Collision in Spring Mass System!

Classical Mechanics Level 3

A spring (spring constant \(k\)) having one end attached to rigid wall & other end attached to a block of mass \(m\) kept on a smooth surface as shown in figure. Initially spring is in its natural length at \(x = 0\), now spring is compressed to \(x = -a\) and released. (Coefficient of restitution \(e = \dfrac 1 2\) ). If velocity of block just after first collision is \( a\sqrt{\frac{nk}{16m}}\) . Find the value of \(n\).

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