A uniform rope of linear mass density λ and length l is coiled on a smooth horizontal surface. One end is pulled up with constant velocity v. The maximum power delivered by the agent in pulling up the rope is

Instantaneous force = mg + (vdm/dt),where mg is the force needed for supporting the weight of already hanging section of the rope and v dm/dt is the force needed to supply momentum to the portion of the rope which is to be pulled up.  (mg)max=(λl)g,dmdt=λvHence, Fmax=λlg+λv2 and Pmax=λlg⁡v+λv3