restructure section on physics

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