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| -rw-r--r-- | assets/manual/manual.tex | 39 |
1 files changed, 27 insertions, 12 deletions
diff --git a/assets/manual/manual.tex b/assets/manual/manual.tex index 8aade44..d503586 100644 --- a/assets/manual/manual.tex +++ b/assets/manual/manual.tex @@ -1375,14 +1375,37 @@ The speed restriction table for trains includes the speed limits for a train, wh \addcontentsline{toc}{part}{Appendices} \section{Physics}\label{s:physics} +This section is mainly intended as a reference that is provided for convenience. -\subsection{Train acceleration} +\subsection{Movement} +This section will use $x$ as the position and \( s = \Delta x \) as the distance. + +\begin{align*} + v(T) &= v_0 + \int_0^T a(t) dt \\ + x(T) &= x_0 + \int_0^T v(t) dt \\ + s(T) &= \Delta x = \int_0^T v(t) dt +\end{align*} + +\subsubsection{Constant acceleration} +\begin{align*} + v(T) &= v_0 + \int_0^T a(t)dt = v_0 + aT \\ + x(T) &= x_0 + \int_0^T v(t)dt = x_0 + v_0T + \frac{1}{2}aT^2 \\ + s(T) &= 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 not that slopes are not taken into consideration. -With the following constants: - +\subsubsection{Acceleration constants} \begin{tabular}{|c|r|r|} \hline Lever & $a_{\text{all}}$ & $a_{\text{locomotive}}$ \\ @@ -1393,15 +1416,7 @@ With the following constants: $3$ & $0$ & $0$ \\ $4$ & $0.5$ & $1.5$ \\ \hline -\end{tabular}\\ - -Please note that, as shown in the equation above, slopes are not taken into consideration. - -The time needed to accelerate from $v_0$ to $v_1$ can be calculated as follows: -\[ t = \frac{v_1-v_0}{a} \] - -The distance needed to accelerate from $v_0$ to $v_1$ can be calculated as follows: -\[ s = \frac{v_1^2 - v_0^2}{2a} \] +\end{tabular} \ifx\HCode\undefined \printindex |