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\noindent {{\sf Name:} Derek Teaney} \\
\noindent {{\sf Lab Section:} 01 } \\
\noindent {{\sf Date:} 01/01/01} \\
\begin{center}
{\sf Projectile Motion}
\end{center}
\section{Introduction }
{\small \sf The purpose here is to convince the TA that you understood
how the lab worked. Needlessly philosophical or lengthy remarks will cost you points. } \\
The purpose of this lab was to measure the properties of
projectile motion. A schematic of the
apparatus is shown below {\small \sf (you could/should simply draw this by hand. I used X-fig which is free) }
\begin{center}
\vspace{1.5in}
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% This is how I actually made the lab report
%
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% \includegraphics[height=1.5in]{launch.pdf}
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A small metal ball was released from a ramp at the edge of the
table of height $h$
The initial velocity $v_o$ of
the ball was measured by measuring the time it took
for the ball to cross the photogate detector and knowing effective diameter of the ball.
The final distance $x$ that the ball landed was recorded as a function
of the initial velocity $v_o$. In the Newtonian theory of projectile motion these quantities
are related by
{ \small \sf (If you are using some program like word where
entering formulas is time consuming, simply leave a bit of space and
write the formula by hand) }
\[
x = v_o \sqrt{\frac{2h}{g} } \, .
\]
This formula is compared to the measured data
in what follows.
\section {Recorded and Derived Data}
{\small \sf The purpose here is to record all relevant numbers
and how they were obtained. The raw data should be in the lab
notebook. Often when making plots we need derived quantities, e.g. if you know the length and width you could determine the area $A=LW$. you should explain how you propagated the errors in $L$ and $W$ to determine the error in $A$. This is described in the error analysis writeup. The data in this ``Lab" is totally
made up.} \\
Before data taking started,
the height of the table was $h=(1.01 \pm 0.005)\,{\rm m} $ measured
using ruler stick. The error was estimated to the nearest half centimeter.
The diameter of the ball $D_{\rm eff}=(1.20\pm 0.05)\,{\rm cm} $ was determined by using the SALT translation stage and the photogate detector.
Specifically, the every turn of the nob of the translation stage advanced the
stage by 1/28 of an inch. After the photogate detector registered off, an
additional 13.5 turns were registered before the light crossed the photogate
detector again. The uncertainty was estimated based on repeating the process.
To vary the velocity of the projectile, the ball
was released from five different positions along the ramp. For
a given ramp position the time the photogate detector was
off $t_{\rm stop}$ was recorded by the photogate electronics. This together with
the effective diameter of the ball was sufficient to determine the projectile
velocity for each ramp position
\[
v_o= \frac{D_{\rm eff}}{t_{\rm stop} } \, .
\]
The uncertainty in $t_{\rm stop}$ was small and is neglected. The uncertainty
in the initial velocity is then entirely due to $D_{\rm eff}$, {\it i.e.}
$\Delta v_o = \Delta D_{\rm eff}/t_{\rm stop}$. These initial velocities
and uncertainties are recorded in Table~\ref{datatab}.
The distance $x$ from the edge of the table to the landing point recorded
for every release using a the ruler stick and a plumb line.
To estimate the uncertainty in this number, the process
was repeated several times and the full data set is recorded in
the notebook.
A tabular summary of the launch stopping times and ranges $x$ for each
launch position is given in Table~\ref{datatab}.
\begin{table}
\begin{center}
\begin{tabular}{l|c|c||c}
Ramp Setting & $x$ (m) & Time $t_{\rm stop}$ (s) & $v_o (m/s)$
\\ \hline \hline
1 & 0.042 $\pm$ 0.003 & 0.1373 & 0.087 $\pm$ 0.003 \\ % 0.040 $\pm$ 0.001 \\
2 & 0.059 $\pm$ 0.003 & 0.0924 & 0.130 $\pm$ 0.005 \\ %0.059 $\pm$ 0.002 \\
3 & 0.087 $\pm$ 0.003 & 0.0758 & 0.158 $\pm$ 0.007 \\ % 0.072 $\pm$ 0.003 \\
4 & 0.089 $\pm$ 0.003 & 0.0660 & 0.181 $\pm$ 0.008 \\ %0.085 $\pm$ 0.003 \\
5 & 0.096 $\pm$ 0.003 & 0.0584 & 0.205 $\pm$ 0.009 \\ %0.093 $\pm$ 0.004 \\
\end{tabular}
\end{center}
\caption{Summary of data taken. \label{datatab} }
\end{table}
\section {Analysis and Conclusions}
According to the Newtonian theory the projectile location is
given by
\begin{equation}
\label{eqnewt}
x = v_o \sqrt{\frac{2h}{g} } \, .
\end{equation}
To derive this, we first note the time the ball is in the air is found by the
formula
\[
h = \frac{1}{2} g t^2\, , \qquad \mbox{so,} \qquad t = \sqrt{\frac{2h}{g} } \, .
\]
Then since the $x$ and $y$ directions are independent we can
simply multiply by the $x$ velocity $v_o$ to obtain the formula given
above.
Fig.~1 shows a graph of the measured landing distance $x$ as
a function of the initial velocity $v_o$.
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\vspace{1.5in}
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%
% This is how I made the real laber report
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%\begin{figure}
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%\includegraphics[height=2.8in]{fake_proj.pdf}
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%\caption{\label{fig1} A graph of the projectile range versus its initial
%velocity. }
%\end{figure}
Also shown is the Newtonian prediction of Eq.~\ref{eqnewt}. (In drawing this
theoretical curve we have neglected the uncertainty in $h$ which is small
compared to the spread of data points.) The theoretical curve is slightly, but
systematically, under the data points. This could arise due a number of
reasons. First, it is difficult to avoid disturbing the photogate detector
between calibration step (where $D_{\rm eff}$ is measured) and the measurement
step (where $v_o$ is determined). Such disturbances can systematically
underestimate the initial velocity. Additionally, the track was not exactly
level. If the launch angle is somewhat larger than $90^o$, this launch angle
could systematically bias the comparison to the theoretical curve. Overall,
the data agree with the Newtonian theory within these systematic uncertainties.
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