\input{labpreamble}
\begin{document}
\subsubsection*{PHSX 101L\hfill Experiment 7\hfill Name:\underline{\hspace*{2in}}}

\underline{Title:} Velocity Measurement Using a Ballistic Pendulum

\underline{Object:} To determine the velocity of a 
projectile by two independent methods: (1) using a ballistic pendulum, 
and (2) measuring a trajectory; and then to compare the results.

\underline{Theory:} When a projectile (of mass $m_{b}$) is 
fired into a ballistic pendulum (of mass $m_{c}$) the impact causes 
the pendulum and embedded projectile (total mass $m_{b}+m_{c}$) to 
swing up to a maximum height $h$.  Kinetic energy is not conserved in 
the inelastic collision, but momentum is.  And after the collision 
mechanical energy is conserved as the kinetic energy of the pendulum 
immediately after the impact is changed completely into potential 
energy at the maximum height.  These conservation laws can be used to 
derive an expression for the initial velocity $v_{o}$ of the 
projectile.

Conservation of momentum \emph{during} the collision gives:
\begin{equation}\label{consmom}
m_{b}v_{o}=(m_{b}+m_{c})v
\end{equation}
where $m_{b}$ is the mass of the ball, $v_{o}$ is the muzzle velocity 
of the ball, $m_{c}$ is the mass of the catcher, and $v$ is the 
velocity of the catcher (and ball) after collision.

Conservation of mechanical energy {\em after} the collision gives:
\begin{equation}\label{consenerg}
\frac{1}{2}(m_{b}+m_{c})v^{2} = (m_{b}+m_{c})gh.
\end{equation}
Where $g$ is the acceleration of objects in free-fall (which we 
called $a_{g}$ in a previous lab) and $h$ is height 
above the starting point.
Solving (\ref{consenerg}) for $v$ we get:
\begin{equation}\label{vf}
 v = \sqrt{2gh}.
\end{equation}

\vspace{-.2in}
Substitute (\ref{vf}) into (\ref{consmom}):
\begin{equation}
m_{b}v_{o}=(m_{b}+m_{c})\sqrt{2gh}
\end{equation}
\begin{equation}\label{answer}
v_{o}=\left({m_{b}+m_{c}\over m_{b}}\right)\sqrt{2gh}.
\end{equation}

\vspace{-.1in}
The velocity of a projectile can also be determined from measurements 
made on its free-flight trajectory fired horizontally.  In these formulae $x$ 
is the horizontal range of the projectile, and $y$ is the vertical 
distance through which it falls, and $t$ is the time of flight (which 
is eliminated in equation \ref{answer2}).
\vspace{-.1in}
\begin{eqnarray*}
x = v_{o}t & \hspace{1in} & y = \frac{1}{2}gt^{2}\\
v_{o} = x/t & \hspace{1in} &  t = \sqrt{{2y\over g}}
\end{eqnarray*}
\begin{equation}\label{answer2}
v_{o}=\sqrt{{gx^{2}\over 2y}}
\end{equation}

\goodbreak


\underline{Apparatus:} \ \ Make a drawing of the apparatus used.
\vspace{1.7in}

\underline{Procedure and Results:} 
\vspace{-.15in}

\hspace*{.4in}For the pendulum method of determining $v_{o}$:
\begin{enumerate}
\vspace{-.15in}

\item Determine mass of ball $m_{b}$:\hfill 
\underline{\hspace{1.2in}}

\item Determine mass of catcher $m_{c}$:\hfill 
\underline{\hspace{1.2in}}

\item Fire the projectile into the catcher and measure the rise $h$:\hfill 
\underline{\hspace{1.2in}}

\item Calculate $v_{o}$ from equation \ref{answer}:\hfill 
\underline{\hspace{1.2in}}

\vspace{.3in}
For the projectile method of determining $v_{o}$:

\item Measure vertical distance of fall $y$:\hfill 
\underline{\hspace{1.2in}}

\item Fire the projectile ten times and find the average horizontal 
range $x$:\hfill 
\underline{\hspace{1.2in}}

\item Use equation \ref{answer2} from trajectory theory to calculate 
$v_{o}$:\hfill 
\underline{\hspace{1.2in}}

\vspace{.2in}
\item Percent difference between the velocities determined by 
the two methods:\hfill 
\underline{\hspace{1.2in}}

\end{enumerate}

\underline{Questions:} 
\vspace{-.25in}
\begin{enumerate}

\item Do the two methods of determining velocity agree within 
expected error limits?  Explain.
\vspace{.5in}

\item Which method do you consider to be most reliable?  Why?
\vspace{.5in}

\end{enumerate}

\noindent
\underline{Conclusions:} 


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