The sixth assigment projectile motion

(level 1-2)

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物基一班
陈天懿 2013301020146

Abstract

The projectile motion is one of the well known physics phenomena to us. It is a relatively simple problem when not considered about the air drag, which can be solved by direct integrate. However, once we introduced the air drag, the direct analysis solution becomes too complicated to be derived. We can only get the numerical solution in this situation. In this assigment, I'm going to solve the problem using Euler method.

What's more, we will then consider the practicle application of the projectile motion model in the motion of a cannon. We will use numerical method to figure out the relationship between the cannon's firing range, its muzzle velocity and its firing angle. Deeply, we will calculate the minimum muzzle velocity of the cannon in hitting different targets.

Background

The projectile motion's kinetic equation can be easily wrote down as

$$\begin{cases} \frac{d^2x}{dt^2}=0\\ \frac{d^2y}{dt^2}=-g \end{cases}$$
This set of equations are easy to compute, but in reality, we have to add the other terms on the right side of the equations such as air drag. We can constract the equations as follow
$$\begin{cases} \frac{d^2x}{dt^2}=f_x(x,y,v_x,v_y,t)\\ \frac{d^2y}{dt^2}=-g+f_y(x,y,v_x,v_y,t) \end{cases}$$
Among which $f_x(x,y,v_x,v_y,t)$and$f_y(x,y,v_x,v_y,t)$are forces exerted on unit mass projectile along $x$ and $y$ directions. In many situations, we get the exact vlaue of air drag as follows:
$$\begin{cases} f_x=-B(1-\frac{ay}{T_0})^{\alpha}vv_x\\ f_y=-B(1-\frac{ay}{T_0})^{\alpha}vv_y\\ \end{cases}$$
Among which $B$ and $\alpha$ are constants, $\alpha$ is
2.5 to air. $a$ is the constant got from experiment, $a$ is approximately $6.5\times10^-3K/m$. Togather thes equations decide the projectile's motion.

Euler method

We can use Euler method to solve the problem, first integrate the 2nd order differential equation to get the 1st order equation, substitute the differential with $\Delta$, we get

$$\begin{cases} x_{i+1}=v_{x,i}{\Delta}t+x_i\\ v_{x,i+1}=f_x{\Delta}t+v_{x,i}\\ y_{i+1}=v_{y,i}{\Delta}t+y_i\\ v_{y,i+1}=(-g+f_x){\Delta}t+v_{y,i}\\ \end{cases}$$
thus we can use the place, velocity at $t_i=i{\Delta}t$, we can get the place and velocity of the next moment $t_{i+1}=(i+1){\Delta}t$. Do this repeatly, we can get the motion situatuion of the projectile of the whole time.

Main

Solution of the kinetic equations with air drag
Set the initial conditions with $v_0=700m/s$, projectile angle $\theta_0=45^\circ$. We get the motion trajectory as follows:

Differences between trajectories of motion with and without air drag

We can clearly see from the diagram that the trajectories are drastically different with and without air drag. After analize different projectile angles, we conlude:

• Air drag greatly reduces the firing rang of the cannon, so if we want to improve the range, we must reduce the air drag, which can be mannaged by adjust the shape of the shells. For example, streamline is proved can reduce the air drag greatly.
• What's more, we can see from the figure that the air drag can affect the relation between maximum range and projectile angle. When the air drag is taken into account, the cannon will reach its maximum range when firing angle is $\theta=43^\circ$. We have to adjust the firing angle in actual use in order to fire the cannon furthest.

Precise aiming system
In practical situation, the target's location are not known by the artilleryman, we hope to design a system that can hit the target with both unknown vertical and horizontal distance. Considering the cannon has a fixed muzzle velocity, we will discuss the firing angle and the target's relationship.
In order to hit the target, we have to scan the possible firing angle. To get the outcome as far as possible, we use the so called "dichotomy method" to accelerate the whole process. That is, after each small range scan, we choose the angle which is closest to the target, then perform the next scan, this will shorten the calculation time. The aming process is listed as follows:

Conclusion

This assignment disscusses the trajectory of a projectile with air drag, and designs an aiming system for the cannon, complete the teacher's homework

see codes here
see codes here

Reference

Computational Physics-Nicholas J. Giordano, Hisao Nakanishi