\documentclass[a4paper]{article}
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\title{Impact Crater Lab}
\author{Will Wrathall}
\date{\today}
\begin{document}
\maketitle
\begin{abstract}
The impact crater of a small metal ball of 63.7 grams (0.0637kg) is dropped from 8 different heights, ranging from 0.20m to 0.90m was observed. A mean was measured for the craters diameter. Using the equation E=mg$\Delta$h given that we have m, and g is a constant of 9.81 we can find the kinetic energy of the ball on impact. The relationship between crater diameter, D, and impact energy, E, is given by D=kE$^n$ where K is constant and n is found by the gradient of the graph and is also constant. This can be modified to give $\log D = n\log E + \log k$.
\end{abstract}
\section{Introduction}
In this experiment we measured the effect of dropping a metal ball bearing in to a sandbox in order to measure the diameter of the resultant crater and therefore measure the kinetic energy of the ball on impact. This required the use of a meter stick, metal ball bearing, sand, container and ruler. We could then plot a graph of D against E or in this case $\log D$ against $\log E$ and find the gradient, which is the constant n. Therefore using the D=kE$^n$ we can find k.
\section{Data}
.
\begin{table}
\centering
\begin{tabular}{c|c|c|c|c|c|c|c|c|c}
Height/m & Diameter/m & 2 & 3 & Mean & E & logE & log(D/m)\\\hline
0.20 & 0.068 & 0.059 & 0.064 & 0.064 & 0.125 & -0.903 & -0.699\\
0.30 & 0.071 & 0.070 & 0.068 & 0.070 & 0.188 & -0.727 & -0.523\\
0.40 & 0.076 & 0.076 & 0.070 & 0.074 & 0.250 & -0.602 & -0.398\\
0.50 & 0.070 & 0.084 & 0.075 & 0.076 & 0.312 & -0.505 & -0.301\\
0.60 & 0.077 & 0.087 & 0.070 & 0.078 & 0.375 & -0.426 & -0.222\\
0.70 & 0.081 & 0.082 & 0.081 & 0.081 & 0.437 & -0.359 & -0.155\\
0.80 & 0.089 & 0.083 & 0.077 & 0.083 & 0.500 & -0.301 & -0.097\\
0.90 & 0.092 & 0.093 & 0.089 & 0.091 & 0.562 & -0.250 & -0.046\\
\end{tabular}
\caption{\label{tab:widgets}Impact Crater.}
\end{table}
.
\section{Graph}
\includegraphics[width=\textwidth]{Impact_Craters_Lab.png}
Therefore gradient(n)=1
\section{Uncertainty}
\begin{table}[h!]
\centering
\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c|c}
Min Height & Max Height & $E_min$ & $E _max$ & logE$_min$ & logE$_max$\\\hline
0.199 & 0.201 & 0.124 & 0.126 & -0.907 & -0.900\\
0.299 & 0.301 & 0.187 & 0.188 & -0.728 & -0.726\\
0.399 & 0.401 & 0.250 & 0.251 & -0.602 & -0.600\\
0.499 & 0.501 & 0.312 & 0.313 & -0.506 & -0.504\\
0.599 & 0.601 & 0.374 & 0.376 & -0.427 & -0.425\\
0.699 & 0.701 & 0.437 & 0.438 & -0.361 & -0.360\\
0.799 & 0.801 & 0.499 & 0.501 & -0.302 & -0.300\\
0.899 & 0.901 & 0.562 & 0.563 & -0.250 & -0.249\\
\end{tabular}
\end{table}
\begin{table}[h!]
\centering
\begin{tabular}{c|c|c|c}
Uncertainty & Percentage Uncertainty\\\hline
0.0035 & 0.39\\
0.0010 & 0.14\\
0.0010 & 0.17\\
0.0010 & 0.20\\
0.0010 & 0.23\\
0.0005 & 0.14\\
0.0010 & 0.33\\
0.0005 & 0.20\\
\end{tabular}
\end{table}
\begin{table}
\centering
\begin{tabular}{c|c|c|c|c|c}
logD$_min$ & logD$_max$ & Uncertainty for logD & log(mean D) & Percentage U in logD\\\hline
0.059 & 0.068 & 0.0045 & -0.699 & -0.64\\
0.068 & 0.071 & 0.0015 & -0.523 & -0.29\\
0.070 & 0.076 & 0.0030 & -0.398 & -0.75\\
0.070 & 0.084 & 0.0070 & -0.301 & -2.33\\
0.070 & 0.087 & 0.0085 & -0.222 & -3.83\\
0.081 & 0.082 & 0.0005 & -0.155 & -0.32\\
0.077 & 0.089 & 0.0060 & -0.097 & -6.19\\
0.089 & 0.093 & 0.0020 & -0.046 & -4.35\\
\end{tabular}
\end{table}
\end{document}
