\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) measuring a trajectory, and (2) using a 
ballistic pendulum; and then to compare the results.

\underline{Theory:} When a projectile (of mass $m$) is fired into a ballistic 
pendulum (of mass $M$) the impact causes the pendulum and embedded 
projectile (total mass $M+m$) 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 $v$ of the projectile.

Conservation of angular momentum during the collision gives:
\begin{eqnarray*}
\Sigma L_{i} & = & \Sigma L_{f} \\
Rmv & = & I\omega \\
v & = & \frac{I\omega}{Rm}
\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.

Conservation of mechanical energy after the collision gives:
\begin{eqnarray*}
\Sigma E_{i} & = & \Sigma E_{f}  \\
\frac{1}{2}I\omega^{2} & = & (M+m)gh \\
\omega^{2} & = & \frac{2gh(m+m)}{I} \\
\omega & = & \sqrt{\frac{2gh(m+m)}{I}}
\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.

Combining these two results into one equation we get:
\begin{equation}
 v = {\sqrt\frac{2gh(m+M)}{R^{2}m^{2}}}.
\end{equation}

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

\vspace{-.15in}

\underline{Apparatus:} \ \ Make a drawing of the apparatus used.
\vspace{1.2in}

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

\item Determine mass of projectile $m$:\hfill 
\underline{\hspace{1.2in}}

\item Determine mass of pendulum $M$:\hfill 
\underline{\hspace{1.2in}}

\item Find center of mass of pendulum $L$:\hfill 
\underline{\hspace{1.2in}}

\item Determine the rotational inertia of pendulum (including 
projectile) $I$:\hfill 
\underline{\hspace{1.2in}}

\item Fire projectile into pendulum and measure maximum angle of 
swing $\theta$:\hfill 
\underline{\hspace{1.2in}}

\item Calculate rise of center of mass of pendulum $h = L(1-
cos \theta)$:\hfill 
\underline{\hspace{1.2in}}

\item Use equation from pendulum theory to calculate projectile 
velocity $v$:\hfill 
\underline{\hspace{1.2in}}

\item Fire projectile off table and measure horizontal range of 
trajectory $x$:\hfill 
\underline{\hspace{1.2in}}

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

\item Use equation from trajectory theory to calculate projectile's 
velocity $v$:\hfill 
\underline{\hspace{1.2in}}

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

\end{enumerate}
\vspace{-.2in}

\underline{Questions:} 
\vspace{-.2in}
\begin{enumerate}

\item Do the two methods of determining velocity agree within 
expected error limits?  Explain.
\vspace{.3in}

\item Which method do you consider to be most reliable?  Why?
\vspace{.3in}

\end{enumerate}

\underline{Conclusions:} 

\end{document}
