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\Lab{Acceleration}
\SecMark
\label{accel}
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%*==== ACCELERATION ====
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\subsection{Essentials}
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%*=== Essentials ===
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\subsubsection{Main ideas}
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%*== Main ideas ==
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\Goal{
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\begin{itemize}
\item
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Geometric introduction of $\rhat$ and $\phat$.
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\item
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Geometric introduction of unit tangent and normal vectors.
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\end{itemize}
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}
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\subsubsection{Prerequisites}
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%*== Prerequisites ==
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\Req{
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\begin{itemize}
\item
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The position vector $\rr$.
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\item
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The derivative of the position vector is tangent to the curve.
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\end{itemize}
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}
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\subsubsection{Warmup}
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%*== Warmup ==
See the prerequisites. It is possible to briefly introduce these ideas
immediately preceding this activity.
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\subsubsection{Props}
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%*== Props ==
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\begin{itemize}
\item
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\end{itemize}
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\subsubsection{Wrapup}
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%*== Wrapup ==
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\begin{itemize}
\item
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Emphasize that $\rhat$ and $\phat$ do not live at the origin! Encourage
students to use the figure provided, which may help alleviate this confusion.
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\item
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Point out to the students that $\rhat$ and $\phat$ are defined everywhere
(except at the origin), whereas $\TT$ and $\NN$ are properties of the curve.
It is only on circles that these two notions coincide; $\rhat$ and $\phat$ are
adapted to round problems, and circles are round! Symmetry is important.
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\item
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Emphasize that $\{\rhat,\phat\}$ can be used as a basis (except at the
origin). Point out to the students that their answer to the last problem
gives them a formula expressing $\rhat$ and $\phat$ in terms of $\ii$ and
$\jj$. When comparing these basis vectors, they should all be drawn with
their tails at the same point.
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\end{itemize}
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\newpage
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\subsection{Details}
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%*=== Details ===
We have had success helping students master the idea of
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``direction of bending''
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%*"direction of bending"
by describing the curve as part of a pickle jar; the principal unit
normal vector points at the pickles!
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\subsubsection{In the Classroom}
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%*== In the Classroom ==
The easiest way to find $\NN$ is to use the dot product to find vectors
orthogonal to $\TT$, then normalize. Students must then use the
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``direction of bending''
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%*"direction of bending"
criterion to choose between the two possible orientations.
Finding $\NN$ in this way requires the student to give names to the its
unknown components. This is a nontrivial skill; many students will have
trouble with this. This is a good example of the general skill discussed in
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Section~\ref{naming}.
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%*Section 11.1.
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\subsubsection{Subsidiary ideas}
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%*== Subsidiary ideas ==
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\Sub{
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\begin{itemize}
\item
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Dividing any vector by its length yields a unit vector.
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\item
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Using the dot product to find vectors perpendicular to a given vector.
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\end{itemize}
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}
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\subsubsection{Homework}
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%*== Homework ==
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\HW{
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\begin{itemize}
\item
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Some students will not be comfortable unless they work out the components of
$\rhat$ and $\phat$ with respect to $\ii$ and $\jj$. Let them.
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\end{itemize}
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}
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\subsubsection{Essay questions \None}
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%*== Essay questions ==
%* (none yet)
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\Essay{
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}
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\subsubsection{Enrichment}
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%*== Enrichment ==
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\Rich{
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\begin{itemize}
\item
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What units does a unit vector have? Do $\rhat$ and $\phat$ have the same
units?
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\end{itemize}
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}
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