\appendix
\part*{Appendices}
\addcontentsline{toc}{part}{Appendices}

\section{Physics}\label{s:physics}
This section is mainly intended as a reference that is provided for convenience.

\subsection{Movement}

\begin{align*}
  v &= \int a\, dt \\
  x &= \int v\, dt \\
\end{align*}

\subsubsection{Constant acceleration}
\begin{align*}
  v &= v_0 + at \\
  x &= x_0 + v_0t + \frac{1}{2}at^2
\end{align*}

In certain cases, the starting velocity $v_0$ and the target velocity $v_1$ are known:
\begin{align*}
  t &= \frac{v_1 - v_0}{a} \\
  s &= \frac{v_1^2 - v_0^2}{2a}
\end{align*}

\subsubsection{Acceleration of a train}
The acceleration of a train is calculate as follows:
\[a = a_{\text{all}} + a_{\text{locomotive}}\cdot\frac{n_{\text{locomotives}}}{n_{\text{wagons}}}\]
Please note that slopes are not taken into consideration.

\subsubsection{Acceleration constants}
\begin{tabular}{|c|r|r|}
  \hline
  Lever & $a_{\text{all}}$ & $a_{\text{locomotive}}$ \\
  \hline
  $0$ & $-10$ & $0$ \\
  $1$ & $-3$ & $0$ \\
  $2$ & $-0.5$ & $0$ \\
  $3$ & $0$ & $0$ \\
  $4$ & $0.5$ & $1.5$ \\
  \hline
\end{tabular}

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