Exercise 3.2. [This problem is challenging!] Two identical particles of mass m are connected by a light spring with stiffness k (neglect the spring's mass) and equilibrium length 2l. Everything is lined up on the x-axis. Let the position of particle 1 be $x_1(t)$ and the position of particle 2 be $x_2(t)$. If at time $t = 0$, the positions are $x_1(0) = -l$ and $x_2(0) = l$, and the velocities are non-zero with $v_1(0) = v_1 \neq 0$ and $v_2(0) = v_2 \neq 0$, determine the following:
(a)
$v_1, v_2$ and $t$.
(b) The stretch vector of the spring can be defined as $s(t) = x_2(t) - x_1(t) - 2l$. The spring will oscillate as its COM position moves. Assuming that
$s(t) = C \cos(\omega t + \phi)$
find the values of $\phi$, $C$, and $\omega$, assuming that you are given $v_1, v_2, l, m$, and $k$.
