Chapter 15, Problem 16P

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 ω. (a) Show that the position and velocity of the object for all time can be written as

$\begin{array}{l}x(t)=x_i\cos \omega t+\left(\frac{v_i}{\omega }\right)\sin \omega t\\ v(t)=-x_i\omega \sin \omega t+v_i\cos \omega t\end{array}$

(b) Using A to represent the amplitude of the motion, show that

$v^2-ax=v_i^2-a_i{x}_i={\omega }^2{A}^2$