From 528685ecf6d6c1ded844627528f3ff232c073d8b Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Tue, 23 Jul 2019 09:31:54 +1000 Subject: [PATCH] [spec] add kinematics and vector functions to collated notes --- spec/spec-collated.pdf | Bin 371859 -> 356126 bytes spec/spec-collated.tex | 42 +++++++++++++++++++++++++++++++++++++++-- 2 files changed, 40 insertions(+), 2 deletions(-) diff --git a/spec/spec-collated.pdf b/spec/spec-collated.pdf index 9714796099c07445488f0159bfd3c9f571620a91..4c80f9f38c1601df31a0e0929c0d41674c770047 100644 GIT binary patch delta 178924 zcmV)SK(fD+*%qGO6|g)H0yr^~F(xQ~?LFCY97l5R_=q57`@GiYqZzA$b?1 z9BYSG55vI_94#=Q27n~$>z!HEb#+(wTp&ZV{Xoo>U6qxUcV-nybKQ{U#n~6%o&E7E z)7Z``rkd~GG*WfS^2SgOug!N?&G+a3ePQ_dYO%d&Ip^n;`PWzPE?Q>JSIhN(;%0sS zaM4nx6+eHrx!SIN@gU#aZP&M}?Zpq@ef`I;gvw^&iguD1K*7yG;b}t1`SZVi`?qht z{4f9XfBfgSUwrxU6+V6T<#%U4olyWM4UH%g7>|f%d3*N#52U$*-@a~0$E|68ageu- zFtjrk-rh8?&MwbLzG$i%N;@lmCCiqrwd@!d9;@efx9@INzXCQ|p2KQ6J-@qozqz}G 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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} -- 2.47.1