A particle of mass \( m \) is attached to one end of a massless spring of force constant \( k \)...

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A particle of mass \( m \) is attached to one end of a massless 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 wall. After this collision
(a) the speed of the particle when it returns to its equilibrium position is \( u_{0} \)
\( \mathrm{P} \)
(b) the time at which the particle passes through the equilibrium position for the first time is
\( t=\pi \sqrt{\frac{m}{k}} \)
(c) the time at which the maximum compression of the spring occurs is \( t=\frac{4 \pi}{3} \sqrt{\frac{m}{k}} \)
(d) the time at which the particle passes through the equilibrium position for the second time is
\[
t=\frac{5 \pi}{3} \sqrt{\frac{m}{k}}
\]
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