Tauberian identities and the connection to Wile E. Coyote physics

Abstract

The application of the motion of a vertically suspended mass-spring system released under tension is studied focusing upon the delay timescale for the bottom mass as a function of the spring constants and masses. This ``hang-time", reminiscent of the Coyote and Road Runner cartoons, is quantified using the far-field asymptotic expansion of the bottom mass' Laplace transform. These asymptotics are connected to the short time mass dynamics through Tauberian identities and explicit residue calculations. It is shown, perhaps paradoxically, that this delay timescale is maximized in the large mass limit of the top ``boulder". Experiments are presented and compared with the theoretical predictions. This system is an exciting example for the teaching of mass-spring dynamics in classes on Ordinary Differential Equations, and does not require any normal mode calculations for these predictions.