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Because I currently write with no +%% paragraph skip at all, I'm using the math-mode space skip to +%% separate the section from the text (of the previous section). A +%% ghost section is cute---check it out. Notice that the +%% responsibility of adding a period after the bold title is the +%% user's. +\newcommand{\ghostsection}[1]{{\vskip\the\belowdisplayskip\noindent{\bf #1}}} +%% Yeah, TeX is very cryptic---I know. It's really a different world. +%% --8<-------------------------------------------------------->8--- + +%% A shortcut to \dfrac. +\newcommand{\fr}{\dfrac} +%% A shortcut to writing "Solution". +\newcommand{\sol}{{\em Solução}} + +%% --8<---------------cut here---------------start------------->8--- +%% Bibliografia et cetera. +\usepackage[backend=bibtex]{biblatex} +\addbibresource{refs.bib} +%% --8<---------------cut here---------------end--------------->8--- \title{Physics} \author{Ruth Hernández \& husband} @@ -201,6 +246,222 @@ foi superada por Galileu e Newton. Ainda está em construção, meu amor. Um beijo apertado da sua mocinha \texttt{<3} -\selectlanguage{english} +Está uma construção bonita e muito gostosa de se ver. Um beijo +delicado do seu namorado. {\tt <3} +\section{Medições} + +Tudo em física tem direção e unidade porque a ciência é a arte de +medir. Medir significa comparar uma quantidade a uma referência, que +chamamos de unidade. +% +\begin{quote} + ``Science, we are often taught, is measurement.''~\cite[capítulo + 2, página 14]{tea} +\end{quote} +% +É bom hábito computar aritmética sempre com as +unidades---requer paciência de escrever e cancelá-las---e prestando +atenção à direção em que as coisas se movem já que em física tudo tem +direção (exceto as coisas escalares). + +Como não podemos usar calculadora na prova, precisamos ganhar +intimidade com computar números. Quando estivermos incertos sobre a +solução de um problema, usemos a calculadora pra rapidamente saber se +realmente o raciocínio está certo. Depois disso, demos uma olhada e +vejamos o que tem de difícil nas contas e procuremos facilitá-las. +Por exemplo, descartemos toda a super-precisão desnecessária, façamos +arredondamentos e usemos os teoremas que facilitam contas---o que é +uma arte admirável. 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 +% + $$1 \text{ h} = 3600 \text{ s} \text{ implica } \fr{1 \text{ h}}{3600 \text{ s}} = 1 $$ +% +Agora, se você quer converter 1800 segundos pra horas, basta +multiplicar pela razão 1 acima, obtendo +% + $$1800 \text{ s} = 1800 \text{ s } \fr{1 \text{ h}}{3600 \text{ s}} = \fr{ 1800\text{ h}}{3600} = \fr{1}{2} \text{ h},$$ +% +como desejado. + +Observe que se você quer conversão pras horas, então sua razão terá as +horas no numerador porque assim os segundos no denominador cancelarão +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 +problema do IF. Existe algum jeito mais simples de computar as +respostas aqui? Como só conhecemos uma única fórmula, sempre a +aplicamos, mas você verá à frente que minha gatinha tinha um +Torricelli na manga. + +\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 +depois disso até parar? Quanto tempo levará pra percorrer essa +distância adicional? \sol. Não sei resolver. Qual é o meu problema? +Não sei que fórmula usar. Sei obter a aceleração do carro. Ele está +freiando, então tem uma aceleração negativa. Aceleração é a variação +da velocidade sobre o tempo. Então temos $a = (72 - 36)/(0 - 5) = +-7.2$ km/h. Aqui já cabe uma pergunta---devemos responder $-7.2$ km/h, +que tem dois algarismos significativos ou devemos dizer $-7$ km/h, já +que os dados contêm apenas um significativo? Digo que devemos +responder $-7$ km/h pra respeitar o sistema internacional. Se a prova +for discursiva, explicar esse fato; se não, marcar a melhor opção---é +o mais sensato a fazer aqui. (Mas note que nesse problema +especialmente, usar o $-7.2$ é o melhor a fazer porque as contas vão +dar mais certo usando-o. A gente só deve respeitar o sistema +internacional quando reportamos os valores finais.) + +Okay, temos a aceleração---e agora? Lembro perfeitamente da fórmula +% + $$x(t) = x_0 + v_0 t + \fr{1}{2}a t^2,$$ +% +mas nem sei como usá-la pra resolver esse problema. + +Ah, espere um momento! Sei como descobrir $x(5)$ porque sei a +velocidade e a posição em $t=0$ e tenho a aceleração. O carro então +depois de 5 segundos está em $x = 0.075$ km, ou seja, posição $75$ +metros, conta que obtive na calculadora (primeiro incorretamente, +depois incorretamente de novo e por último corretamente). Podemos +agora descobrir em que posição ele estará quando sua velocidade chegar +a zero? Não tenho uma fórmula pra velocidade. Ah, compute a derivada +de $x(t)$ pra obter +% + $$v(t) = x'(t) = v_0 + at.$$ +% +Aos 5 segundos, a velocidade inicial do carro vai ser 36 km/h e +queremos $t$ tal que $v(t) = 0$, obtendo o problema +% + $$0 = \fr{36}{3600} + \left(\fr{-7.2}{3600}\right)t$$ +% +cuja solução dá +% + $$t = \fr{36}{7.2} = 5 \text{ segundos}.$$ +% +Note que não escrevemos as unidades aqui por economia de tempo. A +computação de 36/7.2 foi feita por calculadora. No papel, faríamos +$360 \div 72$, que é uma conta fácil de fazer. A conta que não é +fácil de fazer nesse problema é a conta que nos leva ao fato de que +nos 5 segundos iniciais, o carro percorre $0.75$ km, mas nem pensei em +como facilitá-la. + +Anyway, o carro percorre mais 5 segundos até parar. Portanto, pra +obter a distância adicional percorrida, queremos +% + $$x(10) - x(5)= 0.1 - 0.075 = 0.025 \text{ km},$$ +% +ou seja, o carro ainda se desloca por mais 25 metros, o que levou mais +5 segundos. Note que a segunda pergunta do problema foi respondida +primeiro---vida normal. +\end{exercise*} + +\begin{remark*} +Eis a importância de se usar calculadora. Mesmo usando-a, errei o +valor obtido em $x(5)$ e depois acertei em $x(10)$. Usando Python, +consegui descobrir isso rapidamente, mas teria levado talvez horas pra +descobrir se estivesse calculando na mão. O cansaço decorrente não +compensa o desgosto gerado uma vez que estamos realmente preocupados +em cobrir terreno porque temos uma prova a fazer em breve. Parece boa +a estratégia de refazer as contas (quando elas estiverem certas) +observando o que podemos facilitar na computação. (LOL---tinha erro na +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 +% + $$v^2 = v_0^2 + 2a \Delta x$$ +% +Sendo assim, veja como podemos aplicá-la no problema +abaixo---agradecimentos à gatinha mais linda do mundo. + +\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 $y(t) = 0 + v_0 t + (1/2) (-9.8) +t^2$ 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 +$y(t)$, obtendo +% +\begin{align*} +y(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{v_0^2}{2 \cdot 9.8} + = \fr{v_0^2}{9.8} \left(1 - \fr{1}{2}\right) + = \fr{v_0^2}{9.8} \cdot \fr{1}{2} +\end{align*} +% +Como queremos altura de 50m, obtemos a quadrática +% +\begin{align*} + 50 &= \fr{v_0^2}{9.8}\\ + 100 &= \fr{v_0^2}{9.8}\\ + 980 &= v_0^2\\ + 31.30 &= v_0, +\end{align*} +% +como desejado. 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 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. + +O problema é o seguinte. Quando usamos o valor $v_0 = 31.30$ m/s, a +gente tá fazendo uma aproximação. Com 6 algarismos significativos, +$v_0 = 31.3049$. Quando montamos a quadrática com $v_0 = 31.30$, o +erro de arredontamento fez uma quadrática que não tinha raiz. Não +deveríamos ter-nos interrompido: bastava procurar pelo ponto máximo +dela que encontraríamos $t = 3.1938 \approx 3.2$, de onde obteríamos a +resposta correta $t = 6.4$ (ida e volta). Se em vez de $v_0 = 31.30$, +tivéssemos arredondado pra $v_0 = 31.305$, encontraríamos a quadrática +$-4.9t^2 + 31.305 t - 50 = 0$, que tem raiz $t \approx 3.18$. +\end{exercise*} + +\begin{remark*} +O fato é que esse problema precisa ser resolvido do zero +novamente. Tem formas muito mais fáceis de se raciocinar aí. A gente +resolveu o problema colocando a origem em $y = 0$, mas poderíamos +colocá-la no ápice da altura (ou seja, a 50 m do chão), onde $v_0 = +0$. Assim, o termo linear da quadrática some e obtemos a equação +% +\begin{align*} + -50 &= v_0 t + \fr{1}{2} a t^2\\ + &= 0 \cdot t + \fr{1}{2} (-9.8) t^2\\ + 50 &= \fr{1}{2} (9.8) t^2\\ + 100 &= (9.8) t^2\\ + t &= \sqrt{\left(\fr{100}{9.8}\right)}\\ + t &= 3.1943, +\end{align*} +% +como desejado. Note que agora a posição desejada é $-50$ porque +colocamos o ápice da bola como $y = 0$ e então ela vai cair 50 metros, +ficando no lado negativo do eixo $y$. +\end{remark*} + +\begin{remark*} +Sabe o que é curioso? Errei a aritmética quando computei o comentário +acima. Em vez de dividir 100 por 9.8, eu multipliquei 100 por 9.8, +obtendo 980, cuja raiz é exatamente $v_0$, a velocidade com que a bola +precisa partir pra chegar com velocidade zero a 50 metros. Sim, rolou +um momento de perplexidade. +\end{remark*} + +\printbibliography + +\selectlanguage{english} \end{document} diff --git a/refs.bib b/refs.bib new file mode 100644 index 0000000..f04b6cd --- /dev/null +++ b/refs.bib @@ -0,0 +1,69 @@ +@book{tea, + author = {David Salsburg}, + title = {The lady tasting tea: how statistics revolutionized science + in the twentieth century}, + year = {2001}, + publisher = {W.~H.~Freedom and Company New York}, + ISBN = {0-7167-4106-7} +} + +@book{eti, + author = {Steven Schwartzman}, + title = {The words of mathematics. An etymological + dictionary of mathematical terms used in English.}, + year = {1980}, + publisher = {The Mathematical Association of America}, + ISBN = {0-88385-511-9}, +} + +@book{knuthv1, + author = {Knuth, Donald E.}, + title = {The Art of Computer Programming, volume 1, fundamental algorithms}, + edition = {3}, + year = {1997}, + ISBN = {0-201-89683-4}, + publisher = {Addison-Wesley Longman Publishing Co., Inc.}, + address = {Boston, MA, USA}, +} + +@book{mccoy, + author = {Neal H. McCoy and Gerald J. Janusz}, + title = {Introduction to Modern Algebra}, + year = {1992}, + publisher = {Wm. C. Brown Publishers}, + ISBN = {0-697-08570-8}, + edition = {5}, +} + +@book{hardy-pure, + title = {A course of pure mathematics}, + author = {G.~H. Hardy}, + publisher = {Cambridge University Press}, + isbn = {}, + year = {1967}, + edition = {10}, +} + +@book{stewart, + title = {Calculus}, + author = {James Stewart}, + publisher = {Brooks Cole}, + ISBN = {978-0-495-01160-6}, + year = {2007}, + series = {}, + edition = {6}, + volume = {}, + } + + +@book{calculo, + title = {Cálculo}, + author = {James Stewart}, + publisher = {Cengage Learning}, + isbn = {978-85-221-0968-5}, + year = {2009}, + series = {}, + edition = {6}, + volume = {1}, + } +