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\begin{document}

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\title{\ifnum\TeacherEdition=1 TEACHER VERSION  \else STUDENT VERSION \fi\\BALL BOUNCE OFF INCLINED RAMP}





\markboth{Ball Bounce off Inclined Ramp}{Ball Bounce off Inclined Ramp}






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

\makeStitlePDFLaTex



\begin{abstract}
Students build two projectile motion models (1) a one-dimensional model for a vertically falling ball from a fixed distance until it hits an inclined ramp and (2) a two-dimensional projectile motion model of the ball bouncing off the ramp. The purpose of the activity is to determine the angle of the ramp which  permits the ball to bounce OUT the
farthest. 
\end{abstract}


\section*{SCENARIO DESCRIPTION}


We drop a ball onto an inclined plane (ramp) of slope $m$ and watch it bounce.  Suppose the coefficient of restitution is 0.5,  applied only to the component of the velocity which is perpendicular to the plane of the ramp.  We wonder what angle the ramp should be to give  the bounce the maximum horizontal distance along the level ground.

We keep the action in a vertical plane in which the first coordinate ($x$) represents horizontal distance of the ball in meters from the pivot point of the flat inclined ramp (along the direction of the ramp) and the second coordinate ($y$) represents the height of the ball in meters. Place the ball at position $(10, 40)$ with  the ramp's base at $(0, 0)$ and assume the slope of the ramp is a positive value, $m$.  Thus the ramp has equation $y = m x$. For  a drop from a height of 40 meters above the ground (not necessarily above the ramp though) we seek   the slope $m$ of the ramp which will permit the ball to bounce farthest along the ground, i.e. maximize horizontal range.

\teacher{
\section*{NOTES FOR TEACHERS}


We offer  a complete analysis in a Mathematica notebook, 3-045-T-Mma-RampBounce.nb and an accompanying pdf file for those who cannot read Mathematica files.

The basic idea is to use a one-dimensional projectile motion model for the falling ball until it hits the ramp, then use the coefficient of restitution (i.e. the percent of the impact velocity at which the ball leaves the ramp following impact). Then a two-dimensional projectile motion model is used to give the trajectory of the ball after impact and until it strikes the ground. All of this is in terms of the slope, $m$, of the ramp. Finally, the point at which the ball strikes the ground is computed in terms of $m$ and optimized as a function of $m$. The optimal slope  for the parameters in this model is 0.399503 and this converts to an angle of 21.78 degrees.



\subsection*{ISSUES RELATED TO THE PROBLEM}

\subsubsection*{Prerequisites}

Students need to know about one and two-dimensional projectile motion (without resistance) in order to model the falling ball and the bouncing ball.  They need to know how to determine the point of intersection of two curves.  They further need to know about projections of vectors in order to take the component of the bounce which is perpendicular to the plane of the ramp. Finally, students may have to look up the meaning and significance of the coefficient of restitution.

\subsubsection*{Time allotment - time management}

Introducing this problem and giving student class time in small groups will get them going in 15 minutes and then give them two nights over which they can work on the problem and reflect on it as well as do their write-up for class submission.   Pieces include one dimensional projectile motion, projection of component vectors,  and finally two-dimensional projectile motion.

\subsubsection*{Expectations} 

We would expect the students to put these pieces together and to experiment with a graphical solution to affirm their intuitions.  We see them having trouble with the projection vectors, specifically with regard to applying the coefficient of restitution correctly to only the component of the velocity which is perpendicular to the ramp.

\subsubsection*{Future payoffs}

The ability to organize the information in this problem and to apply several different principals and two differential equations for projectile motion in one context will give the students more confidence when they see complex or real problems.  Further the interdependence of one component of the problem with another will show the students the importance of consistency, good notation, building a model of a complex situation, and continuity.

\subsubsection*{Extensions}

\begin{itemize}
\item [] Another variation on this scenario is to always insist that the ball falls 40 meters BEFORE it hits the ramp.  Do not always start the falling ball at $(10, 40)$ but start it at 40 meters above the point on the ramp which is under it, i.e. start the falling ball  at position 
$(10, m 10 + 40)$.

\item [] We could seek to ascertain what angle will give the ball the maximum speed when it strikes the ground;  maximum magnitude of velocity.
\end{itemize}

\subsubsection*{References and Sources}

The basic idea for this problem comes from the text {\it Vector Mechanics for Engineers - Statics and Dynamics, Fifth Edition\/} by F. P. Beer and E. R. Johnston, Jr.  McGraw-Hill:  New York, 1988.  Problem 13.151 on page 638.



\section*{ACKNOWLEDGEMENT}


The production of this material was supported by the National Science Foundation under Division of Undergraduate Education grant DUE-9352849 from 1993-1996 entitled Development Site for Complex, Technology-Based Problems in Calculus with Applications in Science and Engineering and the Arvin Foundation of Columbus IN. This material was produced by Brian Winkel while he was on the faculty of  Department of Mathematics, 
Rose-Hulman Institute of Technology, Terre Haute IN USA under the original full title, {\it Maximum Horizontal Bounce Distance  of Ball Bouncing On Inclined Ramp\/}.  
}



\end{document}
