In recent years, it has become well known that rational Bézier and B-spline curves in the space of dual quaternions correspond to rational Bézier and B-spline motions. However, the influence of weights of these dual quaternion curves on the resulting rational motions has been largely unexplored. In this paper, we present a thorough mathematical exposition on the influence of dual-number weights associated with dual quaternions for rational motion design. By deriving the explicit equations for the point trajectories of the resulting motion, we show that the effect of real weights on the resulting motion is similar to that of a rational Bézier curve and how the change in dual part of a dual-number weight affects the translational component of the motion. We also show that a rational Bézier motion can be reparameterized in a manner similar to a rational Bézier curve. Several examples are presented to illustrate the effects of the weights on rational motions.