\input epsf
\nopagenumbers
\magnification=\magstep1

\noindent {\bf Problem statement} A particle $P$ moves in the
plane. The {\it Rectangular Observer}\/ computes the magnitude of the
speed and acceleration of $P$ by observing the $x$- and
$y$-coordinates of $P$ (as functions of time) and using these
formulas: ${\rm speed}=\sqrt{\bigl({{dx}\over{dt}}\bigr)^{\!2}+
\bigl({{dy}\over{dt}}\bigr)^{\!2}}$ and ${\rm
acceleration}=\sqrt{\bigl({{d^2x}\over{dt^2}}\bigr)^{\!2}+
\bigl({{d^2y}\over{dt^2}}\bigr)^{\!2}}\,$.

\medskip

\noindent The other observer, the {\it Polar Observer}, finds it
easier and more natural to measure the polar coordinates $r$ and
$\theta$ of $P$ (as functions of time), using herself as the origin,
of course. The {\it Polar Observer}\/ computes ${{{dr}\over{dt}}}$,
${{{d\theta}\over{dt}}}$, ${{{d^2r}\over{dt^2}}}$, etc. What formula
should the {\it Polar Observer}\/ use to compute the speed of the
particle $P$? Deduce your answer from the formula for speed displayed
above and the relationships among $x$, $y$, $r$, and
$\theta$. Finally, if $r$ and ${{d\theta}\over{dt}}$ are constant,
derive this formula valid for uniform circular motion: $
\hbox{acceleration}=(\hbox{speed})^2/r.$

\vfil\eject\end