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{\LARGE The Coefficient of Restitution \\
\bigskip{\it 8.01 ESG Physics Experiment}} \\
\vspace{3in} {\Large Juliet C. Bourne '92 \\
\bigskip \bigskip December, 1988}
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\newpage
{\Large \noindent \underline{Introduction}} \\
\bigskip \\
{\large \it  Purpose:} \\
\medskip \\
1. To observe the elasticity of collisions \\
2. To determine the coefficient of restitution \\
\medskip \\
{\large \it Procedure:} \\

The procedure for this experiment was as follows:

(1) I took a ping-pong ball, which is plastic and hollow,
and a foosball, which is plastic and solid.    

(2) Then I chose three different surfaces on which to bounce the balls.  
The first was a wooden table, which was pretty solid and fairly
smooth. The second surface was the carpeted floor in the ESG commons
room. The carpet is fairly short (maybe an inch). The third surface
was the linoleum (or some such tile material) floor in the
computer/physics room at ESG.

(3) I took a meterstick (well, actually a yardstick since I could not
find a meterstick), and dropped each ball on each surface
approximately ten times each. Since it was very difficult to determine
how high the ball bounced, even with the yardstick directly behind it,
I had an assistant (my tutor) watch how high it bounced and then we
compared how high we each thought it bounced.  When we seemed to be
consistently agreeing with the other's value, we assumed that we were
getting a fairly accurate measurement.  Five of the agreed upon values
were used in the calculations.
\bigskip \\
For the procedure on how the final results were calculated, see the
section entitled \underline{Theory (Basis for Calculations)}.

\newpage
{\Large \noindent \underline{Data and Results}} \\
\noindent
\bigskip \\
ball: \underline{ foosball } \\
surface: \underline{ wood } \\
inital height: \underline{ 36 in } \\
\begin{tabular}{||c|c|c|c||}\hline\hline
Trial& Height${}_f$ (in)& ${}\varepsilon$& Average ${}\varepsilon$ \\ 
\hline\hline
1& 12.000& 0.577&   \\ \cline{1-3}
2& 12.250& 0.583&  \\ \cline{1-3}
3& 12.125& 0.580& 0.580 \\ \cline{1-3}
4& 12.000& 0.577&  \\ \cline{1-3}
5& 12.250& 0.583& \\ \hline\hline
\end{tabular}
\bigskip \\
ball: \underline{ ping-pong } \\
surface: \underline{ wood } \\
inital height: \underline{ 36 in } \\
\begin{tabular}{||c|c|c|c||}\hline\hline
Trial& Height${}_f$ (in)& ${}\varepsilon$& Average ${}\varepsilon$ \\ 
\hline\hline
1& 25.000& 0.833&   \\ \cline{1-3}
2& 25.250& 0.837&  \\ \cline{1-3}
3& 25.125& 0.835& 0.835 \\ \cline{1-3}
4& 25.000& 0.833&  \\ \cline{1-3}
5& 25.250& 0.837& \\ \hline\hline
\end{tabular}
\bigskip \\
ball: \underline{ foosball } \\
surface: \underline{ carpet } \\
inital height: \underline{ 36 in } \\
\begin{tabular}{||c|c|c|c||}\hline\hline
Trial& Height${}_f$ (in)& ${}\varepsilon$& Average ${}\varepsilon$ \\ 
\hline\hline
1& 12.125& 0.580&   \\ \cline{1-3}
2& 12.250& 0.583&  \\ \cline{1-3}
3& 12.125& 0.580& 0.581 \\ \cline{1-3}
4& 12.000& 0.577&  \\ \cline{1-3}
5& 12.375& 0.586& \\ \hline\hline
\end{tabular}

\newpage \noindent
ball: \underline{ ping-pong } \\
surface: \underline{ carpet } \\
inital height: \underline{ 36 in } \\
\begin{tabular}{||c|c|c|c||}\hline\hline
Trial& Height${}_f$ (in)& ${}\varepsilon$& Average ${}\varepsilon$ \\ 
\hline\hline
1& 15.750& 0.661& \\ \cline{1-3}
2& 15.625& 0.659& \\ \cline{1-3}
3& 15.125& 0.648& 0.654 \\ \cline{1-3}
4& 15.250& 0.651& \\ \cline{1-3}
5& 15.250& 0.651& \\ \hline\hline
\end{tabular}
\bigskip \\
ball: \underline{ foosball } \\
surface: \underline{ linoleum } \\
inital height: \underline{ 36 in } \\
\begin{tabular}{||c|c|c|c||}\hline\hline
Trial& Height${}_f$ (in)& ${}\varepsilon$& Average ${}\varepsilon$ \\ 
\hline\hline
1& 19.000& 0.726& \\ \cline{1-3}
2& 19.250& 0.731& \\ \cline{1-3}
3& 18.875& 0.724& 0.727\\ \cline{1-3}
4& 19.000& 0.726& \\ \cline{1-3}
5& 19.125& 0.729& \\ \hline\hline
\end{tabular}
\bigskip \\
ball: \underline{ ping-pong } \\
surface: \underline{ linoleum } \\
inital height: \underline{ 36 in } \\
\begin{tabular}{||c|c|c|c||}\hline\hline
Trial& Height${}_f$ (in)& ${}\varepsilon$& Average ${}\varepsilon$ \\
\hline\hline
1& 26.125& 0.852& \\ \cline{1-3}
2& 26.000& 0.850& \\ \cline{1-3}
3& 25.875& 0.848& 0.850 \\ \cline{1-3}
4& 26.000& 0.850& \\ \cline{1-3}
5& 25.875& 0.848& \\ \hline\hline
\end{tabular}
\medskip \\

\newpage
{\Large \noindent \underline{Theory (Basis for Calculations)}} \\
\bigskip \\

In non-ideal collisions, meaning that they are not perfectly elastic,
energy is lost, or, rather, changed into the energy of the deformation
that occurs in collisions.  In the case of the collision of a ball and
the earth (dropping a ball on the floor, more precisely), this energy
loss is observed in the form of the ball not reaching its original
height after ``colliding'' with the floor.  The coefficient of
restitution ${}\varepsilon$, the fraction of energy that is ``saved''
after the collision, can be deriived from momentum considerations.

After the collision of the ball and the earth (the floor), the ball
noticeablely loses energy since it does not reach its original height.
This indicates a loss in potential energy and therefore a loss in
kinetic energy since the potential at the top of the bounce is
completely converted into kinetic energy at the bottom, right before
it collides with the floor.  By the fact that there is a loss in
kinetic energy it can be deduced that the maximum velocity just after
the collision is less than the maximum velocity just before the
collision.  Therefore, the momentum ($mv_f$) of the ball after the
collision is a fraction of the momentum ($mv_i$) of the ball before
the collision.  The momentum change of the earth is totally
negligible.

\begin{eqnarray}
m_1v_{1i} + m_2v_{2i} &=& m_1v_{1f} + m_2v_{2f} \\
\varepsilon(m_1v_i) &=& m_1v_f \\
\varepsilon &=& \frac{m_1v_f}{m_1v_i} \\
\varepsilon &=& \frac{v_f}{v_i} 
\end{eqnarray}

Through kinetic and potential energy considerations, the velocities of
the ball just before and just after the collision can be determined in
terms of the initial and final height, respectively. 

\newpage
\begin{eqnarray}
\frac{1}{2}mv_i^2 &=& mgh_i \\
v_i^2 &=& 2gh_i \\
v_i &=& \sqrt{2gh_i} \\
\frac{1}{2}mv_f^2 &=& mgh_f \\
v_f^2 &=& 2gh_f \\
v_f &=& \sqrt{2gh_f}
\end{eqnarray}

Therefore, by substituting for ${}v_i$ (equation 7) and ${}v_f$
(equation 10) into equation 4, we get the definition of
${}\varepsilon$.

\begin{eqnarray}
\varepsilon &=& \frac{\sqrt{2gh_f}}{\sqrt{2gh_i}} \\
\varepsilon &=& \sqrt{\frac{h_f}{h_i}}
\end{eqnarray}

On the basis of this equation, ${}\varepsilon$ can be calculated from
the initial height before the collision and the final height after the
collision of the ball. 

\newpage
{\LARGE \noindent \underline{Conclusions (Summary of Results)}}

The values for the coefficient of restitution for the different cases
seem to be sensible results for the type of surfaces and balls
involved in the collisions.  In the case of the first surface, wood,
the results were pretty much as expected. The foosball had a very low
coefficient of restitution; it loses an average of 43 percent of its
velocity after the collision. 

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