From 7d2d0cdf2010e74a3b2ad4ee4c3eed452981c884 Mon Sep 17 00:00:00 2001 From: Charlie Root Date: Wed, 8 Apr 2026 23:19:17 -0300 Subject: [PATCH] How to convert units. --- calculus.pdf | Bin 935220 -> 905080 bytes calculus.tex | 63 +++++++++++++++++++++++++++++++++++++++++++++++++++ 2 files changed, 63 insertions(+) diff --git a/calculus.pdf b/calculus.pdf index 8f740771190acd3f3f4c0c12517994e176db072a..670e28bb5001f8687fbba77f12b293bcc3529a9c 100644 GIT binary patch delta 322463 zcmV($K;yr(iaGe6HwYzAL`E$!E;q3VbO!-3m%-`*DSy=(*^=D2?cHC|Pg!+kS`iN& zom3t+E@vy=I8HXslf4g|mS%=qI>tI?UGw$9MM>S3*W;wTm6U5N5(GgI009u}WOutu zcHccqKJ)$4n`giKvo$-F$6BR!_vU7&jfqpEcG=Oy$^xj|y^a3Xjn%L}7-gca zpQd^m_J2RVc?mS^PO12qrW$D|9UGJH4!VeoqF@rw?_6K|3MT6;`k~vyVASnZG$JIA ze!~Ml`k=?p0)Kug7il8wAZ*7NAjUHjUX z-OzD=3XQVDZxP*8eeas8-NQylH}K#)lPYeVy0-Ke%VuV^>{~(`e2X$uS@eE#*hr%m zRPZsEjDFucChuCHt7<<~+;7}iH3%C0$;JDF)yL?0*Hp-XYdT@?p4HNGFYMqz8UZn| ziho%Qv?XH<@P!N-8=b0toHtzY;54|tBJzmQI)+xweP8*}mB7iYy2X4@;CY^cYfGbttw&@Z1gqoY0VL4fo+9Z 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Viver sem calculadora é excelente desenvolvimento, mas usemo-na sempre que útil---quando em pressa, por exemplo. +\ghostsection{Como se converte unidades}. Uma estratégia muito útil em +converter unidades é considerar a relação entre duas unidades como uma +fração de valor 1. Por exemplo, se 1 hora é 3600 segundos, então +% + $$1 \text{ h} = 3600 \text{ s} \text{ implica} \fr{1 \text{ h}}{3600 \text{ s}}$$ +% +Agora, se você quer converter 1800 segundos pra horas, basta +multiplicar pela razão 1 acima, obtendo +% + $$1800 \text{ s} = 1800 \text{ s} \fr{ \text{ h}}{3600 \text{ s}} = \fr{ 1800\cancel{\text{s}}\text{h}}{3600\cancel{\text{s}}} = \fr{1}{2} \text{ h}.$$ +% +Observe que se você quer conversão pras horas, então sua razão terá as +horas no numerador porque assim os segundos no denominador cancelará +os segundos que estão no numerador. Por ``segundos'' quero dizer a +constante ``s'' que representa a unidade em si. + +Assim você converte qualquer coisa com facilidade. + +\ghostsection{De volta ao movimento unidimensional}. + +Eis o primeiro exercício do IF. + \begin{exercise*} Um motorista freia seu carro uniformemente de forma que a velocidade cai de 72 pra 26 km/h em 5 segundos. Que distância o carro percorre @@ -7048,6 +7070,47 @@ fórmula Python, então tive que corrigir novamente. Agora é enfática a importância de calculadora aqui, isto é, de um computador mesmo.) \end{remark*} +\ghostsection{A equação de Torricelli}. A equação de Torricelli +relaciona a velocidade diretamente com a aceleração sem envolver o +tempo. Ela é +% + $$v^2 = v_0^2 \cdots$$ +% +Não lembro. Preciso da minha gatinha de novo. O fato é que Torricelli +nos economiza um tremendo passo num problema que vale a pena +documentar---até porque estou encontrando um aparente erro no +problema. + +\begin{exercise*} +Com que velocidade uma bola deve ser lançada verticalmente pra cima +pra que a bola alcance a altura máxima de 50m? Quanto tempo ela ficará +no ar? Desenhe os gráficos $y$, $v$ e $a$ {\em versus} $t$. Indique +nos dois primeiros gráficos o instante em que ela alcança os +50m. \sol. Computei a derivada de $x(t) = 0 + v_0 t + (1/2) (-9.8) +t^2$ pra obter $v(t) = v_0 - 9.8 t$ e assim isolei $t = +v_0/9.8$. Então consegui eliminar $t$, substituindo $v_0/9.8$ em +$x(t)$, obtendo +% +\begin{align*} +x(t) &= \fr{v_0^2}{9.8} - \fr{9.8}{2} \fr{v_0^2}{9.8^2}\\ + &= \fr{v_0^2}{9.8} - \fr{1}{2} \fr{v_0^2}{9.8}\\ + &= \left(\fr{v_0^2}{9.8} - \fr{1}{2} \fr{v_0^2}{9.8}\right) +\end{align*} +% +Como queremos altura de 50m, obtemos uma quadrática com uma única +solução positiva que é $v_0 = 31.30 \text{m}$, o que casa com o +gabarito. Agora, quando prosseguimos pra perguntar quanto tempo leva +pra essa bola alcançar esses 50m, obtemos a quadrática $-4.9t^2 + +31.30 t - 50 = 0$, que não tem raiz. Como pode? O problema enfatiza +que ele quer o tempo. E o problema não tem como estar errado. Queremos +lançar uma bola pra cima até atingir 50m e contabilizar o tempo---isso +é perfeitamente físico. Estamos errados em algum lugar. Como nossa +$v_0$ casa com o gabarito, é na produção da quadrática que estamos +errados---aritmética! (A gente erra muita aritmética.) Mas a +aritmética parece correta, então física é uma coisa muito perplexante. +\end{exercise*} + + \newpage \tableofcontents \newpage \printbibliography