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\begin{document}

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\title{\ifnum\TeacherEdition=1 TEACHER VERSION \else STUDENT VERSION \fi\\Rocket Flight}




\markboth{Rocket Flight}{Rocket Flight}

\author{Brian Winkel, Director SIMIODE\\
Cornwall NY  12518  USA\\ 
Director@SIMIODE.org
}

\makeStitlePDFLaTex


\begin{abstract} We offer an opportunity to build a mathematical model using Newton's Second Law of Motion and a Free Body Diagram to analyze the forces acting on the rocket of changing mass in its upward flight under power and then without power followed by   its fall to earth.
 \end{abstract}





\section*{SCENARIO DESCRIPTION}



Suppose we have a small  rocket with a full 100 liter fuel tank. Fuel (with a mass density of .98 kg/l) is burned at a steady rate of 3 liters per second and can provide a constant thrust force of 5900 Newtons as it burns. The rocket and  fuel tank has a mass of 400 kg with no fuel in it.  The shape design of the rocket causes a resistance proportional to its velocity during flight of 2 $v(t)$ where $v(t)$ is in m/s and the constant 2 for proportionality of resistance   is  2 N/(m/s).  We aim the rocket straight up and launch it \ldots 5 \ldots 4 \ldots 3 \ldots 2 \ldots 1 \ldots Launch \ldots We have lift off!!!

\begin{enumerate}
\item How long will the rocket burn and hence provide thrust? We call this time interval the {\it burn period\/}.
\item Build a differential equation model for the powered flight (during the burn period) of the rocket based on all the External Forces and Newton's Second Law of Motion. 

\begin{center}
\parbox{13 true cm}{Here are some issues to consider. Construct a Free Body Diagram - very important. See Figure~\ref{fig:RocketFBD} for hints. What are reasonable initial conditions? The mass $m(t)$ is now a function of time. What is that function? There is an external constant force due to thrust during the burn. Lots of stuff changing except the acceleration due to gravity, $g$.}
\end{center}


\begin{figure}[!h]
\centering

  \includegraphics[width=4.5cm]{RocketFBD.png}


\parbox{8true cm}{\caption{\label{fig:RocketFBD} Outline of Free Body Diagram for upward rocket flight with fuel burn thrust looking for completion.}}
\end{figure}


\item Solve the differential equation   and plot the rocket's altitude over the burn period and until the rocket reaches its apex or highest point. Also plot the velocity over the burn period. 

\item  Model what happens to the rocket after the burn period? 
\item  Determine maximum height of this rocket?
\item Plot the rocket's altitude over the free falling period. Also plot the velocity over the free falling period.  We can assume the rocket turns around and heads downward with its front first as it did in the rising flight. 
\item How long before it strikes the earth?  That is, how long is the total flight? We may assume this is a vertical flight only.
\item With what velocity does the spent rocket strike the ground? Ouch!
\end{enumerate}
 

In the above model we  have moved away from constant coefficient differential equations. Moreover, we have introduced additional, important external forces (thrust and weight) to the model of the rocket. These forces will make our differential equation into a {\it non-homogeneous} differential equation.  




\teacher{
\section*{NOTES FOR TEACHERS}

This rocket flight has three time regimes (a) upward with fuel burn thrust and fuel on board, (b) upward without fuel burn thrust and no fuel on board, and (c) downward withoutd fuel burh thrust and no fuel on board. Each of them has a slightly different Free Body Diagram and direction needs to be treated carefully.

We have places a Mathematica notebook, 1-128-T-Mma-RocketFlight-TeacherVersion, and an accompanying pdf version for those who do not use Mathematica, in which we offer a full analysis.

This can be an in class activity with completion at home. We have found that assembling Free Body Diagrams is often difficult for students who have not seen them or practiced them and so class time devoted to this for each of the three time regimes would be advisable and then letting students perform the solutions of the resulting differential equations and the accounting of the velocity and height  through all time regimes.

We have used Mathematica to solve the three differential equations (one for each of the three time regimes) and we show the equation for the rocket being thrust upward (\ref{eq:Up}) here:
\begin{equation}\label{eq:Up}
m(t)\cdot v'(t)= - m(t)\cdot 9.8- 2\cdot v(t)+ 5900\, 
\end{equation}

\noindent where $m(t) = 400 + 100\cdot 0.98 - 3\cdot 0.98\cdot t$ is the changing mass of the rocket, $400$ being the mass of the rocket itself, $100\cdot 0.98$ is the full load of fuel,  and $3\cdot 0.98\cdot t$ is the spent fuel as of time $t$.

This differential equation  (\ref{eq:UpExp}) looks formidable if by hand solution strategy, say integrating factor,  is to be pursued:
\vspace*{-.5 true cm}

\begin{equation}\label{eq:UpExp}
(400 + 100\cdot 0.98 - 3\cdot 0.98\cdot t) \cdot v'(t)= - (400 + 100\cdot 0.98 - 3\cdot 0.98\cdot t) \cdot 9.8- 2\cdot v(t)+ 5900\,  ,
\end{equation}
 \noindent but the solution in Mathematica is,
 \begin{equation}\label{eq:solUp}
 v(t) = 30.6511 (169.388\, - t)-248.016 (169.388\, -
   t)^{0.680272}+2950\, .
 \end{equation}
 
 In Figure~\ref{fig:TotalFlightHeight} we offer a plot of the height vs time for the entire rocket flight.


\begin{figure}[!h]
\centering

  \includegraphics[width=7cm]{TotalFlightHeight.png}


\parbox{8true cm}{\caption{\label{fig:TotalFlightHeight} Plot of height vs time for rocket flight over all three time regimes.}}
\end{figure}

} 
\end{document}


