From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Sat, 27 Jul 2019 07:44:41 +0000 (+1000)
Subject: [spec] re-render graphs \& add kinematics formulas
X-Git-Tag: yr12~79
X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/990a3f22a94c5170b86016dcb4d3b592a1bca58c

[spec] re-render graphs \& add kinematics formulas
---

diff --git a/spec/spec-collated.pdf b/spec/spec-collated.pdf
index 15ea285..2af5693 100644
Binary files a/spec/spec-collated.pdf and b/spec/spec-collated.pdf differ
diff --git a/spec/spec-collated.tex b/spec/spec-collated.tex
index 0acbd24..d75883f 100644
--- a/spec/spec-collated.tex
+++ b/spec/spec-collated.tex
@@ -1161,9 +1161,23 @@
 
       \subsection*{Constant acceleration}
 
+      \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 }  % TODO need to fix centering here
+          \begin{tabular}{ l r }
               \hline & no \\ \hline
               \(v=u+at\) & \(x\) \\
               \(v^2 = u^2+2as\) & \(t\) \\
@@ -1179,9 +1193,12 @@
           &= \sqrt{v_x^2 + v_y^2 + v_z^2}
         \end{align*}
 
-        \textbf{Distance travelled between \(t=a \rightarrow t=b\):}
+        \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}
 
         \[ \boldsymbol{r}(t) = x \boldsymbol{i} + y \boldsymbol{j} + z \boldsymbol{k} \]