A mass hanging on a spring is given a positive initial velocity, V from equilibrium. The ensuing displacement, x = x(t), is governed by mx''= -ax | x'| - kx; x(0) = 0, x' (0) = V where x is displacement, -ax |x ' | is a nonlinear damping force, and kx is a linear restoring force.
(a) Given that the units of x and t are length and time, what are the units of m, k, a?
(b) What are the possible time scales and on which physical processes are they based?
(c) Assume the restoring force is small compared to the damping force. Non-dimensionalize the model using appropriate time and spatial scales (i.e., such that the remaining parameter appears in the (non-dimensional) restoring term.)