A particle of mass mis attached to one end of a mass-less spring of force constant k, lying on a frictionless horizontal plane. The other end of the spring is fixed. The particle starts moving horizontally from its equilibrium position at time t = 0 with an initial velocity $$u_0$$. When the speed of the particle is 0.5 $$u_0$$,it collides elastically with a rigid wail. After this collision,

the speed of the particle when it returns to its equilibrium position is $$u_0$$.

the time at which the particle passes through the equilibrium position for the first time is $$t = \pi \sqrt{\frac{m}{k}}$$.

the time at which the maximum compression of the spring occurs is $$t = \frac{4 \pi}{3} \sqrt{\frac{m}{k}}$$.

the time at which the particle passes through the equilibrium position for the second time is $$t = \frac{5 \pi}{3} \sqrt{\frac{m}{k}}$$.