\usepackage{graphicx}
\usepackage{wrapfig}
\usepackage{tikz}
+\usepackage{tkz-fct}
\usepackage{tikz-3dplot}
\usepackage{pgfplots}
\usetikzlibrary{calc}
\begin{tikzpicture}
\begin{axis}[yticklabel style={yshift=1.0pt, anchor=north east},x=0.1cm, y=1cm, ymax=2, ymin=-2, xticklabels={}, ytick={-1.5708,1.5708},yticklabels={\(-\frac{\pi}{2}\),\(\frac{\pi}{2}\)}]
\addplot[color=orange, smooth] gnuplot [domain=-35:35, unbounded coords=jump,samples=350] {atan(x)} node [pos=0.5, above left] {\(\tan^{-1}x\)};
- \addplot[->, gray, dotted, thick, domain=-35:35] {1.5708};
- \addplot[->, gray, dotted, thick, domain=-35:35] {-1.5708};
+ \addplot[gray, dotted, thick, domain=-35:35] {1.5708} node [black, font=\footnotesize, below right, pos=0] {\(y=\frac{\pi}{2}\)};
+ \addplot[gray, dotted, thick, domain=-35:35] {-1.5708} node [black, font=\footnotesize, above left, pos=1] {\(y=-\frac{\pi}{2}\)};
\end{axis}
\end{tikzpicture}
\columnbreak
\section{Kinematics \& Mechanics}
\subsection*{Constant acceleration}
- {\centering \begin{tabular}{ l r } % TODO need to fix centering here
- \hline & no \\ \hline
- $v=u+at$ & $x$ \\
- $s = {1 \over 2}(v+u)t$ & $a$ \\
- $s=ut+{1 \over 2}at^2$ & $v$ \\
- $s=vt-{1 \over 2}at^2$ & $u$ \\
- $v^2=u^2+2as$ & $t$ \\ \hline
- \end{tabular}}
-
- \[ v_{\text{avg}} = \frac{\Delta\text{position}}{\Delta t} \]
- \begin{align*}
- \text{speed} &= |{\text{velocity}}| \\
- &= \sqrt{v_x^2 + v_y^2 + v_z^2}
- \end{align*}
- \textbf{Distance travelled between \(t=a \rightarrow t=b\):}
- \[= \int^b_a \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \cdot dt \]
-
+
+ \begin{itemize}
+ \item \textbf{Position} - relative to origin
+ \item \textbf{Displacement} - relative to starting point
+ \end{itemize}
+
+ \subsubsection*{Velocity-time graphs}
+
+ \begin{itemize}
+ \item Displacement: \textit{signed} area between graph and \(t\) axis
+ \item Distance travelled: \textit{total} area between graph and \(t\) axis
+ \end{itemize}
+
+ \[ \text{acceleration} = \frac{d^2x}{dt^2} = \frac{dv}{dt} = v\frac{dv}{dx} = \frac{d}{dx}\left(\frac{1}{2}v^2\right) \]
+
+ \begin{center}
+ \renewcommand{\arraystretch}{1}
+ \begin{tabular}{ l r }
+ \hline & no \\ \hline
+ \(v=u+at\) & \(x\) \\
+ \(v^2 = u^2+2as\) & \(t\) \\
+ \(s = \frac{1}{2} (v+u)t\) & \(a\) \\
+ \(s = ut + \frac{1}{2} at^2\) & \(v\) \\
+ \(s = vt- \frac{1}{2} at^2\) & \(u\) \\ \hline
+ \end{tabular}
+ \end{center}
+
+ \[ v_{\text{avg}} = \frac{\Delta\text{position}}{\Delta t} \]
+ \begin{align*}
+ \text{speed} &= |{\text{velocity}}| \\
+ &= \sqrt{v_x^2 + v_y^2 + v_z^2}
+ \end{align*}
+
+ \noindent \textbf{Distance travelled between \(t=a \rightarrow t=b\):}
+ \[= \int^b_a \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \cdot dt \]
+
+ \noindent \textbf{Shortest distance between \(\boldsymbol{r}(t_0)\) and \(\boldsymbol{r}(t_1)\):}
+ \[ = |\boldsymbol{r}(t_1) - \boldsymbol{r}(t_2)| \]
+
\subsection*{Vector functions}
\[ \boldsymbol{r}(t) = x \boldsymbol{i} + y \boldsymbol{j} + z \boldsymbol{k} \]