ROCKET MOTION

Aaron Koga

Physics 305

10 April 2007

Introduction

\begin{figure} \epsfxsize =3.1in \epsfbox{rocketFig.eps} \end{figure} Figure 1: Figure showing the forces on a rocket and a sketch of the trajectory through the atmosphere.

Rockets are subject to three forces, the thrust force ($F_{p}$), the drag force ($F_{b}$), and gravity ($F_{g}$). A sketch of these forces is shown in FIG 1 along with a sketch of a rocket trajectory. These forces are given by the following equations.


\begin{displaymath} $F_{p}=pg\theta(t)$ \end{displaymath} (1)

Here $p$ is the thrust force, $g$=9.8 m/s$^{2}$, and $\theta(t)$ is a step function that is initially on and turns off at a time $t_{o}$, as specified by the rocket design.
\begin{displaymath} $F_{b}=-\frac{1}{2}\rho(\vert\vec{r}\vert)C_{d}Av^{2}\hat{v}$ \end{displaymath} (2)


\begin{displaymath} $\rho(\vert\vec{r}\vert)=\rho_{0}e^{-\vert\vec{r}\vert}$ \end{displaymath} (3)

In the equation for $F_{b}$, $\rho$ is the density of air, $C_{d}$ is the coefficient of drag, $A$ is the cross-sectional area.
\begin{displaymath} $F_{g}=-\frac{GMm(t)}{\vert\vec{r}\vert^{3}}\vec{r}$ \end{displaymath} (4)


\begin{displaymath} $m(t)=m_{0}-m_{f}\frac{t-t_{b}}{t_{b}}$ \end{displaymath} (5)

The constants in the expression for $F_{g}$ are $G$, the gravitational constant and $M$, the mass of the earth. The mass of the rocket, $m(t)$, is dependent the launch mass, $m_{o}$; the total mass of fuel, $m_{f}$; and the burn out time (when all the fuel is burnt), $t_{b}$. [1]

C Program for Simulation

A C Program, missile.c (see also source code: 3vector.h, FRK4-3D.h, and missile.h ) was written and used to simulate the flight of rocket. The program used the Runge-Kutta method of fourth order (RK4) to solve the differential equations of motion for the rocket, as shown in EQ 1-5.

This program determined the density of the atmosphere by reading in data from a file for the density at known heights.

Simulation


Table I: Table of physical quantities used in rocket simulation. $v_{0}$ is initial velocity
Physical Quantity Value
$C_{d}$ 0.25
diameter 1.32 m
$p$ 26,050 kg
$m_{0}$ 16,000 kg
$m_{f}$ 12,912 kg
$t_{b}$ 110 s
$v_{0}$ 0.1 m/s


The flight of the rocket was simulated using the physical parameters shown in TABLE I. The initial velocity was chosen to be 0.1 m/s because the program needed a non-zero quantity to point the rocket in the correct direction (even though the rocket should theoretically start with no velocity).

\begin{figure} \epsfxsize =3.1in \epsfbox{missile.eps} \end{figure} Figure 2: Trajectories for a few different starting angles.
Figure 3: Plots of the distance traveled (top) and velocity (bottom) as functions of time for a starting angle of 88.75$^{o}$, accounting for curvature of the earth. \begin{figure} \epsfxsize =3.1in \epsfbox{missileDR.eps} \epsfxsize =3.1in \epsfbox{missileV.eps} \end{figure}
\begin{figure} \epsfxsize =3.1in \epsfbox{missileR_zoom.eps} \end{figure} Figure 4: Figure showing the distance traveled as a function of starting angle.
\begin{figure} \epsfxsize =3.1in \epsfbox{missileR_cVnc.eps} \end{figure} Figure 5: Comparison of the distance traveled when accounting for the curvature of the earth (red) and not accounting for the curvature of the earth (green) .

The program was run for different starting angles, taking into account the curvature of the earth. The trajectory of the rocket is shown for a few different starting angles in FIG 2. The distance and velocity as functions of time are shown for a particular starting angle in FIG 3. The range of the rocket as a function of starting angle is shown in FIG 4. From this plot, the maximum distance occurs at a starting angle of about 89.99$^{o}$. To study the effect of the curvature of the earth, the program was run with the assumption that the earth is flat. FIG 5 shows the effect of the curvature, which is only noticeable at angles very close to 90$^{o}$..

The parameters shown in TABLE I are characteristic of a North Korean Nodong missile. Therefore, the program was used to assess North Korea's ability to hit Tokyo with this missile. Assuming a distance of 1286 km from Pyongyang to Tokyo, the results of the simulation suggest that North Korea could hit Tokyo if they use a starting angle of 89.76$^{o}$.

Bibliography


1
P. Gorham. http://www.phys.hawaii.edu/~gorham/P305/Missile.html

APPENDIX