To determine what effect a spring's mass has on simple harmonic motion, consider a spring of mass \(m\) and relaxed length \(\ell_{\text {spring }}\). The spring is oriented horizontally, and one end is attached to a vertical surface. When the spring is stretched a distance \(x\), the potential energy of the system is \(\frac{1}{2} k x^{2}\). If the free end of the spring is moving with speed \(v\) when the free end is at \(x\), calculate the kinetic energy of the spring in terms of \(m\) and \(v\). (Divide the spring into many infinitesimal elements each of length \(d \ell_{\text {segment }}\) and located at position \(\ell_{\text {scgment }}\), then determine the speed of each piece in terms of \(t_{\text {segment }}\),\(d \ell_{\text {segment }}, m, v\), and \(x\) and integrate from \(\ell_{\text {segment }}=0\) to \(\ell_{\text {scgment }}=\ell_{\text {spring }}+x\).)