mirror of https://github.com/E-Almqvist/hsf
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e5b047071e
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elalmqvist@gmail.com --- \url{https://wych.dev} |
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\begin{document} |
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\title{Anteckningar 2022-05-05} |
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\author{Elias Almqvist} |
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\maketitle |
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\newpage |
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\section{Fallande kropp i atmosfär} |
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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: |
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$$ |
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F_g = -mg |
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$$ |
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$$ |
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F_\gamma = \gamma(v) |
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$$ |
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$$ |
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\implies F = \Delta F = F_g - F_\gamma = -mg - \gamma(v) |
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$$ |
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|
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För att få den resulterande accelerationen $a$ dividerar vi med $m$: |
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$$ |
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a = \frac{F}{m} \because F = ma |
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$$ |
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$$ |
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a = \frac{-mg - \gamma(v)}{m} = -\left(g + \frac{\gamma(v)}{m}\right) |
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$$ |
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|
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Vi kan också skriva om $a$: |
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$$ |
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a = \frac{dv}{dt} = \dot{v} = -\left(g + \frac{\gamma(v)}{m}\right), \quad +\frac{\gamma(v)}{m} |
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$$ |
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$$ |
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\therefore \dot{v} + \frac{1}{m} \gamma(v) = -g |
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$$ |
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|
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Nu varierar ekvationen beroende på vad $\gamma(v)$ är för något. Men vi kan anta att $F_\gamma \propto v$ vilket ger: |
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$$ |
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\dot{v} + \frac{1}{m} kv = -g \quad | \quad k \in \mathbb{R} |
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$$ |
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$$ |
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\implies v_h = Ce^{-\frac{k}{m} t} \quad | \quad C \in \mathbb{R} |
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$$ |
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$$ |
||||
\implies v_p = \frac{-mg}{k} |
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$$ |
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$$ |
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\therefore v_a = v_p + v_h = Ce^{-\frac{k}{m} t} - \frac{mg}{k} |
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$$ |
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|
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Vi har nu den en generell lösning. Behöver bara lösa vad $C$ är givet ett villkor. etc etc. \\ |
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Exempel: |
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$$ |
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v(0) = 0 \implies C - \frac{mg}{k} = 0 \implies C = \frac{mg}{k} |
||||
$$ |
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$$ |
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\therefore v = \frac{mg}{k} \left( e^{\frac{-k}{m}t} - 1 \right) |
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$$ |
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|
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\end{document} |
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Reference in new issue