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\newpage
\title{\ifnum\TeacherEdition=1 TEACHER VERSION \else STUDENT VERSION \fi\\River Crossing}






\markboth{River Crossing}{River Crossing}









\author{Brian Winkel, Director SIMIODE\\
Chardon OH 44024 USA\\
BrianWinkel@simiode.org}


\makeStitlePDFLaTex


\begin{abstract}
Students develop a model of a river crossing in a boat with thrust using Newton's Second Law of Motion from a Free Body Diagram they construct. The model is thence a system of one second order linear and a second order nonlinear differential equations. Students must construct a river current model from some general notions and three data points on the flow magnitudes to incorporate into the model. With a given set of parameters we see where the boat lands on the other side of the river and then we seek changes in  parameters to be sure the boat arrives at a specified landing point on the other side of the river? We offer a number of ``What if'' issues once the model has been developed beyond the obvious question of, ``Where does the boat land downstream on the East bank knowing its West bank starting point?'' For example, given different river current parameters  in what direction might the boat head in order to arrive at the same landing point under each set of conditions? If we make the flow of the river asymmetric about its central line or axis how does this change the landing spot on the East bank, if all other things are left the same? If we fix the heading of the boat then for various river current parameters  what thrust is necessary to arrive at the same landing point under each set of conditions? 
\end{abstract}


\section*{SCENARIO DESCRIPTION}

In Figure~\ref{fig:6-12-RiverCrossing} we see a depiction of a boat crossing a river. We shall model such a crossing using a Free Body Diagram which will enable us to apply Newton's Second Law of Motion.  Basically, Newton's Second Law of Motion says that the mass times the acceleration ($m\cdot s''(t)$) of a body is equal to the sum of all external forces which apply to that body. A good Free Body Diagram will enable us to see all these external forces and then build a model (a differential equation  model) to describe the position of a body, in this case a boat.

\begin{figure}[!t]
  \begin{center}
   \includegraphics[width=9 true cm]{6-12-RiverCrossing.eps}

\parbox{11.9 true cm}{\caption{
      \label{fig:6-12-RiverCrossing}
      Depiction of river crossing showing upstream river current profile with strength and direction at bottom, resulting curved path of the boat, boat, tangential arrow indicating thrust of the boat's engine, and river banks on each side of the river.}}
  \end{center}

\vspace*{-.7 true cm}
\end{figure}

Let us look at the situation. The straight river section is  a constant 400 m wide and flowing   South to North with maximum velocity or current  of 0.9 m/s in the center. The stream velocity is symmetric about the center of the river and decreases to effectively 0 m/s at the bank contact point due to friction.

\begin{itemize}
  \item [1)] Offer several models of  the current velocity profile across the river and defend each one. We are looking for something like $W(x)$  is the velocity in m/s as a function of the distance $x$ m from the West side of the river. Distance is measured perpendicular to the river bank. We will need this in considering how the boat travels across the river, for the velocity of the water in the river will carry the boat along.
\end{itemize}


\vspace*{.2 true cm}

Let us agree to identify the variables in question as follows.
\begin{itemize}
\item 
$x(t)$ is the horizontal   position in meters from the West bank in a perpendicular direction to the bank of the boat at time $t$ in  seconds and thus $ 0 \leq x(t) \leq 400$ for $ t \geq 0$.
\item $y(t)$ is the vertical position in meters from the starting coordinate on the West bank in a  direction parallel to the shore of the boat at time $t$ in  seconds and $y(t)>0$ signifies the boat is North of its starting position while $y(t)<0$ signifies the boat is South of its starting position. 
\end{itemize}

Here is some other information which should help in  model building.

\begin{itemize}
\item The mass of the boat is $m = 225$ kilograms.  
\item The constant thrust of the boat's engine is $T = 65$ Newtons.
\item The coefficient of drag the boat experiences as it travels through water is $c = 25$ N/(m/s). 
\item  The heading of the boat  is $\alpha = 10^{\circ}$ measured off the horizontal West-East line.
\end{itemize} 

\subsection*{Newton's Second Law of Motion applied to a Free Body Diagram}

Newton's Second Law of Motion says that  for a body of given mass  the mass times the acceleration is equal to the sum of the external forces acting on the body. This is true in both the horizontal ($x$) and vertical ($y$) components of our boat as it crosses the river.  Thus we are looking for something like these two differential equations:

\begin{eqnarray} \label{eq:NSL} \nonumber
m\, x''(t) &=& \sum_{i} F_{x\, i} \\
m\, y''(t) &=& \sum_{i} F_{y\, i}  
\end{eqnarray}
where $F_{x\, i}$ is a force in the $x$ direction acting on the boat and  $F_{y\, i}$ is a force in the $y$ direction acting on the boat. We sum over all the forces acting on the boat in each direction.

 \begin{itemize}
   \item [2)] Draw a Free Body Diagram of the boat, i.e. isolate the boat and consider it as a point mass.  Then, first list all the forces acting on the boat in each of the coordinate directions, $x$ and $y$.  For  example in which direction and in which direction ONLY does the force of the current act on the boat? When does the drag coefficient play a role? 
   \item [3)] Define the magnitude and direction of the forces acting on the free body diagram of the boat and define their functional form in terms of $x$, $y$, and $t$ according to the information we are given about the boat crossing.
   \item [4)] Complete the differential equations of motion, (\ref{eq:NSL}), with each force and its appropriate sign and give initial conditions on position and velocity in each direction, $x$, and $y$.  
   \item [5)] Solve the differential equations you obtain in activity (4) and plot your solution over sufficient time so the boat gets to the other side of the river. Report out the time for the journey and the vertical coordinate of the landing place on the other side of the river. Explain why your answer is reasonable or not.  If not, then  change your model, being sure to check that you have all the forces acting on the boat accounted for and your signs and numerical values are correct. 
   \item [6)] Using your base model from    activities (4)-(5) and your  current velocity profile across the river, $W(x)$, i.e.  the North flow  rate in m/s at distance $x$ m from the West side of the river, increase (and then decrease) the total flow rate say by 1\%, 5\%, 10\%, 20\%. For each of these different flow rates determine the new heading angle, $\alpha$, the boat person needs to use to land at the same spot as was found in activity (5).  Determine a relationship between the percent change in your flow stream and the heading angle, $\alpha$, and then use it to predict $\alpha$ for a given percent increase in the flow stream you had not computed, all to land at the same spot as was found in activity (5). Hint: How do you know how to increase the TOTAL flow rate by a fixed percent? Key word is TOTAL!
\item [7)] Holding all else constant, as prescribed before activity (2), just change the total flow rate by various percentages (increase and decrease) and relate the landing position to the percent change, i.e.  try to establish a functional relationship between the percent increase in the flow rate and the downstream landing coordinate.
\item [8)]  Show that if we multiply the vertical velocity profile across the river, $W(x)$, by $x$ and a constant, say $\beta$, to keep the TOTAL flow rate the same that the flow rate is faster near the East bank and that if we multiply $W(x)$ by $400 -x$ and a constant to keep the TOTAL flow rate the same that the flow rate is faster near the West bank. Do each, solve the resulting differential equations,  and then see how the downstream landing coordinate on the East bank changes. Comment on your find. Incidentally, such flow rate changes across a river occur at turns or bends, but can occur in straight flows due to river bottom configurations. 

 Hint:  In one case you will need to find $\beta$ such that $$\int_0^{400} W(x)\, dx = \beta \int_0^{400}x  \,W(x)\, dx\, . $$
\end{itemize}   

\subsection*{Across the river - straight shot}

Consider the following model which may have been used in the previous section:

\vspace*{-.4 true cm}
\begin{eqnarray}\label{eq:straighta} 
m x''(t) &=& T \cos(\alpha) - c\, x'(t)\, ,\\  
m y''(t) &=& T \sin(\alpha) - c\, y'(t) + c \, W \sin\left(\frac{\pi x(t)}{L}\right)\, .  \label{eq:straightb} 
\end{eqnarray}

\begin{itemize}
  \item [9)] Use almost the same numbers used above in model.
  \begin{itemize}
\item [$\bullet$] The mass of the boat is $m = 225$ kilograms.  
\item  [$\bullet$] The constant thrust of the boat's engine is $T = 65$ Newtons.
\item    [$\bullet$] The coefficient of drag the boat experiences as it travels through water is $c = 25$ N/(m/s). 
\item    [$\bullet$] The heading of the boat  is $\alpha$ measured off the horizontal West-East line, but is to be determined.
\item  [$\bullet$] The maximum speed of the current $W = 0.9$ m/s. 
\end{itemize} 
Solve the equation for $x(t)$ in  (\ref{eq:straighta}), replace $x'(t)$ in the equation for $y(t)$ in (\ref{eq:straightb}) and then proceed to determine the heading $\alpha$ which will assure a ``straight shot'' crossing, i.e., $y(tL) = 0$ at the time of landing, $tL$, on the opposite shore of the river. Confirm your result.
\end{itemize}  


\section*{NOTES FOR TEACHERS}

The source idea for the problem of a river crossing is \cite[p.~170, Exercise 7-15]{McClamroch1980} However, the river crossing notion is something students see in physics courses and do in many variation, although usually  in a simpler approach of obtaining resultant vectors in an introductory physics.

This is an opportunity for students to build a set of differential equations as a model from first principles and then put the model to use to determine some additional information; information, which should have an intuitive meaning in the context of the physical situation. 

An essential step in building the model is to use a Free Body Diagram in which all the forces acting on the boat (broken into horizontal - East-West components and vertical - North-South components) are ascertained so that Newton's Second Law of Motion may be applied.  If students have had some physics or mechanics experiences this should be an experience they have had, albeit, as with all student experience they might need some assistance and review of these skills. If students have not seen this approach for model building it is time they did see it,  as many applications of differential equations in the physical world  benefit from such an approach. We have found that it is easy to ``tell the story'' of this standard Newtonian mechanics approach several times in our course and students eventually do absorb the method. Always, the accounting involved in establishing direction and summing forces needs detailed attention.
 
We have made a number of  comments on solution strategies in  a {\it Mathematica\/} file, 6-12-T-Mma-RiverCrossing-TeacherVersion.nb (and .pdf). Of course, any computer algebra system is appropriate and the notions and commands we offer, while specific to {\it Mathematica\/} in our file, are fully notated  and may be reproduced in other systems.  We have some referee comments that could be helpful.

On Activity (5) one referees says, as to the question ``What is reasonable?'' This is physics and there is a precise answer based on the definition and constraints put on the problem posed. Suggest questions like ``given the physics where do you expect the boat to arrive.''  This point here is critical and the students need to make assessments like the following.  They need to understand the importance, direction  and magnitude of each of  the forces acting on the vehicle and is this properly reflected in their  mathematical model. What are the magnitude of the coefficients in the equations and the relative importance of each of the terms? Recall that  $\alpha$ is small and can be approximated.  The students should have to look at the differential equations and glean things like the thrust to mass ratio, the roll of the drag (e.g., for a thrust of 65N and a drag coefficient of 25N  at a velocity of less than 3 m/s the drag force equals the thrust). $\alpha$ is a small angle so its cosine is about one and its sine is about  zero.  So the $x$ progress and velocity is controlled by the drag and the $y$ position and velocity are basically a result of the vertical boat velocity and current velocity. If then they look at the coefficients in the $y$ equation and the average of a $\sin$ function they can even glean an estimate the relative impact  boat velocity vs. that of the current. This can be made easier for them to analyze if they are asked to recast the equations in $V_x = x'(t)$ and $V_y = y'(t)$ as an alternative perspective. This will allow them to make estimates without actually even resorting to a solution of the differential equation. By they way, there is an analytical solution to the $V_x$ form of the equation. 

\begin{thebibliography}{99}


\bibitem{McClamroch1980} McClamroch, N. H. 1980.  {\it  State Models of  Dynamic Systems:  A Case Study Approach\/}. New York:  Springer-Verlag.  Exercise 7-15 on page 170 and text on p.~120-121.



\end{thebibliography}









\end{document}
 
