Two identical horizontal spring-mass systems, each consisting of a block of mass \(m\) attached to an ideal spring of force constant \(k\), rest on a frictionless surface.

System 1 is displaced to position \(x = A\) and released from rest at time \(t = 0\), oscillating with total mechanical energy \(E\) and maximum speed \(v_{1,\max}\).

System 2 is initially held at rest at position \(x = \dfrac{1}{2}A\), and at time \(t = 0\) receives an instantaneous horizontal impulse that sets it into oscillation with total mechanical energy \(2E\).

What is the ratio of the speed of the block in System 2 immediately after the impulse to the maximum speed of the block in System 1?