The ZK Spiral Family of Trajectories and Forces

Particle trajectories in the form of a logarithmic spiral with specifiedangular time dependence, "ZK spirals," are shown to be analytic solutions formotion in non-central, but simple force power-laws. Each ZK spiral is aparticular solution to a single associated force law. The position and velocityare determined as analytic functions of time in terms of the growth parameter,specified power law, and the initial conditions. Four examples, each for adifferent force law and relevant to the advanced classroom, are presented: thewell-known attractive central inverse cube force; a bead on a rigid,horizontal, frictionless wire in the shape of the spiral trajectory; a carmoving in a changing radius turn; and a known solution for a powered rocketwith variable thrust in a Newtonian gravitational field. In this last case,general expressions for position and velocity of powered transfer orbits withinthe solar system are presented along with a new analytic expression for therocket mass as a function of time or distance from the Sun. These results canbe extended to include powered flight solutions to a generalized Lambert'sproblem for circular orbits in attractive central force laws of the form F(r) =- Fo/(r^q), for q greater than or equal to 1.