\input{labpreamble}
\begin{document}

\subsubsection*{PHSX 221L\hfill  Measuring Velocity\hfill Name:\underline{\hspace*{2in}}}

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

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

\noindent
\underline{Theory:} When a projectile (of mass $m_{1}$) is fired into a ballistic 
pendulum (of mass $m_{2}$) the impact causes the pendulum and embedded 
projectile (total mass $m_{1}+m_{2}$) to swing up to a maximum height.  
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 $v_{1i}$ of the projectile.

\noindent
Conservation of momentum \emph{during} the collision gives:
\begin{equation}\label{consmom}
m_{1}v_{1i}=(m_{1}+m_{2})v_{f}
\end{equation}
where $m_{1}$ is the mass of the ball, $v_{o}$ is the muzzle velocity 
of the ball, $m_{2}$ is the mass of the catcher, and $v$ is the 
velocity of the catcher (and ball) after collision.

\noindent
Conservation of mechanical energy {\em after} the collision gives:
\begin{equation}\label{consenerg}
\frac{1}{2}(m_{1}+m_{2})v_{f}^{2} = (m_{1}+m_{2})gh.
\end{equation}
Where $g$ is the acceleration of objects in free-fall and $h$ is height 
above the starting point.

% \noindent
% Conservation of angular momentum \emph{during} the collision gives:
% \begin{eqnarray}
% \Sigma L_{i} & = & \Sigma L_{f} \\ \label{angmom}
% Rm_{1}v_{1_{i}} & = & I\omega
% \end{eqnarray}
% where $I$ is the moment of inertia and $R$ is the distance from a 
% horizontal axis of rotation of the pendulum to the projectile.
% 
% \noindent
% Conservation of mechanical energy \emph{after} the collision gives:
% \begin{eqnarray}
% \Sigma E_{i} & = & \Sigma E_{f}  \\ \label{mecheng}
% \frac{1}{2}I\omega^{2} & = & (m_{1}+m_{2})gh
% \end{eqnarray}
% where $h=d(1-\cos\theta)$ is the vertical rise of the center of mass of the pendulum 
% (including projectile), and $d$ is the distance from the horizontal 
% axis to the center of mass.

\noindent
Since the collision is inelastic, only a fraction $F$ of the initial 
kinetic energy of the projectile is transferred to the system after 
impact.
\begin{equation}\label{KE}
F\,\left(\frac{1}{2}\right) m_{1}v_{1i}^{2} = \frac{1}{2}(m_{1}+m_{2})v_{f}^{2}
\end{equation}
where $v_{f}$ is the velocity of the combined mass after the collision.  
If conservation of linear momentum is used (equation \ref{consmom}), it can be 
shown that
\begin{equation}\label{KEfrac}
F={m_{1}\over m_{1}+m_{2}}.
\end{equation}

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

\item\label{inertia} Solve (\ref{consenerg}) for $v_{f}$ and then
(\ref{consmom}) for $v_{1i}$ and verify by comparison to Example 9.6 
in Serway.

% \item Show that the expression in procedure \ref{inertia} above 
% reduces to the expression given in Example Problem 9.10 in Serway if 
% $I=(m_{1}+m_{2})R^{2}$.
% 
\item Using equations of motion in both the $x-$ and $y-$direction, 
derive an expression for the initial velocity of the projectile in 
terms of the distance of fall and the horizontal range (and $g$) by 
eliminating $t$ between the equations.

\item Fire the projectile into the ballistic pendulum a number of 
times and calculate the initial projectile velocity from your derived 
equation.

\item Fire the projectile off onto the floor a number of times and 
note the horizontal range and the height from which it was fired.  
Calculate the initial projectile velocity from your derived equation.

\end{enumerate}

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

\item Compare the two results.  (Compute percent difference.)

\item How precise was your data?

\item Note that Equation \ref{KE} may be solved for $F$, thus giving 
the ratio of the two kinetic energies.  Compute the kinetic energies 
before and after impact, and find the fraction $F$ by computing the 
ratio of the two energies.  Compare this value of $F$ with the 
theoretical value predicted by Equation \ref{KEfrac}.  (Find the 
percent difference between the two values.)

\end{enumerate}


\noindent
\underline{Conclusions:} 
Summarize and evaluate your experiment.  Include mention of likely sources of 
error and how you dealt with them.

\end{document}
