【目標】

  1. 「$\mathrm{km/h}$」と「$\mathrm{m/s}$」の相互変換ができる．
  2. 組立単位の相互変換ができる．
  3. 有効桁数，科学表記，単位の接頭語などを意識して単位変換ができる．

【単位変換のやり方】

【その1】分かっている関係式を順番に代入

・分子「$\newcommand{\sn}{\@ifstar{\snstar}{\snnostar}}\newcommand{\snstar}[3]{#1\,\mathrm{#3}}\newcommand{\snnostar}[3]{#1\!\times\!10^{#2}\,\mathrm{#3}} % \sn*{1}{}{km}=\sn*{1\, 000}{}{m}$」

・分母「$\newcommand{\sn}{\@ifstar{\snstar}{\snnostar}}\newcommand{\snstar}[3]{#1\,\mathrm{#3}}\newcommand{\snnostar}[3]{#1\!\times\!10^{#2}\,\mathrm{#3}} % \sn*{1}{}{min}=\sn*{60}{}{s}$」「$\newcommand{\sn}{\@ifstar{\snstar}{\snnostar}}\newcommand{\snstar}[3]{#1\,\mathrm{#3}}\newcommand{\snnostar}[3]{#1\!\times\!10^{#2}\,\mathrm{#3}} % \sn*{1}{}{h}=\sn*{60}{}{min}$」

$$ \newcommand{\sn}{\@ifstar{\snstar}{\snnostar}}\newcommand{\snstar}[3]{#1\,\mathrm{#3}}\newcommand{\snnostar}[3]{#1\!\times\!10^{#2}\,\mathrm{#3}} % \begin{align*} \sn*{5}{}{m/s} &= \dfrac{5\, \mathrm{m}}{1\ \mathrm{s}} \\[12pt] &= \dfrac{5 \times (\frac{1}{1\, 000}\ \mathrm{km})}{1 \times (\frac{1}{60}\ \mathrm{min})} \\[12pt] &= \dfrac{5 \times \frac{1}{1\, 000}\ \mathrm{km}}{1 \times \frac{1}{60} \times (\frac{1}{60}\ \mathrm{h})} \\[12pt] &= \dfrac{\frac{5}{1\, 000}}{\ \frac{1}{60 \times 60} \ } \ \mathrm{\frac{km}{h}} \\[12pt] &= \dfrac{\ 5 \times 60 \times 60 \ }{1\, 000} \ \mathrm{\frac{km}{h}} \\[12pt] &= \sn*{18}{}{km/h} \end{align*} $$

【その2】単位が約分される様な分数を変換先を見据えてかける

$$ \newcommand{\sn}{\@ifstar{\snstar}{\snnostar}}\newcommand{\snstar}[3]{#1\,\mathrm{#3}}\newcommand{\snnostar}[3]{#1\!\times\!10^{#2}\,\mathrm{#3}} % \begin{align*} \sn*{5}{}{m/s} &= \dfrac{5\ \fcolorbox{blue}{aliceblue}{m}}{1\ \fcolorbox{red}{seashell}{s}} \times \underbrace{\dfrac{\rule[-4pt]{0pt}{14pt} 1\,\mathrm{km}}{\rule[-4pt]{0pt}{14pt} 1\, 000 \fcolorbox{blue}{aliceblue}{m}}}{=1} \times \underbrace{\dfrac{\rule[-4pt]{0pt}{14pt} 60 \fcolorbox{red}{seashell}{s}}{\rule[-4pt]{0pt}{14pt} 1 \fcolorbox{green}{honeydew}{min}}}{=1} \times \underbrace{\dfrac{\rule[-4pt]{0pt}{14pt} 60 \fcolorbox{green}{honeydew}{min}}{\rule[-4pt]{0pt}{14pt} 1\,\mathrm{h}}}_{=1} \\[28pt] &= \dfrac{5 \times 60 \times 60}{1\, 000} \ \mathrm{\frac{km}{h}} \\[12pt] &= \sn*{18}{}{km/h} \end{align*} $$

【その3】分母・分子に分かりやすい数をかける

$$ \newcommand{\sn}{\@ifstar{\snstar}{\snnostar}}\newcommand{\snstar}[3]{#1\,\mathrm{#3}}\newcommand{\snnostar}[3]{#1\!\times\!10^{#2}\,\mathrm{#3}} % \begin{align*} \sn*{5}{}{m/s} &= \dfrac{5\ \mathrm{m}}{1\ \mathrm{s}} \\[12pt] &= \dfrac{2 \times 5\ \mathrm{m}}{2\ \mathrm{s}} \\[12pt] &= \dfrac{3 \times 5\ \mathrm{m}}{3\ \mathrm{s}} \\ & \ \ \vdots \quad(\sn*{3\,600}{}{s}=\sn*{1}{}{h}を狙って)\\[12pt] &= \dfrac{3\,600 \times 5\ \mathrm{m}}{3\, 600\ \mathrm{s}} \\[12pt] &= \dfrac{18\, 000\ \mathrm{m}}{1 \mathrm{h}} \\[12pt] &= \sn*{18}{}{km/h} \end{align*} $$

Q1.【単位変換】

次の単位変換を行え．（通常表記，分数を用いて良い）

（１）$1\,\mathrm{\hskip6pt h \hskip6pt}= \underline{\hskip10.5pt \phantom{60} \hskip10.5pt}\,\mathrm{min}$ （２）$1\,\mathrm{min}= \underline{\hskip10.5pt \phantom{60} \hskip10.5pt}\,\hskip6pt\mathrm{s}$ （３）$1\,\mathrm{\hskip6pt h \hskip6pt}= \underline{\ \ \phantom{3600}\ \ }\,\hskip6pt\mathrm{s}$

（４）$1\,\mathrm{min}= \underline{\ \ \phantom{3600}\ \ }\,\hskip6pt\mathrm{h}$ （５）$1\,\mathrm{\hskip6pt s \hskip6pt }= \underline{\ \ \phantom{3600}\ \ }\,\mathrm{min}$ （６）$1\,\mathrm{\hskip6pt s \hskip6pt}= \underline{\ \ \phantom{3600}\ \ }\,\hskip6pt\mathrm{h}$