Section 6.2 Relative Motion Analysis: Velocity
We first aim to relate the velocity of point \(P\) relative to \(\cE\) to that relative to \(\cF\text{.}\) We start with the equality \(\br_{OP}= \br_{OB}+\br_{BP}\) which we differentiate relative to \(\cE\) to obtain:
\begin{equation*}
\vel_{P/\cE} = \vel_{B /\cE} + \left( {d\over dt} \br_{B P}
\right)_{\cE}
\end{equation*}
The derivative of
\(\br_{BP}\) in
\(\cE\) can be obtained from that in
\(\cF\) according to formula
(3.1.3)
\begin{equation*}
\left( {d\over dt} \br_{B P}\right)_{\cE}
=
\left( {d\over dt} \br_{B P} \right)_{\cF}
+
\bom_{\cF/\cE}\times \br_{BP}
=
\vel_{P/\cF} + \bom_{\cF/\cE}\times \br_{BP}
\end{equation*}
Hence,
\begin{equation*}
\vel_{P/\cE} = \vel_{P/\cF}+\vel_{B /\cE} +\bom_{\cF/\cE}\times \br_{BP}
\end{equation*}
The sum of the last two terms, \(\vel_{B /\cE} +\bom_{\cF/\cE}\times \br_{BP}\text{,}\) is then recognized as the transport velocity \(\vel_{P\in \cF / \cE}\) of point \(P\) by \(\cF\) (relative to \(\cE\)). We can now state the following result.
Theorem 6.2.1. Change of Referential (Velocity).
Given two referentials \(\cE\) and \(\cF\) in relative motion, the velocity \(\vel_{P/\cE}\) of a particle \(P\) relative to \(\cE\) can be found from its velocity \(\vel_{P/\cF}\) relative to \(\cF\) according to the formula
\begin{equation}
\vel_{P/\cE} = \vel_{P/\cF} + \vel_{P\in \cF / \cE} \tag{6.2.1}
\end{equation}
where \(\vel_{P\in \cF / \cE}\) is known as the transport velocity of \(P\) by \(\cF\) relative to \(\cE\text{.}\) The velocities vectors \(\vel_{P/\cE}\) and \(\vel_{P/\cF}\) are commonly called absolute velocity and relative velocity, respectively.
