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mode 100644 index 0000000..92b1978 --- /dev/null +++ b/ma5/notes/fall/fall.out @@ -0,0 +1 @@ +\BOOKMARK [1][-]{section.1}{\376\377\000F\000a\000l\000l\000a\000n\000d\000e\000\040\000k\000r\000o\000p\000p\000\040\000i\000\040\000a\000t\000m\000o\000s\000f\000\344\000r}{}% 1 diff --git a/ma5/notes/fall/fall.pdf b/ma5/notes/fall/fall.pdf new file mode 100644 index 0000000..b3fe8e3 Binary files /dev/null and b/ma5/notes/fall/fall.pdf differ diff --git a/ma5/notes/fall/fall.synctex.gz b/ma5/notes/fall/fall.synctex.gz new file mode 100644 index 0000000..2b166bd Binary files /dev/null and b/ma5/notes/fall/fall.synctex.gz differ diff --git a/ma5/notes/fall/fall.tex b/ma5/notes/fall/fall.tex new file mode 100644 index 0000000..5ed729a --- /dev/null +++ b/ma5/notes/fall/fall.tex @@ -0,0 +1,105 @@ +\documentclass{article} + +\usepackage[margin=2cm]{geometry} +\usepackage{titlesec} +\usepackage{titling} +\usepackage[hidelinks]{hyperref} +\usepackage{multicol} +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{amssymb} +\usepackage{braket} + +\titleformat{\section} +{\Large\bfseries} +{} +{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}