Comment by SAI_Peregrinus

Hooke's law only accounts for the force the spring applies, not the mass of the spring itself. Once you have to account for the mass of the spring, how that mass distribution changes as the string stretches & compresses, and how that alters the momentum it quickly loses its simplicity. That's far too difficult to do by hand, but it's what the real world does so I'd rather say that Hooke's law is a first-order approximation of spring force, not a perfect law for describing linear spring behavior.