\begin{problem*}{12.7}
The initial position, velocity, and acceleration of an objectmoving in
simple harmonic motion are $x_i$, $v_i$, and $a_i$; the angular
frequency of oscillation is $\omega$.  \Part{a} Show that the position
and velocity of the object for all time can be written as
\begin{align}
  x(t) &= x_i \cos \omega t + \p({\frac{v_i}{\omega}}) \sin \omega t \;, \\
  v(t) &= -x_i\omega \sin \omega t + v_i \cos \omega t \;.
\end{align}
\Part{b} Using $A$ to represent the amplitude of the motion, show that
\begin{equation}
  v^2 - ax = v_i^2 - a_i x_i = \omega^2 A^2 \;.
\end{equation}
\end{problem*}

\begin{solution}
\end{solution}
