Simple Harmonic Motion

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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 x0 , are oscillating with amplitude aand angular frequency ω . Thus, their positions at time t are given by x1(t)=(x0+d)+asin‌ωt and x2(t)=(x0−d)− a sin ωt , 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 t0 . Finally, particles 1 and 2 acquire a center of mass speed vcm and oscillate with amplitude b and the same angular frequency ω.