An experiment is designed to determine the unknown spring constant \(k\) of an ideal horizontal spring. A cart of known mass \(m\) is placed against the spring, compressing it by a distance \(x\) from its equilibrium length along a horizontal track of negligible friction. The cart is released from rest, and a motion sensor records its final speed \(v\) after it completely separates from the spring. The trial is repeated for several different initial compression distances \(x\). Which quantities should be plotted on the vertical and horizontal axes to yield a linear graph, and what is the resulting expression for \(k\) in terms of the slope \(S\) of the best-fit line?