Question

Two particles, 1 and 2, each of mass 𝑚, are connected by a massless spring, and are on a horizontal frictionless plane, as shown in the figure. Initially, the two particles, with their center of mass at $𝑥_0$, are oscillating with amplitude 𝑎 and angular frequency 𝜔. Thus, their positions at time 𝑡 are given by $x_1 (t) = (x_0 + d) + \alpha \ sin \omega t$ and $x_2t = (x_0 -d) − \alpha sin \ wt,$ respectively, where $𝑑 > 2𝑎.$ Particle 3 of mass 𝑚 moves towards this system with speed 𝑢0 = 𝑎𝜔/2, and undergoes instantaneous elastic collision with particle 2, at time $𝑡_0$. Finally, particles 1 and 2 acquire a center of mass speed $𝑣_{cm}$ and oscillate with amplitude 𝑏 and the same angular frequency 𝜔.