        pl.xlabel('x ($km$)')
        pl.ylabel('y ($km$)')
        pl.legend()
        pl.show()
a = trajectories()
a.run()
a.show_results()

Result

• result for different angles
• result compare with situation whose g is a constant 1
• result compare with situation whose g is a constant 2
• result compare with situation whose g is a constant and ideal situation
  
  

We can see that the influence on the g is not a constant is very small when v not so big.

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Conclusion

• With the help the of method of Euler, we can solve many physics problem by computer easy. Also the problem can be very difficult with a lot of factors, but we can change the code to have the right answer with computer. It's important for us to have a better using of the method of Euler.

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Thanks For

• Lesson plan of Cai Hao:Chapter 2