A particle of mass m is 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 `0.5u_0`, it collides elastically with a rigid wall. After this collision

Options

A. the speed to the particle when its returns to its equilibrium is `u_(0)`.
B. the time is at which paricle passes through the equilirbium position for the first time is `t = pisqrt((m)/(k))`.
C. the time at which the maximum compression of the spring occurs is `t = (4pi)/(3)sqrt((m)/(k))`.
D. the time at which the particles passes throuhout the equilirbium position for the second time is `t = (5pi)/(3)sqrt((m)/(k))`.

Answer

A, D

Explanation

`x = A sinomegat`
`v = Aomegacosomega t = (omegaA)/(2)`
`rArr cosomegat = (1)/(2)`
`omegat = (pi)/(3) rArr t = (2pi)/(3) = (pi)/(3) sqrt((m)/(k))`
for (C) time `= (2pi)/(3) sqrt((m)/(k)) + (pi)/(2)sqrt((m)/(k))`
`= (5pi)/(6)sqrt((m)/(k))`
for (D) time `= (2pi)/(3)sqrt((m)/(k)) + pisqrt((m)/(k)) = (5pi)/(3),sqrt((m)/(k))`