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## speed of Earth around Sun

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from \cite{1999_Tipler_Llewellyn}, page 9

Eq. \ref{eq:1131405} is an initial equation. $$v = \frac{d}{t} \label{eq:1131405}$$ Change variable $$d$$ to $$C$$ in Eq. \ref{eq:1131405}; yields Eq. \ref{eq:7360652}. $$v = \frac{C}{t} \label{eq:7360652}$$ Eq. \ref{eq:7959026} is an initial equation. circumference of a circle $$C = 2 \pi r \label{eq:7959026}$$ Change variable $$C$$ to $$C_{\rm Earth\ orbit}$$ and $$r$$ to $$r_{\rm Earth\ orbit}$$ in Eq. \ref{eq:7959026}; yields Eq. \ref{eq:5753220}. $$C_{\rm Earth\ orbit} = 2 \pi r_{\rm Earth\ orbit} \label{eq:5753220}$$ Change of variable $$C$$ to $$C_{\rm Earth\ orbit}$$ and $$v$$ to $$v_{\rm Earth\ orbit}$$ and $$t$$ to $$t_{\rm Earth\ orbit}$$ in Eq. \ref{eq:7360652}; yields Eq. \ref{eq:5320197}. $$v_{\rm Earth\ orbit} = \frac{C_{\rm Earth\ orbit}}{t_{\rm Earth\ orbit}} \label{eq:5320197}$$ Substitute LHS of Eq. \ref{eq:5753220} into Eq. \ref{eq:5320197}; yields Eq. \ref{eq:9129246}. $$v_{\rm Earth\ orbit} = \frac{2 \pi r_{\rm Earth\ orbit}}{t_{\rm Earth\ orbit}} \label{eq:9129246}$$ Eq. \ref{eq:9494155} is an assumption. $$t_{\rm Earth\ orbit} = 1 {\rm year} \label{eq:9494155}$$ Multiply RHS of Eq. \ref{eq:9494155} by 1, which in this case is $$\frac{365 {\rm days}}{1 {\rm year}} \frac{24 {\rm hours}}{1 {\rm day}} \frac{60 {\rm minutes}}{1 {\rm hour}} \frac{60 {\rm seconds}}{1 {\rm minute}}$$; yields Eq. \ref{eq:9417128} $$t_{\rm Earth\ orbit} = 3.16 10^7 {\rm seconds} \label{eq:9417128}$$ Substitute LHS of Eq. \ref{eq:9417128} into Eq. \ref{eq:9129246}; yields Eq. \ref{eq:1441436}. $$v_{\rm Earth\ orbit} = \frac{2 \pi r_{\rm Earth\ orbit}}{3.16\ 10^7 {\rm seconds}} \label{eq:1441436}$$ Eq. \ref{eq:4133484} is an assumption. $$r_{\rm Earth\ orbit} = 1.496\ 10^8 {\rm km} \label{eq:4133484}$$ Substitute LHS of Eq. \ref{eq:4133484} into Eq. \ref{eq:1441436}; yields Eq. \ref{eq:8698819}. $$v_{\rm Earth\ orbit} = \frac{2 \pi \left( 1.496\ 10^8 {\rm km} \right)}{3.16\ 10^7 {\rm seconds}} \label{eq:8698819}$$ Simplify Eq. \ref{eq:8698819}; yields Eq. \ref{eq:1001745}. $$v_{\rm Earth\ orbit} = 29.8 \frac{{\rm km}}{{\rm sec}} \label{eq:1001745}$$ Eq. \ref{eq:1001745} is one of the final equations.