Serway & Jewett — Oscillatory Motion: Problem 8

The initial position, velocity, and acceleration of an object moving in simple harmonic motion are \(x_i\), \(v_i\), and \(a_i\); the angular frequency of oscillation is \(\omega\). (a) Show that the position and velocity of the object for all time can be written as \[x(t) = x_i \cos \omega t + \left( \frac{v_i}{\omega} \right) \sin \omega t\] \[v(t) = -x_i \omega \sin \omega t + v_i \cos \omega t\] (b) Using \(A\) to represent the amplitude of the motion, show that \[v^2 - ax = v_i^2 - a_i x_i = \omega^2 A^2\]