\section{M-5~~Projectile Motion}

\noindent OBJECTIVE:  To find the initial velocity and predict the range of a 
projectile. \\ 
\noindent APPARATUS:
\begin{enumerate}
\item[] \latex{\vspace*{-.05in}}Ballistic pendulum with spring gun and plumb bob, projectile, single pan balance, elevation stand.
\end{enumerate}\latex{\vspace*{-.05in}}

\noindent PART I.  BALLISTIC PENDULUM  \vspace*{-.1in}
\begin{figure}[h]
%%\begin{center}
%\begin{minipage}[b]{0.6\linewidth} 
\centering
\includegraphics[height=2.70in]{figs/l103/m05-3rev.eps}
\caption{The spring gun.} \vspace*{.1in}  
%\end{minipage} 
%\begin{minipage}[b]{0.4\linewidth}
%%\end{center}
%%\end{figure} \\
%%\begin{figure}[h]
%%\begin{center}
\centering\includegraphics[height=1.70in]{figs/l103/m05-r4rev.eps}
\caption{A side view of the catcher} %%\end{center}
%\end{minipage}
\end{figure} \vspace*{-.1in} 
\noindent INTRODUCTION: 
\begin{enumerate}
\item[] \latex{\vspace*{-.05in}}
A properly aligned spring gun shoots a ball of mass $m$ into a pendulum catcher of mass $M$.  See Figs.~1 and 2.  The pendulum traps the ball; thereafter the two move together in {\bf pure translation}. Since ball and 
pendulum have no relative motion, the collision is inelastic and thus does not conserve mechanical energy (where does it go?).  Of course linear momentum is still conserved, and hence momentum of ball before impact, $mu$, equals the momentum of ball plus pendulum after impact,
$(m+M)V$: 
\begin{equation} mu=(m+M)V \end{equation} 
where $u$ = ball's velocity before impact and $V$ = initial velocity of combined pendulum plus ball.  \\

To find $V$, note that although the impact doesn't conserve mechanical energy, the motion {\bf after} impact is almost frictionless and thus conserves mechanical energy. Hence the kinetic energy of the ball plus pendulum at $A$ in Fig. 2, just after impact, equals the potential energy of the two at the top of  the swing (at $B$).  Thus 
\begin{equation}
\frac{1}{2}(M+m)V^{2}=(M+m)gh~. ~~~~\mbox{ Hence,  } ~~~~V=\sqrt{2gh}~. \end{equation}
\end{enumerate}
(An on-line prelab quiz is available if you wish to test your knowledge.)

\noindent  %\newpage
\begin{rawhtml}
<HR> <P>
    <a name="Demo"><h3><u> Pre-lab Quiz </u></h3></a>
     <blockquote> If you are familiar with conservation of energy and momentum then you should be able to complete this brief quiz before pro ceding. (You may skip this part if you haven't covered these topics in class as yet. <br>
     <font color="ff0000">Read the Introduction section carefully, especially if
     you are unfamiliar with the concepts of conservation of energy and momentum. </font>
     </blockquote>

<ul><ul>
    <li><a target="_blank" href="../js/q207_m05.html">Expt. M05 Quiz</a>

</ul></ul>
</P>
<HR>
\end{rawhtml}

ALIGNMENT:
\begin{enumerate}
\item[] \latex{\vspace*{-.05in}}If properly aligned, our bifilar type 
of suspension for the pendulum (see Figs.~1 and 2) prevents rotation of the bob of mass $M$.  Hence the 
motion is pure translation.  To ensure proper alignment, adjust the three knurled screws on the base so that
\begin{enumerate}
\item[A.] \latex{\vspace*{-.05in}}A plumb bob hangs parallel to the vertical axis of the protractor, and
\item[B.] \latex{\vspace*{-.05in}}The uncocked gun axis points along the axis of the cylindrical bob.  (You may need to adjust the lengths of the supporting strings.)
\end{enumerate}  \end{enumerate} \latex{\vspace*{-.05in}}

\noindent SUGGESTED PROCEDURE:~ \medskip \\
\begin{rawhtml}  
<IMG SRC="../gif/caution.gif" ALIGN=center> <br>
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\hspace*{.3in} \framebox[5.2in]{\parbox[b]{5in}{\smallskip {\em PRECAUTIONS:} 
Some spring guns have three different cocking stops.  For consistent results be sure to recock the gun to the same stop.\smallskip \par }}

\begin{enumerate} 
\item \latex{\vspace*{-.05in}}Find $m$ and $M$ by weighing on the single pan balance.  
\item \latex{\vspace*{-.05in}}Find the height $h$
by measuring angle $\theta$ and  the length $L$ of the 
pendulum.\\  {\em NOTE:} $h=L-L\cos\theta$ and $L$ is not the length of the 
string, (see Fig. 2). \\
\latex{\hspace*{.3in}\parbox{4.5in}}{
Suggestions on finding $\theta$:  Find an approximate $\theta$, and 
then make a masking tape slit on the back of the protractor at this approximate $\theta$ so your eye can 
locate the proper viewing area to find a more accurate $\theta$ on subsequent firings.  To avoid parallax (\hyperref{Appendix D}{Appendix }{}{parallax}) in reading the  protractor choose a line of sight determined by the string in front of the protractor and the string in back of the protractor.  Careful observation after a little practice will enable you to get $\theta$ to within a degree.}
\item Calculate the initial velocity $V$ of the combined pendulum bob and ball.
\item Calculate the initial velocity of the ball, $u$, as it leaves the gun.
\item Estimate the uncertainty in $u$.  
[Hint:  Since the largest uncertainty is likely $\Delta V$, then $\Delta h$ is important.  
While $h$ is a function of the measured $L$ and $\theta$, the uncertainty in the angle measurement, $\Delta \theta$, will probably dominate.  To manually calculate the uncertainty in the resulting 
trigonometric function, sometimes it is simpler to calculate separately the function for $\theta+\Delta \theta$ and for $\theta-\Delta \theta$.  Remember to use absolute, not relative errors when propagating errors through
addition \hyperref{error propagation}{(See ``Errors and Uncertainties'' in the
manual}{.)}{error_addition} 
\end{enumerate} 

\input{toolbar/m05_tb1}


%\renewcommand{\theenumii}{\roman{enumii}}
\noindent PART II.  RANGE MEASUREMENTS 
\noindent HORIZONTAL SHOT: 
\begin{enumerate} 

\item After finding $u$, (the velocity of the ball leaving the 
gun) predict the impact point on the floor for the ball when shot horizontally from a position on the table.
\item To check your prediction experimentally:

\begin{figure}[h] 
\centering 
\includegraphics[height=2.3in]{figs/l103/m05-6rev.eps}
\end{figure}


\begin{enumerate} 

\item[(i)] \latex{\vspace*{-.04in}}Use the plumb bob to check that the initial velocity is horizontal.

\item[(ii)] \latex{\vspace*{-.04in}}Measure all distances from where the ball starts free fall (not 
from the cocked position).  All measurements of course refer to the bottom of the ball so $x = 0$ corresponds to the radius of the ball beyond the end of the gun rod.  Check that the gun's recoil does not change $x$.  

\item[(iii)] \latex{\vspace*{-.04in}}Fasten (with masking tape) a piece of computer paper at the 
calculated point of impact, and just beyond the paper place a box to catch the ball on the first bounce.  
\item[(iv)] \latex{\vspace*{-.04in}}Record results of several shots.  (The ball's impact on the paper leaves 
a visible mark.)  Estimate the uncertainty in the observed range.
\item[(v)] \latex{\vspace*{-.04in}}Is the observed range (including uncertainty) within that predicted.  
\end{enumerate} 
\item Work backwards from the observed range to calculate the initial velocity $u$. This $u$ is probably more accurate than the value obtained with the ballistic pendulum.
\end{enumerate}

\noindent ELEVATED SHOT:  
\begin{enumerate}
\item Use the stand provided to elevate the gun at an angle above the horizontal.  The plumb bob will give the angle of elevation, e.g.\ 90$^\circ$ - protractor reading.  For the elevated gun, be sure to include the additional initial height above the floor of the {\bf uncocked} ball.

\item Before actually trying a shot at an angle, again predict the range but use the  value of $u$ which you found from the horizontal shot.  (See item 3 above).

\item Make several shots, record the results and compare with predictions.
\end{enumerate}

\noindent QUESTION:
\begin{enumerate}
\item[] \latex{\vspace*{-.05in}} From the measured values of $u$ and $V$ in
Part I of this experiment, calculate the kinetic 
energy of the ball before impact, $\frac{1}{2}mu^{2}$ and the ball and 
pendulum together after impact, $\frac{1}{2}(m+M)V^{2}$.
What became of the difference?
\end{enumerate}\latex{\vspace*{-.05in}}
%\input{toolbar/m05_tb2}

\noindent {\em OPTIONAL:}
\begin{enumerate}
\item[] \latex{\vspace*{-.15in}}
$$\mbox{1.~~Show that for momentum to be conserved:~~~} \frac{KE_{\mbox{\tiny before impact}}}
{KE_{\mbox{\tiny after impact}}}=\frac{m+M}{m}    \mbox{~~~~~~~~~~~~}$$ 
2. Find the spring constant $k$ of the gun from 
$\frac{1}{2}mu^{2}=\frac{1}{2}kx^{2}$.
\end{enumerate}\latex{\vspace*{-.05in}}



