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\title{\ifnum\TeacherEdition=1 TEACHER VERSION  \else STUDENT VERSION \fi\\ Space Flight For Recolonization}





\markboth{Space Flight For Recolonization}{Space Flight For Recolonization}

\author{Brian Allen and Karoline Hood\\
United States Military Academy\\
West Point, NY 10996 USA\\
}


\makeStitlePDFLaTex


\begin{abstract}
This project is a combination of differential equations, multi-variable calculus, and vector calculus with the use of technology to model colonization of a new planet.   Students will solve a system of second order differential equations to model a planets orbit and also solve a differential equation to model Kepler's Second Law.  In addition they will solve two optimization problems with the sum of square error and Lagrange multipliers.  They will also use vector calculus to solve for work using line integrals in both a conservative and non-conservative fields. Included in this project is a Mathematica Notebook with solutions.
\end{abstract}


\

\rule{\textwidth}{.5pt}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% GOALS

\textbf{\textsc{Goals:}}
\begin{enumerate}
\item Students clearly communicate their mathematical model and solution methods in written form.
\item Students explore aspects of the problem from a different disciplinary perspective to supplement their mathematical modeling.
\item Students develop and solve a mathematical model for a real world problem using multivariable calculus.
\item Students confront the ambiguity of problem solving and understand the necessity of making assumptions in formulating a mathematical model and the impact these assumptions have on their solutions.
\item Students apply technology learned in the course to analyze and solve a multivariable calculus model and develop an appreciation for how technology enhances their problem solving capabilities.

\end{enumerate}
\section*{SCENARIO DESCRIPTION}
The year is $2045$, you are a Colonel in the Army working for U.S. Army Space and Missile Defense Command. Due to the depletion of resources on Earth and the achievement of nuclear weapons by various countries with competing interests, we have been tasked with finding a new planet to colonize in order to eventually move the United States to a new solar system, which we can fortify and defend. Before colonizing the new planet, we must obtain a better idea of the planet's orbit.  In addition, we need to understand the amount of work required to travel to the new planet along with the amount of radiation exposure.  We also need to develop a design for the rocket to take us to the new planet.

\setlength{\parindent}{.5 cm}


%\textit{pgs.876-877 and pgs. 880-881 give a nice discussion of the math used in Task 1 and 2 and will comprise the basis on which App Day 1 will be developed.}

\noindent \textbf{TASK 1: Exploring Second Order Systems and Multivariable Calculus}



Let $\textbf{x} = \langle x_1,x_2,x_3\rangle$ be the position vector of  planet Aldrin (named after Buzz Aldrin USMA class of 1951 and former astronaut) orbiting the star named Borman (named after Frank Borman USMA class of 1950 and former astronaut). We are interested in solving Newton's Law of Gravitational Pull, describing how any two objects exert a gravitational force of attraction on each other. The direction of the force is along the line joining the objects. The magnitude of the force is proportional to the product of the masses of the objects, and inversely proportional to the square of the distance between them known as the Inverse Square Laws (http://theory.uwinnipeg.ca/physics/circ/node7.html). 

\noindent Therefore we can rewrite Newton's Law of Gravitational Pull as follows,
\begin{align}\label{NewtVector}
m\textbf{x}'' = -\frac{mMG}{|\textbf{x}|^3}\textbf{x},
\end{align}
\noindent where,
\begin{center}
$m = $ the mass of planet of Aldrin ($6.39 \times 10^{23}$ kg is the mass of Mars which we will use as an approximation to the mass of Aldrin),
\\$M= $ the mass of the star ($4.018 \times 10^{30}$ kg is the mass of Sirius which we will use as an approximation to the mass of Borman),
\\$G=6.67408 \times 10^{-11} \text{m}^3 \text{kg}^{-1}\text{s}^{-2}$, the Universal Graviational constant. 
\end{center}
 We can calculate,
\begin{align}
\left |\frac{\textbf{x}}{\mid\textbf{x}\mid^3}\right|=\frac{\mid\textbf{x}\mid}{\mid\textbf{x}\mid^3}=\frac{1}{\mid\textbf{x}\mid^2},
\end{align}
which makes clear the name inverse square law.

We can rewrite \eqref{NewtVector} in coordinates as
\begin{equation}\label{NewtCoords}
\begin{cases}
m x_1''&= -\frac{mMG}{(x_1^2+x_2^2+x_3^2)^{3/2}} x_1,
\\ mx_2''&=-\frac{mMG}{(x_1^2+x_2^2+x_3^2)^{3/2}} x_2,
\\ mx_3''&=-\frac{mMG}{(x_1^2+x_2^2+x_3^2)^{3/2}} x_3.
\end{cases}
\end{equation}
Note that we are ignoring the gravitational pull of any other objects in the universe as well as the gravitational pull of planet Aldrin on the star Borman. We are just studying the gravitational pull of the star Borman on the planet of interest.

A solution to the system above with initial conditions $\textbf{x}(0) = \textbf{x}_0$ and $ \textbf{x}'(0) = \textbf{v}_0$ represents the motion of a planet (star, satellite, astronaut, etc.), under the influence of gravity, that passes through the point $\textit{x}_0$ with velocity $\textbf{v}_0$. 
%\textbf{A larger discussion of Newton's Law of gravity should be added here.}
\begin{itemize}
\vspace{-.1in}	

\item[i.] Solve the second order system of ODEs using a numerical solver, with the initial values suggested below, to get an idea of sample solutions. %Hint: For solutions d and e you will need to include in your plot the command {\kf{BoxRatios}}.    \\


%\textit{For this part, if you allow Mathematica to solve over too large of a  value the solution will not be very good due to error. Explore the solutions through the suggested parameter choices below. For all of these examples, I am assuming that $MG = 2.567502 \times 10^{20} m^3s^{-2}$}

\begin{itemize}
\item[a.] $x_1(0) =10^{9},x_2(0)=0,x_3(0)=0,x_1'(0)=0,x_2'(0)=.9\times 10^5,x_3'(0)=0$ with $0\le t \le 5000$,
\item[b.] $x_1(0) =10^{9},x_2(0)=0,x_3(0)=0,x_1'(0)=0,x_2'(0)=.9\times 10^5,x_3'(0)=0$ with $0\le t \le 500,000$,
\item[c.] $x_1(0) =10^{8},x_2(0)=0,x_3(0)=0,x_1'(0)=10,x_2'(0)=10^{5},x_3'(0)=10^{5}$ with $0\le t \le 5000$,
\item[d.] $x_1(0) =10^{8},x_2(0)=0,x_3(0)=0,x_1'(0)=10^{5},x_2'(0)=0,x_3'(0)=0$ with $0\le t \le 50$,
\item[e.] $x_1(0) =10^{8},x_2(0)=10^{8},x_3(0)=10,x_1'(0)=10^{8},x_2'(0)=10^{8},x_3'(0)=100$ with $0\le t \le 500$.

\end{itemize}

\item[ii.] Discuss the differences in solutions to initial value problems i.a., i.c., i.d., i.e., suggested above. What could these solutions represent physically?   \\

\item[iii.] Notice the initial conditions in 1.a. gives you an ellipse, as expected, but initial conditions i.b. does not look like an ellipse. This is due to error in the numerical solver for this nonlinear, second order system. Use the $x_1(t), x_2(t)$, and $x_3(t)$  solutions from Part i.b., found using  a numerical solver, to verify our hypothesis about error by computing and plotting the energy from $0\le t \le 500,000$.  You will see that the energy of the solution, defined as
\begin{align*}
E(x_1(t),x_2(t),x_3(t)) = \frac{1}{2}\left (x_1'(t)^2 + x_2'(t)^2 + x_3'(t)^2 \right ) - \frac{MG}{\sqrt{x_1(t)^2+x_2(t)^2+x_3(t)^2}},
\end{align*}
\noindent is not constant for the initial conditions of Part i.b. We know by the principle of Conservation of Energy that the energy of the solution should be constant. What does this tell you about the validity of your solution using a numerical solver.% {\kf{NDSolve}}? 
\end{itemize}

\noindent \textbf{TASK 2: Finding and Parameterizing the Orbit of the Planet of Interest}

 The orbit of a planet must stay in a 2-dimensional plane so we may assume that our orbits are contained in the $xy$-plane. We also specify that the orbit should be elliptical given by the following equation in polar coordinates,

\begin{equation}\label{orbitparam}
r(\theta) = \frac{e d}{1+e\cos{\theta}},
\end{equation} 
\noindent where $e$ and $d$ are parameters which represent the \textit{eccentricity} and the \textit{directrix} of the ellipse. Eccentricity is a parameter that determines the amount an ellipse deviates from a perfect circle. \\
\noindent The values range from  $0\le e \le 1$.  See Figure~\ref{fig:ecc} for different eccentricities.
%\textit{(See pgs. 674-688 for a discussion of Conic Sections, Conic Sections in Polar Coordinates and 's Laws)}. Eccentricity is a parameter that determines the amount in which its orbit around another body deviates from a perfect circle.\footnote{https://en.wikipedia.org/wiki/Orbital eccentricity}  The scale ranges from 0 to 1 where 1 is a perfect circle.  See the image below for an example of different eccentricities.\footnote{Stewart, James. Calculus. Cengage Learning, 2015.}
  \begin{figure}[h!]\label{fig:ecc}

\includegraphics[scale=0.5]{5-080-eccentricity.jpg}
\caption{Eccentricities Ranging From 0 to 1 \cite{SC}.}
%\end{centering}
\end{figure}

For a fixed eccentricity, the directrix controls the size of the ellipse. The value of $d$ then should be positive and for the orbits of planets should be fairly large.

Our goal now is to find a particular curve which describes planet Aldrin by determining specific values for the parameters $e$ and $d$. Our team of astronomers has collected the position of planet Aldrin's data below and our job is to find the best fit  to the data given.
% (this needs to be made more realistic. How do the positions of celestial bodies get expressed in coordinates?)


\begin{itemize}	\vspace{-.1in}	
	\item[i.] Find the best values for the parameters $e$ and $d$ by minimizing a SSE function. Do you need to include the $z$-coordinate, from Table 1, when attempting to minimize error? What is the best way for you to reconcile the data given with the model above? Can you constrain the domain over which you try to minimize SSE using physical justification relating to (3)? What is the error associated with your answer and do you think it is reasonable?
	
	\begin{table}[!h]
	\begin{center}
		\begin{tabular}{ | p{4cm} | p{4cm} |  p{3.8cm} | p{3.8cm} |}
			\hline
			Planet Position Data &  \quad $r$-coordinate (meters)& $\theta$-coordinate (radians) & $z$-coordinate (meters) \\ \hline
		$\textbf{d}_1$ & $2.53569 \times 10^8$ & $ \pi/5$ & $3$ \\ \hline
		$\textbf{d}_2$ & $3.39708 \times 10^8$ & $ 2\pi/5$ & $9$ \\ \hline
		$\textbf{d}_3$ & $5.85633 \times 10^8$ & $3\pi /5$ & $7$ \\ \hline
		$\textbf{d}_4$ & $1.41333 \times 10^9$ & $4\pi /5$ & $-4$ \\ \hline
		$\textbf{d}_5$ & $3.07142 \times 10^9$ & $ \pi$ & $17$ \\ \hline
		$\textbf{d}_6$ & $1.41332 \times 10^9$ & $6\pi /5$ & $-23$ \\ \hline
		$\textbf{d}_7$ & $5.85631 \times 10^8$ & $7\pi /5$ & $13$ \\ \hline
		$\textbf{d}_8$ & $3.39718 \times 10^8$ & $8\pi /5$ & $-3$ \\ \hline
		$\textbf{d}_9$ & $2.53566 \times 10^8$ & $9\pi /5$ & $11$ \\ \hline
		$\textbf{d}_{10}$ & $2.31178 \times 10^8$ & $2 \pi$ & $-1$ \\ \hline
		\end{tabular}
\caption{Position of the planet Aldrin measured from the star Borman in cylindrical coordinates where $\textbf{d}_i = (r_i,\theta_i,z_i)$.}
%\caption {Position Data of Planet Aldrin}
	\end{center}
	\vspace{-12pt}
	
\end{table} 

%\textit{Solution: The students have minimized SSE before when trying to find the optimal damping parameter in a spring/mass system model in order to fit the data given. The major difference here of course is that they need to minimize over two variables. The SSE function should be defined as
%$SSE(e,d) = \displaystyle\sum_{i=0}^n(r_i - r(\theta_i))^2 $ where $n$ is the number of data points.
%}


	\item[ii.] What does the gradient of the SSE represent? 
	
	%\textit{Solution: The gradient points in the direction which increases error the most and so $-\nabla SSE$ points in the direction which will minimize error the fastest!}
	\item[iii.] Kepler's Second Law says that the line joining the sun to a planet sweeps out equal areas in equal times which can be used to parameterize the motion of our planet of interest with respect to time. We can derive the following differential equation for the angle as a function of time $\theta(t)$.
	
	\begin{equation}
	\frac{d\theta}{dt} = \frac{h}{r^2}
	\end{equation}
where $h = \sqrt{G M e d}$. Solve (5) using a numerical solver with initial condition $\theta(0) = 0$ and use your solution combined with your results Task 2 i.  to find a parameterization of planet Aldrin.

%\textit{Solution: First the students need to realize to use the equation of the orbit found in Task 2 part i. in order to rewrite the differential equation as $ \frac{d\theta}{dt} = \frac{h(1+e\cos(\theta))^2}{e^2d^2}$ which is a separable equation but unfortunately cannot be solved by hand. Next, the students need to recognize that they should obtain initial conditions from the orbit they obtained in Task 2 part i. Then by solving this equation, using NDSolve (The students have done this a few times before), for $\theta(t)$ they can find a parameterization of the orbit as $ \gamma(t) = (r(\theta(t)),\theta(t))$.}

\item[iv.] Comments For Teacher:  You can have your students create a movie of the planet orbiting the star Borman by modifying the $\bf{sample}$ code below which visualizes the motion of an object along the path $\gamma(t)=\langle r(t),\theta(t)\rangle$ given in polar coordinates. Verify that the motion you see in your movie conforms to Kepler's Second Law of motion.
\vspace{0.25cm}
%{\kf{r[t\_]=$\text{t}^2$}\\
%{\kf$\text{theta}[\text{t}\_]=\text{t}$}\\
%{\kf$\text{gamma}[\text{t}\_]=\{\text{r}%[\text{t}],\text{theta}[\text{t}]\}$}\\
%{\kf$\text{Animate}[\text{Graphics}[\{\text{PointSize}[\text{Large}],\text{Red}, \text{Point}[\{\text{r}[\text{theta}[\text{t}]]\text{ } \text{Cos}[\text{theta}[\text{t}]], \text{r}[\text{theta}[\text{t}]]\text{ } \text{Sin}[\text{theta}[\text{t}]]\}]\}$}\\
%{\kf$,\text{PlotRange}\rightarrow\{\{-5,5\},\{-5,5\}\}, \text{Axes} \rightarrow \text{True}, \text{AxesLabel} \rightarrow \{\text{"x", "y"}\}],\{\text{t},0,20 \text{ } \text{Pi}\}]$}}
\begin{align*}
&r[t\_]=t^2\\
&theta[t\_]=t\\
&gamma[t\_]=\{r[t],theta[t]\}\\
&Animate[Graphics[\{PointSize[Large],Red, Point[\{r[theta[t]]\text{ } Cos[theta[t]], r[theta[t]]\text{ } Sin[theta[t]]\}]\}\\
&,PlotRange\rightarrow\{\{-5,5\},\{-5,5\}\}, Axes \rightarrow True, AxesLabel \rightarrow \{"x", "y"\}],\{t,0,20 \text{ } Pi\}]
\end{align*}
\end{itemize}

\vspace{0.5cm}

\noindent \textbf{TASK 3: Traveling From Our Solar System to the New World}

%\textit{On pg.1073, the book talks about a potential function for a gravitational field. On pg.1088, example 1 shows how to find the work done by a gravitational field. On pg.1093, the book discusses conservation of energy. These are all concepts which are at work behind the scenes in this portion of the project, so it will be a good idea for you to cover these concepts/examples in class.}

We have been assuming that the only force we need to worry about is the force due to the star, but, now we will need to incorporate the force of the star and our Sun. Let the star be located at the origin $(0,0,0)$ and assume that the Sun is located at $\textit{x}_0=(10.1,9.8,8.7)$ light years. \textbf{(Make sure to convert all physical quantities to light years for this portion of the project.)} Then the force due to the gravitational pull of the star is given by, 
\begin{equation}
\textbf{F}_1 = -\frac{m M_1G}{|\textbf{x}|^3}\textbf{x},
\end{equation} 

\noindent where $M_1=1.989 \times 10^{30}$ kg is the  mass of the star Borman and $m = 2000$ kg is the mass of the spaceship. Likewise, the gravitational force of the Sun is given by, 
\begin{equation}
\textbf{F}_2 = -\frac{mM_2G}{|\textbf{x}-\textbf{x}_0|^3}(\textbf{x}-\textbf{x}_0), 
\end{equation}

\noindent where $M_2=4.018 \times 10^{30}$ kg is the  mass of the Sun. So the gravitational force field of interest in order to consider the motion from Earth to planet Aldrin is given by 
\begin{equation}
\textbf{F}=\textbf{F}_1+\textbf{F}_2.
\end{equation}

\begin{itemize}	 
	\item[i.] We say that a vector field $\textbf{F}$ is conservative if there exists a function, called the potential function, such that $\nabla f = \textbf{F}$. In the case where $\textbf{F}$ represents a gravitational field the potential function represents the potential energy of the gravitational field. Is the vector field $\textbf{F}$, defined above, conservative? If yes, can you find a potential function?
	
	
	\item[ii.] Plot the vector field.  You will have to multiply the vector field by a large scale factor in order to get a nice visual.\  Find the work required to travel along a straight line path under the influence of the gravitational field $\textbf{F}$. Explore the conservative property of $\textbf{F}$ by calculating the work along at least two different paths with the same endpoints,
	
\begin{equation*}
Work = \int_{\gamma} \textbf{F}\cdot \textbf{T} ds.
\end{equation*}

	\item[iii.] Are there any other considerations, besides gravity, which may cause you to choose a particular path?
	
	%\textit{We are hoping that the students will use what they know about physics to make interesting suggestions. Some ideas include radiation, electromagnetic fields, other celestial bodies, other enemy space crafts, etc.}
	
	\item[iv.] Radiation in space travel is of great concern to the health of astronauts.  The closer a space shuttle approaches a star the greater the intensity of radiation. Most planets have a layer of atmosphere that provides protection from radiation but as you leave the atmosphere of a planet the protection dissipates. See Figure~\ref{fig:rad} of the Earth's exposure to radiation by the Sun.
\begin{figure}\label{fig:rad}
 \begin{centering}
\includegraphics[scale=0.5]{5-080-Radiation.jpg}
%\end{centering}
\caption{Earth's Exposure to Radiation \cite{Rad}.}
\par\end{centering}
\end{figure}
 
Given the function, 
 \begin{align*}
 R(x,y,z) &= \frac{10.4 \sin^2(100x) \sin^2(100y)\sin^2(100z)}{\sqrt{x^2+y^2+z^2}} \\
 &+ \frac{23.5\sin^2(100(x-x_0))\sin^2(100(y-y_0))\sin^2(100(z-z_0))}{\sqrt{(x-x_0)^2+(y-y_0)^2+(z-z_0)^2}}
 \end{align*}
 ,which represents the radiation at a given point in space in units of Gray (Gy),  calculate line integrals of radiation experienced by a rocket traveling along different paths from our solar system to the new solar system.  Gray  is a derived unit of inoizing radion does used in the International System of Units (SI).  Then provide an interpretation and units analysis of these integral quantities in relation to the risk of astronauts traveling aboard the spacecraft. 
 
 We cannot graph the function $R(x,y,z)$ completely but how can we get a feel for what the function looks like visually and qualitatively?
	
	%\textit{The students should at least calculate the total radiation along a straight line path using a line integral where they will need to take the absolute value of $R(x,y,z)$ while integrating. The students should plot the function in $x y$, $x z$ and $y z$ in order to visualize as well as give a qualitative description of the function.}



\end{itemize}

\vspace{1cm}

\noindent \textbf{TASK 4: Engineering the Design of the Rocket} \\
\vspace{0.25in}
\begin{figure}\label{fig:rock}
\begin{centering}
\includegraphics[scale=0.5]{5-080-pay_load.jpg} %Photo from Wiki
\caption{Example of Multi-Stage Rocket \cite{rock}}
\end{centering}
\par\end{figure}

This portion of the project is based off of a project scenario in (\cite{SC}, pg. 979-980).  We significantly modified the project scenario from \cite{SC} and integrated it into the rest of the project.


%\textit{This section of the project is based on pgs. 979-980 where the book discusses using Lagrange Multipliers in order to find the optimal mass of three stages of a rocket.}\\
The purpose of a two stage rocket is so that once the fuel is exhausted in a structure, the structure which contained the fuel is useless and only adds weight to the vehicle which slows the rocket's acceleration. By dropping the stages, which are no longer needed, the rocket lightens itself.  When a stage drops off, the rocket is still traveling close to the original speed of the whole assembly. Therefore, less total fuel is needed to reach the limits of before earth's atmosphere. In this part, we wish to minimize the total mass $M_T=M_1+M_2$ of a two stage rocket engine, where $M_1$ and $M_2$ represent the masses of the two stages, respectively. This minimization should be carried out with the constraint that we require a final velocity, $v_f$, after the two stages of the rocket have been jettisoned. This will represent the typical velocity throughout our journey from earth to our new planet. If we let $A$ represent the mass of the payload of the rocket, $S$ the structural factor, and $c$ the exhaust speed then we can find the following expression for the final velocity .\\

\begin{equation*}
v_f=c\left[\ln\left(\frac{M_1+M_2+A}{S M_1+M_2+A}\right)+\ln\left(\frac{M_2+A}{S M_2+A}\right)\right ].
\end{equation*}



%In this part we wish to minimize the total mass $M=M_1+M_2+M_3$ of a three stage rocket engine where $M_1,M_2,M_3$ represent the masses of the three stages respectively. This minimization should be carried out with the constraint that we require a final velocity $v_f$, after the three stages of the rocket have been lost, which will represent the typical velocity throughout our journey from earth to our new planet. If we let $A$ represent the mass of the payload of the rocket then we can find the following expression for the final velocity

%\begin{equation*}
%v_f=c\left[\ln \left(\frac{M_1+M_2+M_3+A}{S M_1+M_2+M_3+A}\right)+\ln\left(\frac{M_2+M_3+A}{S M_2+M_3+A}\right)+\ln\left(\frac{M_3+A}{S M_3+A}\right)\right ]
%\end{equation*}

\noindent The final velocity $v_f$ is related to a concept in physics called \textit{escape velocity} $v_e$, which is the velocity that an object would need to travel, away from the source of gravitation, in order to successfully escape the source of gravitation's pull on the object, without the use of thrust. The very minimum speed required, or escape velocity, can be calculated by finding the velocity such that the sum of the kinetic plus potential energy is zero ($ KE +PE =0$) \cite{ev}. The equation for this is
\begin{equation*}
\frac{1}{2}v^2-\frac{GM}{r} = 0.
\end{equation*}
Solving for escape velocity, we have,\\
\begin{equation*}
v_{e}=\sqrt{\frac{2GM}{r}},
\end{equation*}
where \textit{G} is the Universal Gravitational constant, $6.67408 \times 10^{-11} \text{m}^3 \text{kg}^{-1}\text{s}^{-2}$, \textit{M} is the mass of the planet (kg), and \textit{r} is the radius of the planet (meters).  The radius of the Earth, including the Earth's atmosphere, is $6.38\times10^6$ meters and the mass of Earth is $5.98\times10^{24}$ kg.

%It is important to be able to propel the rocket through the atmosphere of the earth and so we should not expect to reach $v_f$ until we are at least a distance of $d_*=?$ from the earth. (Look into this more to make sure what you are saying is consistent with physics)


\begin{itemize}	
	\item[i.] Find the $\textit{v}_\textit{f}$ in $\frac{\text{m}}{\text{s}}$ that is required once we pass through the atmosphere by finding the escape velocity at this point. Which of the solutions form Task 1 Part ii represents a rocket trying to escape a gravitational field?
	
	%\textit{Task 1 part ii e. represents a rocket trying to escape a gravitational field. In this case the object must be able to apply thrust (acceleration) on its own which we do not expect from a planet}
	
	
	
	\item[ii.] Minimize the total mass of the rocket engine, given the final velocity constraint, a structural factor, S, of 0.1, an exhaust speed of $6000$  $\frac{\text{mi}}{\text{h}}$, and payload mass, A, of $600$ pounds. Make sure your final velocity constraint is in units of $\frac{\text{mi}}{\text{h}}$.

\item[iii.] If we only consider the gravitational field due to earth, where we assume the earth is located at the origin for this task, we find the force field
\begin{align*}
\textbf{F}_3=-\frac{M(M_T+A)G}{|\textbf{x}|^3}\textbf{x},
\end{align*}
which is relevant for the design of the rocket in order to escape the gravitational pull of the earth. Notice that $M_T + A$ represents the total mass of the rocket, i.e. the mass of the payload and the two stages, which is the relevant mass for the gravitational pull between the earth and the rocket, according to Newton's law, in the definition of $\textbf{F}_3$. 

Consider the path $\gamma(t) = \frac{1}{\sqrt{3}}\langle t,  t, t\rangle$ on the time interval $6.371\times10^6 \le t \le 6.38\times10^6$ seconds which represents the rocket traveling from the earth to the radius at which we would hit the escape velocity. Remember that the particular speed of the straight line parameterization, from the surface of the earth to outside the earth's atmosphere,  does not effect the calculation of work. Calculate the work due to the gravitational field $\textbf{F}_3$ along the path $\gamma(t)$. How does this calculation of energy relate to the design of the rocket? (A discussion is fine for this last part.)

\end{itemize}

\begin{thebibliography}{99}

\bibitem{SC} Stewart, James. 2015. \textit{Calculus}.Cengage Learning, 8 ed. 

\bibitem{Rad}``The Sun: Earth's Primary Energy Source — The Sun and Earth's Climate — Beyond Weather And The Water Cycle." Beyond Weather \& The Water Cycle. N.p., n.d. Web. 28 Apr. 2017.  \href{http://beyondweather.ehe.osu.edu/issue/the-sun-and-earths-climate/the-sun-earths-primary-energy-source}{http://beyondweather.ehe.osu.edu/issue/the-sun-and-earths-climate/the-sun-earths-primary-energy-source}. 

\bibitem{ev}``Escape Velocity N.p., n.d. "Web. 28 Apr. 2017. \href{http://hyperphysics.phy-astr.gsu.edu/hbase/vesc.html}{http://hyperphysics.phy-astr.gsu.edu/hbase/vesc.html}.

\bibitem{rock}N.p., n.d. Web. 28 Apr. 2017. \href{https://history.nasa.gov/SP-4402/ch2.html}{https://history.nasa.gov/SP-4402/ch2.html}.

\end{thebibliography}


\teacher{
\section*{NOTES FOR TEACHERS}
This is intended to be a group assignment. The primary submission is a hard copy report that should not exceed four (4) pages (not including appendices, title page, etc.). This report will include graphs reflecting student's work. 

The report should include any additional supporting graphs, equations, and computations in the appendices. In addition students should digitally submit any technology files. 

There is also a 15 minute presentation including time for questions to your instructor with no more than five (5) slides (one slide per task).

There is a Mathematica Notebook approved solution with in text comments attached.}



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