Exercise E2 Conversions: Speed, Energy, and Power
Convert the following values step by step:
1. A vehicle speed of $80.00~{\rm {km}\over{h}}$ in ${\rm {m}\over{s}}$
\begin{align*}
v = 80.00~{\rm {km}\over{h}} \qquad \rightarrow \qquad {{\rm {numerical~value}}\over{3.6}} \qquad \rightarrow \qquad
v = 22.22~{\rm { m}\over{s}}
\end{align*}
\begin{align*}
v &= 80.00~{\rm {km}\over{h}} = 80~{\rm {1'000~m}\over{3'600~s}} = 80~{\rm {m}\over{3.6~s}} \\
&= 22.22~{\rm {m}\over{s}}
\end{align*}
\begin{align*}
v = 22.22~{\rm {m}\over{s}}
\end{align*}
2. An energy of $60.0~{\rm J}$ in ${\rm kWh}$ ($1~{\rm J} = 1~{\rm Joule} = 1~{\rm Watt}\cdot {\rm second}$).
\begin{align*}
P = 60.0~{\rm J} = 60.0~{\rm Ws} = 0.06~{\rm kWs} = 0.0000167~{\rm kWh}= 1.67 \cdot 10^{-5}~{\rm kWh}
\end{align*}
\begin{align*}
P = 1.67 \cdot 10^{-5}~{\rm kWh}
\end{align*}
3. The number of electrolytically deposited single positively charged copper ions of $1.2 ~\rm Coulombs$ (a copper ion has the charge of about $1.6 \cdot 10^{-19}{\rm C}$)
\begin{align*}
n_{\rm ions}= {{{\rm charge}}\over{{\rm charge\,per\,ion}}}={{1.2~{\rm C}}\over{1.6\cdot10^{-19}~{\rm C/ion}}}=7.5 \cdot 10^{18}~{\rm ions}
\end{align*}
\begin{align*}
n_{\rm ions}= 7.5 \cdot 10^{18}~{\rm ions}
\end{align*}
4. Absorbed energy of a small IoT consumer, which consumes $1~{\rm µW}$ uniformly in $10 ~\rm days$
\begin{align*}
10 ~{\rm days} &= 10 \cdot {{24 ~\rm h}\over{1 ~\rm day}} \cdot {{60 ~\rm min}\over{1 ~\rm h}} \cdot {{60 ~\rm s}\over{1 ~\rm min}} \\
&= 10 \cdot 24 \cdot 60 \cdot 60~{\rm s} \\
&= 864'000 ~{\rm s} \\ \\
W_{\rm absorbed} &= 1~{\rm µW} \cdot 864'000~{\rm s} \\
&= 864'000~{\rm µWs} \\
&= 864~{\rm mWs} \\
&= 0.864~{\rm Ws} \\
&= 0.864~{\rm J}
\end{align*}
\begin{align*}
W_{\rm absorbed} = 0.864~{\rm Ws} = 0.864~{\rm J}
\end{align*}