From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Mon, 22 Jul 2019 23:31:54 +0000 (+1000)
Subject: [spec] add kinematics and vector functions to collated notes
X-Git-Tag: yr12~84
X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/528685ecf6d6c1ded844627528f3ff232c073d8b?hp=54d401dc1d5c827359ef69866955bdaab0efd5ee

[spec] add kinematics and vector functions to collated notes
---

diff --git a/spec/spec-collated.pdf b/spec/spec-collated.pdf
index 9714796..4c80f9f 100644
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diff --git a/spec/spec-collated.tex b/spec/spec-collated.tex
index f28326a..89d37b5 100644
--- a/spec/spec-collated.tex
+++ b/spec/spec-collated.tex
@@ -1156,5 +1156,43 @@
 
                   \[\implies f(x+h) \approx f(x) + hf^\prime(x)\]
 
-                \end{multicols}
-              \end{document}
+              
+    \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(t)^2 + v_y(t)^2 + v_z(t)^2} \tag{for vector \(v\)}
+      \end{align*}
+      \textbf{Distance travelled between\(t=a \rightarrow t=b\):}
+      \[= \int^b_a \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \cdot dt \]
+      
+      \subsection*{Vector functions}
+
+        \[ \boldsymbol{r}(t) = x \boldsymbol{i} + y \boldsymbol{j} + z \boldsymbol{k} \]
+
+        \begin{itemize}
+          \item If \(\boldsymbol{r}(t) \equiv\) position with time, then the graph of endpoints of \(\boldsymbol{r}(t) \equiv\) Cartesian path
+          \item Domain of \(\boldsymbol{r}(t)\) is the range of \(x(t)\)
+          \item Range of \(\boldsymbol{r}(t)\) is the range of \(y(t)\)
+        \end{itemize}
+
+      \subsection*{Vector calculus}
+
+      \subsubsection*{Derivative}
+
+        Let \(\boldsymbol{r}(t)=x(t)\boldsymbol{i} + y(t)\boldsymbol(j)\). If both \(x(t)\) and \(y(t)\) are differentiable, then:
+        \[ \boldsymbol{r}(t)=x(t)\boldsymbol{i}+y(t)\boldsymbol{j} \]
+
+  \end{multicols}
+\end{document}