\section{MC-11--Elastic and Inelastic Collisions}
 
\subsection{MC-11a~~Collisions Between Rolling Carts}
 
\noindent {NOTE TO INSTRUCTORS:} 
\begin{enumerate}
\item[] This lab uses PASCO carts and avoids many of the problems of the air track. The older version of this lab that uses gliders on the air track can be found in MC-14b. The three procedures outlined below will certainly take more than the three hours allotted. Please do procedure I, then as many 
of the others as time allows.
\end{enumerate}

\noindent {OBJECTIVE:} 
\begin{enumerate}
\item[] In this experiment you will observe elastic and inelastic collisions between two PASCO carts;
the carts are provided with Velcro bumpers on one end, and magnetic
bumpers on the other.
\end{enumerate}


\noindent {THEORY:} 
\begin{enumerate}
\item[] Collisions are a common experience in our lives--the collisions of billiard balls, the collisions of two football players, the collisions of cars.
Various `natural philosophers' from Galileo onwards discussed the laws that 
govern such collisions. As early as the XVII$^{th}$ century it was understood
that collisions between hard bodies, like billiard balls, behaved differently
from collisions like car wrecks, where the two masses stick together after
the collision.  The concept of Momentum was first introduced in mechanics by
Descartes as $momentum = mass \cdot velocity$; he did not get it quite right
however, because he thought of momentum as a {\bf{scalar}} quantity. 
It was Leibnitz that defined momentum as a {\bf{vector}} quantity:
 $$\vec p = m \cdot \vec v.$$ 
\noindent Newton's 2$^{nd}$ Law $\vec F = m \vec a$ can be rewritten
$$\vec F = m {{\Delta \vec v} \over {\Delta t}};$$
 \noindent now $m \Delta \vec v$ is the change in the quantity 
$\vec p = m \vec v$, so one can finally rewrite Newton's II$^{nd}$ law as
 $$\vec F = {{\Delta \vec p} \over {\Delta t}}$$ or
$$\vec F \Delta t= \Delta \vec p. $$
\noindent The quantity $\vec F \Delta t$ is called the {\bf{impulse}} and is equal to
the change in momentum.
\end{enumerate}


\noindent {FUNDAMENTAL CONCEPTS:}
\begin{enumerate}
\item[] A physical quantity is said to be conserved in a process if its
value does not change even though other quantities are changing.
An example of a conserved quantity is energy.  
When a body falls freely its total energy (potential energy plus kinetic 
energy) remains unchanged while the position and velocity of the body 
change with time. 

Linear momentum is conserved if the net force acting on an object is zero.  This follows from the equation which relates the change in momentum to the impulse given to the object.  Clearly if the force $F$ is zero, the impulse is zero and the change in momentum is zero, hence the momentum remains constant.
\end{enumerate}

\begin{enumerate}
\item[]This same principle becomes more useful when it is applied to an {\it {isolated system}} of objects. An isolated system is one where the only forces acting on an object {\it {in}} the system are due to the interaction with another object {\it {within}} the same system: there are no {\it {external}} forces. 


A simple such system may consist of two particles, let us call them A and B, interacting with each other. The force $\vec F_A$ on particle A is equal and opposite to the force $\vec F_B$ acting on particle B: \ \  $\vec F_A = - \vec F_B$.  Because the times during which the forces act are the same, it follows that the changes in momentum of the two particles are also equal and  opposite, so that the total change in momentum is zero.  The conservation of momentum is therefore a consequence of Newton's III$^{rd}$ Law.  In collisions, provided there is no net external force on any of the bodies, the sum of the initial momenta equals the sum of the final momenta:
$$ \vec p_1 +\vec p_2 = \vec {p_1}' +\vec {p_2}' $$ 
\noindent where the unprimed quantities refer to the velocities before the collision, and the primed quantities refer to the velocities after the collision.

 
For  one dimensional processes the physical quantity 
that is conserved is linear momentum.  

A collision is called totally inelastic if the two bodies stick together
after colliding. The conservation of momentum for a {\it {one}} {\it {dimensional}} totally inelastic collision is then:
$$ m_1v_1 + m_2v_2 = (m_1 + m_2)\cdot v'.$$
\noindent Energy is not conserved in inelastic collisions.

A collision of two bodies is called totally elastic if energy is conserved in the process. In this case the result of the collision in one dimension can be calculated by $$ (v_1-v_2) = -(v'_1-v'_2)'. $$
\noindent It is interesting to note that $(v_1-v_2)$  and $(v'_1-v'_2)$ are 
the relative velocities of body \#~1 relative to body \#`2 before and after the collision.
\latex{\vspace*{-.05in}}\end{enumerate}


\noindent {APPARATUS:} 
\begin{enumerate}\setlength{\itemsep}{-1pt}
\item \latex{\vspace*{-.05in}}Personal Computer with monitor, keyboard, and mouse
\item \latex{\vspace*{-.05in}}PASCO dynamic track with magnetic bumpers
\item \latex{\vspace*{-.05in}}PASCO signal interface
\item \latex{\vspace*{-.05in}}Two PASCO carts with a ``picket fence''  
\item \latex{\vspace*{-.05in}}A set of 500 g masses
\item \latex{\vspace*{-.05in}}One meter long ruler
\item \latex{\vspace*{-.05in}}Two photogates (plugged into DIGITAL channels \#1 and \#2): these devices consist of 
\subitem \latex{\vspace*{-.05in}}(a) a source which produces a narrow beam of infrared radiation
\subitem{\latex{\vspace*{-.05in}}(b) an infrared detector at the other side of the gate that senses the radiation.}
\item[] \latex{\vspace*{.05in}}When the beam between the source and detector is blocked, a red Light Emitting Diode (LED) on the top of the gate lights up; concurrently an electrical signal is sent to the Signal Interface which converts the time during which the beam was blocked into the velocity of the fence.  NOTE: for the photogates to work properly, the picket fence must be on the opposite side of the cart closest to the LED.
\end{enumerate} 

\begin{rawhtml}
<HR> <P>
    <a name="Demo"><h3><u> <font color="ff0000">Pre-lab Quiz</font> </u></h3></a>
     <blockquote>You should be able to complete this brief quiz before proceeding.
     </blockquote>

<ul><ul>
    <li><a
    target="_blank" href="../js/q207_m14.html">Expt. M14 Quiz</a>
</ul></ul>
</P>
<HR>
\end{rawhtml}


\noindent {SUGGESTIONS:} 
\begin{enumerate}
\item \latex{\vspace*{-.05in}}Measure the mass of the empty carts using the pan balance; MAKE SURE YOU PUT THE CART ON ITS SIDE ON THE PAN. Record these masses in your lab notebook. Note that the cart with the magnet bumper is somewhat heavier than the cart without  magnet, you may wish to tape some masses on the lighter cart to equalize the weights.
\item \latex{\vspace*{-.05in}}Initiate the PASCO interface software in the usual way. The monitor should now look as shown in Figure 1.
\end{enumerate}
\latex{\vspace*{-.05in}}
\begin{figure}[h]
\centering
\includegraphics[width=7.5cm]{figs/m14_pasco.eps}
%\includegraphics[width=8cm]{figs/fmc8-1.eps}
\caption{The Pasco DataStudio display for M14.}
\end{figure}
\latex{\vspace*{-.15in}}
\begin{enumerate}
\item[] \latex{\vspace*{-.05in}}On the computer monitor you see a table of velocities for each of the two photogates. As seen by a person looking at the computer monitor the photogate on the left is \#~1 and the one on the
right is \#~2. 

\item[3.]  \latex{\vspace*{-.05in}}Level the track so the carts do not move to the right or the left. 
\item[4.]  \latex{\vspace*{-.05in}}Place the photogates 15 cm apart. 
\end{enumerate}

\input{toolbar/m14_tb1}

{\vspace*{-.05in}}
\noindent {\bf{PROCEDURE~I - INELASTIC COLLISION - EQUAL MASSES}}
\vspace*{-.05in}
\begin{enumerate}
\item {\vspace*{-.05in}}Prepare a table like the one below:
\end{enumerate}
\begin{table}[!h]
\begin{center}
\begin{tabular}{|ccc|}
\hline Run \#  & Velocity \# 1 (m/s) & Velocity \#1 (m/s) \\ 
\hline & ~~  & ~~   \\
1 & ~~ & ~~   \\
2 & ~~ & ~~   \\
3 & ~~ & ~~   \\ 
\hline
\end{tabular}
\caption{Sample layout for inelastic collison data.}
\end{center}
\end{table}

\begin{enumerate} 
\item[2.] Install a picket fence on one cart (on the side opposite the LED). Make sure that it is the longest black stripe on the fence that intercepts the LED beam. Put the cart at the left end of the track. Place the othercart just to the right of photogate \#2 as shown below. Check that the carts do not repel each other; the Velcro hooks and loops must be able to stick together easily. 
\end{enumerate}

\begin{figure}[h]
\centering
\includegraphics[width=8cm]{figs/fmc8-2.eps}
\caption{}
\end{figure}

\begin{enumerate}
\item[3.] CLK on REC. Gently push the projectile cart to 
the right toward the first
photogate and the target cart. Catch the carts before they hit the end of the
track. CLK on STOP. Record the velocity of the projectile cart 
before and after the collision in Table 1.
\item[4.] Take two  more runs.
\item[5.] ANALYSIS OF THE DATA.  
\subitem Prepare tables as shown below in Table 2 and 3. 
\begin{table}[!h] 
\centering
\begin{tabular}{cccc}
\hline & Car 1 & Cars 1+2 & \\
Run \#  &  $p_{ini}$ (kg m/s) & $p_{final}$ (kg m/s) & $100 \times \Delta p/p_{ini}$ \\ 
\hline \\
1 & ~~ & ~~ & ~~  \\
2 & ~~ & ~~ & ~~  \\
3 & ~~ & ~~ & ~~  \\ 
\\ \hline
Avg\\
\hline
\end{tabular}
\caption{Momentum}
\end{table}
\begin{table}[!h] \centering
\begin{tabular}{cccc}
\hline & Car 1 & Cars 1+2 & \\
Run \#  & $E_{ini}$ (J) & $E_{final}$ (J) & $100 \times \Delta E/E_ini$ \\ 
\hline \\
1 & ~~ & ~~ & ~~  \\
2 & ~~ & ~~ & ~~  \\
3 & ~~ & ~~ & ~~  \\ 
\\ \hline
Avg\\
\hline
\end{tabular}
\caption{Energy}
\end{table}
\subitem (a) Calculate the initial and final momenta and enter them in Table
2. 
\subitem (b) Calculate the initial and final kinetic energies and enter them 
in Table 3
\subitem[(c)] Calculate and record in Table 2 the percent difference in the momenta: \\
$100 \times (p_{ini} - p_{fin})/p_{ini}$. Calculate the average percent difference in the momentum.
\subitem[(d)] Calculate and record in Table 3 the percent difference in energy: \\
$100 \times (E_{ini} - E_{fin})/E_{ini}$.  Calculate the average percent difference in the energy.
\item QUESTIONS: 
\subitem A. Give a likely physical cause for the observed difference between initial and
final momentum.
\subitem B. What fraction of the energy was dissipated? 
\subitem C. If you could remove the physical cause you listed, what fraction of
energy would you expect to find dissipated?

\end{enumerate}


\noindent {\bf{PROCEDURE~II - ELASTIC COLLISION - EQUAL MASSES}}\\
In this experiment you will be measuring the velocities of two 
bodies with equal masses, before and after a totally elastic collision. 
\begin{enumerate}
\item  Place the photogates about 35 cm apart with one cart just to the left of  photogate \#~2. {\em Make sure the carts will repel each other}. (One cart has one end without any magnets-that end will not repel the other cart.) Place the other cart on the left end of the track. as shown in Figure~3 below. Cart \#~2 should be close to gate \#~2, (not as shown in the figure). Make sure that there is enough space, so that cart \#~2 starts moving
{\bf{after}} cart \#~1 has passed completely through gate \#~1.
\begin{figure}[!h]
\centering
\includegraphics[width=10cm]{figs/fmc8-3.eps}
\caption{}
\end{figure}
\item Prepare a table like Table 4.
\begin{table}[!h] \centering 
\begin{tabular}{ccc}
\hline Run \#  & Velocity \# 1 (m/s) & Velocity \#1 (m/s) \\ 
\hline & ~~ & ~~  \\
1 & ~~ & ~~   \\
2 & ~~ & ~~   \\
3  & ~~ & ~~  \\ 
\\ \hline
\end{tabular}
\caption{}
\end{table}
\item  CLICK on the START icons. Gently push the projectile cart (the one 
on the left) to the right toward the first photogate and the target cart. The projectile cart should stop between the gates and the target will move through photogate \# 2. Catch it before it has time to hit the end of the track; CLK on STOP. Record the velocities in Table 4.  Take two more runs. 
\item  ANALYSIS OF THE DATA
\subitem  Prepare  tables like Tables 5 and 6.
\begin{table}[!h] \centering
\begin{tabular}{cccc}
\hline & Car 1 & Cars 1+2 & \\
Run \# &  $p_{1}$ (kg m/s) & $p_{2}$ (kg m/s) & $100 \times \Delta p/p_{1}$ \\ 
\hline \\
1  & ~~ & ~~ & ~~ \\
2  & ~~ & ~~ & ~~ \\
3  & ~~ & ~~ & ~~ \\ 
\\ \hline
Avg\\
\hline
\end{tabular}
\caption{Momentum}
\end{table}
\begin{table}[!h] \centering
\begin{tabular}{cccc}
\hline & Car 1 & Cars 1+2 & \\
Run \#  & $E_{1}$ (J) & $E_{2}$ (J) & $100 \times \Delta E/E_{1}$ \\ 
\hline \\
1  & ~~ & ~~ & ~~ \\
2  & ~~ & ~~ & ~~ \\
3  & ~~ & ~~ & ~~ \\ 
\\ \hline
Avg\\
\hline
\end{tabular}
\caption{Energy}
\end{table}
\subitem A. Calculate the initial momentum  of Car \#~1 and final momentum
of Car \#~2 and enter them in Table 5. 
\subitem B. Calculate and record the percent difference of the momenta: 
$100 \times (p_{1} - p_{2})/p_{1}$.  
\subitem C. Calculate the average percent difference in the momentum.
\subitem D. Calculate the initial and final kinetic energies and enter them 
in Table 6. 
\subitem E. Calculate
and record the percent difference of the energies: 
$100 \times (E_{1} - E_{2})/E_{1}$. 

\item QUESTIONS 
\subitem What is a reason for the observed difference between $p_1$ and
$p_2$?
\subitem What fraction of 
the energy was dissipated? 
\subitem What fraction of 
the energy should have been dissipated in the absence of friction? 
\end{enumerate}


\noindent{\bf{PROCEDURE~III - ELASTIC COLLISION - UNEQUAL MASSES}} 
\begin{enumerate}
\item The photogates should be about 35 cm apart. Find and record the masses of  two 500g masses using the triple beam balance; place them on the target cart, place this cart just to the left of photogate \# 2 as shown in Figure~3 (not as shown in the figure) . The carts should be placed so they will 
repel each other.  The lighter cart is `the projectile' and should be placed 
on the left end of the track. 
\item Prepare a table like that of Table 7.
\begin{table}[!h] \centering
\begin{tabular}{cccc}
\hline Run \#  &  $V_{1ini}$ (m/s) & $V_{1fin}$ (m/s) & $V_{2fin}$ (m/s) \\ 
\hline \\
1 & ~~ & ~~ & ~~ \\
2 & ~~ & ~~ & ~~  \\
3  & ~~ & ~~ & ~~ \\ 
\hline
\end{tabular}
\caption{}
\end{table}
\item  CLICK on the START icon. Gently push the projectile cart 
(the lighter one on the left) to the right toward the first photogate and the target cart. Try to predict what will happen. Catch the carts before they hit the end of the track. CLICK on STOP. Record the velocities in Table~7
\item Take two more runs. 
\item ANALYSIS OF THE DATA
\subitem Prepare tables like Tables~8 and 9.
\begin{table}[!h] \centering 
\begin{tabular}{ccccc}
\hline & Car 1 ini & Car 1 fin & Car 2  \\
Run \#  &  $p_{1ini}$ (kg m/s) & $p_{1fin}$ (kg m/s) & $p_{2fin}$(kg m/s) & $100 \times \Delta p/p_{1ini}$ \\ 
\hline \\
1 \\
2 \\
3 \\ 
\\ \hline
Avg\\
\hline
\end{tabular}
\caption{Momentum}
\end{table}
\begin{table}[!h] \centering
\begin{tabular}{ccccc}
\hline & Car 1 ini & Car 1 fin  & Car 2 \\
Run \#  & $E_{1ini}$ (J) & $E_{1fin}$ (J) & $E_{fin}$ (J) & $100 \times \Delta E/E_{1ini}$ \\ 
\hline \\
1  & ~~ & ~~ & ~~ \\
2  & ~~ & ~~ & ~~ \\
3  & ~~ & ~~ & ~~ \\ 
\\ \hline
Avg\\
\hline
\end{tabular}
\caption{Energy}
\end{table}
\subitem Calculate the initial and final momentum of each cart 
and enter them in Table~8. Calculate the total momentum of the system $p_{fin}$ after the collision and record it in Table~8. Remember that the final momentum is the sum of the momenta of carts \#~1 and \#~2 after the collision.
\subitem Calculate the initial and final kinetic energies and enter them 
in Table 9. Calculate the total energy of the system $E_{fin}$ after the 
collision and record it in Table 9. Remember that the final energy
is the sum of the energies of carts \# 1 and \# 2 after the collision.

\subitem  Calculate and record the percent difference $100 \times (p_{1ini} - p_{fin})/p_{1ini}$. 
\subitem Calculate and record the percent difference  $100 \times (E_{1ini} - E_{fin})/E_{1ini}$.

\item QUESTIONS: 
\subitem What is a reason for the observed difference?
\subitem Was momentum approximately conserved?
\subitem What fraction of the energy was dissipated? 
\subitem What fraction of the energy should have been dissipated in the absence of friction? 
\end{enumerate}

\noindent {JAVA APPLET:} 
\begin{enumerate} 
\item[] If time permits and you are interested the web version of the lab has
a link to an applet which animates elastic collisions of two point masses.
\end{enumerate}



\begin{rawhtml}
<img border=2 src="../gif/exclaim.gif" align="left"><font size=+2 color="ff0000"> Interesting resources on the web </font>
<table cellpadding=5>
  <tr>
    <td valign=top>
This  <a href="../java/collisionapplet.html" target="_blank"> 2D collision applet </a>
demonstrates <i>conservation of energy and momentum</i>.
<p> 
The applet, courtesy of <a href="http://galileo.phys.virginia.edu/classes/109N/more_stuff/Applets/Collision/applet.html" target="_blank"> Prof. Fowler </a>, animates elastic collisions of two masses. The user enters the ratio of the target mass to the projectile mass and the impact parameter. The applet animates the collision in the target rest frame and, simultaneously, tracks the center of mass frame. 
</td> 
</tr>
</table>


\end{rawhtml}


