From cf93b64e2cc46973c078a51dade9d7cd0fee2c22 Mon Sep 17 00:00:00 2001 From: "E. Almqvist" Date: Thu, 5 May 2022 13:52:49 +0200 Subject: [PATCH] Minor notes --- ma5/notes/fall/fall.aux | 20 ++ ma5/notes/fall/fall.fdb_latexmk | 87 +++++ ma5/notes/fall/fall.fls | 556 ++++++++++++++++++++++++++++++++ ma5/notes/fall/fall.log | 400 +++++++++++++++++++++++ ma5/notes/fall/fall.out | 1 + ma5/notes/fall/fall.pdf | Bin 0 -> 137942 bytes ma5/notes/fall/fall.synctex.gz | Bin 0 -> 7636 bytes ma5/notes/fall/fall.tex | 105 ++++++ 8 files changed, 1169 insertions(+) create mode 100644 ma5/notes/fall/fall.aux create mode 100644 ma5/notes/fall/fall.fdb_latexmk create mode 100644 ma5/notes/fall/fall.fls create mode 100644 ma5/notes/fall/fall.log create mode 100644 ma5/notes/fall/fall.out create mode 100644 ma5/notes/fall/fall.pdf create mode 100644 ma5/notes/fall/fall.synctex.gz create mode 100644 ma5/notes/fall/fall.tex diff --git a/ma5/notes/fall/fall.aux b/ma5/notes/fall/fall.aux new file mode 100644 index 0000000..cf39c4f --- /dev/null +++ b/ma5/notes/fall/fall.aux @@ -0,0 +1,20 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument} +\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined +\global\let\oldcontentsline\contentsline +\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}} +\global\let\oldnewlabel\newlabel +\gdef\newlabel#1#2{\newlabelxx{#1}#2} +\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}} +\AtEndDocument{\ifx\hyper@anchor\@undefined +\let\contentsline\oldcontentsline +\let\newlabel\oldnewlabel +\fi} +\fi} +\global\let\hyper@last\relax +\gdef\HyperFirstAtBeginDocument#1{#1} +\providecommand\HyField@AuxAddToFields[1]{} +\providecommand\HyField@AuxAddToCoFields[2]{} +\@writefile{toc}{\contentsline {section}{\numberline {1}Fallande kropp i atmosfär}{2}{section.1}\protected@file@percent } +\gdef \@abspage@last{2} diff --git a/ma5/notes/fall/fall.fdb_latexmk b/ma5/notes/fall/fall.fdb_latexmk new file mode 100644 index 0000000..567e674 --- /dev/null +++ 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+{0em} +{}[\titlerule] + +\titleformat{\subsection} +{\large\bfseries} +{} +{0em} +{} +\titlespacing{\subsection} +{0em}{2em}{.4em} + +\titleformat{\subsubsection}[runin] +{\bfseries} +{} +{0em} +{} +\titlespacing{\subsubsection} +{0em}{2em}{1em} + +\renewcommand{\maketitle}{ + \begin{center} + {\huge\bfseries\thetitle}\\ + \vspace{1em} + {\Large\theauthor} \\ + \vspace{1em} + elalmqvist@gmail.com --- \url{https://wych.dev} + \end{center} +} + + +\begin{document} + +\title{Anteckningar 2022-05-05} +\author{Elias Almqvist} + +\maketitle +\newpage + +\section{Fallande kropp i atmosfär} +Något som faller har tyngdaccelerationen $g$ i $\frac{m}{s^2}$ och en massa $m$ i kg. $v$ är dess hastighet i $\frac{m}{s}$ och $F$ är kraften som verkas på den i newton. Den har dessutom en luftmotståndsfunktion: $\gamma(v)$. Låt positiva ($v > 0$) gå uppåt: +$$ +F_g = -mg +$$ +$$ +F_\gamma = \gamma(v) +$$ +$$ +\implies F = \Delta F = F_g - F_\gamma = -mg - \gamma(v) +$$ + +För att få den resulterande accelerationen $a$ dividerar vi med $m$: +$$ +a = \frac{F}{m} \because F = ma +$$ +$$ +a = \frac{-mg - \gamma(v)}{m} = -\left(g + \frac{\gamma(v)}{m}\right) +$$ + +Vi kan också skriva om $a$: +$$ +a = \frac{dv}{dt} = \dot{v} = -\left(g + \frac{\gamma(v)}{m}\right), \quad +\frac{\gamma(v)}{m} +$$ +$$ +\therefore \dot{v} + \frac{1}{m} \gamma(v) = -g +$$ + +Nu varierar ekvationen beroende på vad $\gamma(v)$ är för något. Men vi kan anta att $F_\gamma \propto v$ vilket ger: +$$ +\dot{v} + \frac{1}{m} kv = -g \quad | \quad k \in \mathbb{R} +$$ +$$ +\implies v_h = Ce^{-\frac{k}{m} t} \quad | \quad C \in \mathbb{R} +$$ +$$ +\implies v_p = \frac{-mg}{k} +$$ +$$ +\therefore v_a = v_p + v_h = Ce^{-\frac{k}{m} t} - \frac{mg}{k} +$$ + +Vi har nu den en generell lösning. Behöver bara lösa vad $C$ är givet ett villkor. etc etc. \\ +Exempel: +$$ +v(0) = 0 \implies C - \frac{mg}{k} = 0 \implies C = \frac{mg}{k} +$$ +$$ +\therefore v = \frac{mg}{k} \left( e^{\frac{-k}{m}t} - 1 \right) +$$ + +\end{document}