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Por exemplo, se 1 hora é 3600 segundos, então % - $$1 \text{ h} = 3600 \text{ s} \text{ implica} \fr{1 \text{ h}}{3600 \text{ s}}$$ + $$1 \text{ h} = 3600 \text{ s} \text{ implica } \fr{1 \text{ h}}{3600 \text{ s}} = 1 $$ % Agora, se você quer converter 1800 segundos pra horas, basta multiplicar pela razão 1 acima, obtendo % - $$1800 \text{ s} = 1800 \text{ s} \fr{ \text{ h}}{3600 \text{ s}} = \fr{ 1800\cancel{\text{s}}\text{h}}{3600\cancel{\text{s}}} = \fr{1}{2} \text{ h}.$$ + $$1800 \text{ s} = 1800 \text{ s } \fr{1 \text{ h}}{3600 \text{ s}} = \fr{ 1800\text{ h}}{3600} = \fr{1}{2} \text{ h},$$ % +como desejado. + Observe que se você quer conversão pras horas, então sua razão terá as -horas no numerador porque assim os segundos no denominador cancelará +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 exercício do IF. +\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 @@ -7074,42 +7078,85 @@ importância de calculadora aqui, isto é, de um computador mesmo.) relaciona a velocidade diretamente com a aceleração sem envolver o tempo. Ela é % - $$v^2 = v_0^2 \cdots$$ + $$v^2 = v_0^2 + 2a \Delta x$$ % -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. +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 $x(t) = 0 + v_0 t + (1/2) (-9.8) +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 -$x(t)$, obtendo +$y(t)$, 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) +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 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. +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. +\end{remark*} \newpage \tableofcontents \newpage \printbibliography