JEE Advanced 2024 Paper 2

Paragraph for next 2 Questions 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 $\omega$ . Thus, their positions at time $t$ are given by $x_1(t)= (x_0+d) + a \sin \omega t$ and $x_2(t)= (x_0-d) -$ a sin $\omega t$ , respectively, where $d > 2a$ . Particle 3 of mass $m$ moves towards this system with speed $u_0 = \frac{a \omega}{2}$,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 $\omega$.