Increased spectrum efficiency has been demonstrated with the use of cognitive radios, however with increased likelihood of interference to the incumbents of spectrum. Several studies solved the interference problem from the transmitter power control perspective, so as to curtail excessive cognitive interference powers; however, neglecting the effect of secondary terminal mobility. We show by simulation that such assumption of terminal immobility in the power control algorithm would fail in time variant cases resulting in increased levels of interference to the Incumbents as well as serious degradation in QoS within the cognitive radio network. We model the link gain evolution process as a distance dependent shadow fading process and scale up the target signal to interference ratio to cope with time variability. This paper therefore, proposes a mobility driven power control algorithm for cognitive radios based on sensing information, which ensures that the interference limit at the Incumbents is unperturbed at all times while concurrently maintaining the QoS within the cognitive radio network. \textcopyright 2011 IEEE.