@zqbinggong 2016-06-21T02:33:06.000000Z 字数 5301 阅读 753

# the Analysis of Realistic Projectile Motion

张强 2013301020039 the final exam

final.py

## Abstract

In the final exam,I choose the subject about the realistic projectile motion, in which I will consider several problems involving the motion of the objects through the atmosphere,easpecilly the trajectory of the cannnon.

## Background

The Euler method is the numerical approach I choose to deal with motion of the realistic projectile objects.In the begin,we just consider the case of two dimention,and then I will study the more realistic situation where a cannon flies in the sky,with air resistance.And the effec of the astomsphere will be considered.There are two simple approximation to treat the atomosphere as an isothmal ideal gas or an adiabatic ideal gas.

### 1.Projectile motion

1.Kinematic quantities of projectile motion
In projectile motion, the horizontal motion and the vertical motion are independent of each other; that is, neither motion affects the other. This is the principle of ''compound motion'' established by Galileo in 1638.Galileo Galilei, ''[[Two New Sciences]]'', Leiden, 1638, p.249

2.Acceleration
Since there is only in the vertical direction, the velocity in the horizontal direction is constant, being equal to . The vertical motion of the projectile is the motion of a particle during its free fall. Here the acceleration is constant, being equal to g .The g is the Standard gravity|acceleration due to gravity. ( 9.81 m/s^2 near the surface of the Earth) The components of the acceleration are:

3.Velocity
The horizontal component of the [[velocity]] of the object remains unchanged throughout the motion. The downward vertical component of the velocity increases linearly, because the acceleration due to gravity is constant. The accelerations in the x and y directions can be integrated to solve for the components of velocity at any time t , as follows:

The magnitude of the velocity (under the Pythagorean theorem, also known as the triangle law):

4.Displacement

At any time t , the projectile's horizontal and vertical are:

The magnitude of the displacement is:

### 2.Air Drag

As is mentioned above and will be discussed thoroughly in our codes, air drag force plays a critical role in the shells' trajactory. But engineers have come up ideas of rifling to fight against such impedence.
Rifling is a series of spiral line inside the pipe of fire guns like rifles and cannons. When the shells get out of the pipe pushed by the gunpowder at the end of the bore of a gun, the high temperarure will inflate the shell making it rub with the rifling and spin. Such spin motion will effectively reduce the air drag force.

### Air Density Distribution

In the first section we gave the form of air drag force under simple assumptions. Now we come to a stage a little more complex. The force is sill propotional to the density of air. However, the density of air is a decreaseing function of altitude.

### Isothermal Condition

The simplest approximation is to treat the atmosphere as an isothermal ideal gas.One then finds that the pressure p depends on altitude according to $\rho (y)= \rho (0) e^{-mgy / k_{B}T }$

The more realistic approach is to assume that the ait is a poor conductor of heat, and that convection is very slow.
The adiabatic approxination leads to a somewhat different dependence of the density on altitude $\rho = \rho _{0} (1- \frac{ay}{ T_{0} } )$

### 3.the implementation of the program

1.Firstly,the program ask you to input some parameters such as target coordinate,the effective attack range and the "h",which is used to determine the approximation methonds.For d=0,we consider the isothermal condition and d=1 for adiabatic condition.
2.Then,the program run ,calculate the firing speed and angular.
3.In the end, program draw the 2D with matplotlib or 3D trajectory using vpyhton according to the firing speed and angular calculated and outpute the related information.

## Body

### 1. the case of two dimension

and the result are listed in figure1

20000 2000 10 等温近似 49.5 565.773929138
20000 2000 10 绝热近似 48.6 584.738975708
30000 3000 10 等温近似 49.8 733.751512241
30000 3000 10 绝热近似 48.6 779.306132074

According to the results,we draw a conclusion that the cannon needs faster speeds and smaller angular on adiabatic condition to hit the same target.

### 2.the 3D trajectory of the cannon

and the results are listed in the figure2

10000 10000 10000 10 等温近似 37.3478897565 710.409868665
10000 10000 10000 10 绝热近似 36.7478897565 681.75948603
100000 1000 100000 100 等温近似 40.6478897565 1957.58009383
100000 1000 100000 100 绝热近似 40.6478897565 1957.58009383

According to the results,we find that when the target is far away from us,the initial speed ang angular calculated are the same.Besides,when the targer coordiante is set as(10000,10000,10000) where the Y-coordinate is the height of the target,we find that the result is different fron the case of two dimension

## Connclusion

The details of the conclusion are showed in the body,there I just want to say something about program and the similarity and difference between the case of 2D and 3D.

1.The program needs a long time to calculate the firing speed and angular,so you should be patient to wait the outcome information.

2.The 3D case is similar to the 2D case. In deed, we just rotate the firing plane $oxy$ around the y-axis by the angular of $\phi$.

And $\phi=tan \frac{t_z}{t_x}$
$t(x,y,z)$is the target coordinates.At this point,we should get the similar results about the relationship between the different approxiamtion approach used to deal with the effect of the atomsphere.This is problem as far as I can see.Unfortunately,I am at a lost what to do to deal with it.

## Acknowledgement

1.Giodano, N.J., Nakanishi, H. Computational Physics. Tsinghua University Press, December 2007.
2.Wikipedia contributors. "Projectile motion." Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, 5 Mar. 2016. Web. 2 Apr. 2016.
3.会编程的大白熊--Vpython
4. 使用matplotlib的示例：调整字体-设置刻度、坐标、colormap和colorbar等
5.Thanks to Liu Wentao for his template.

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