A new framework for video compressed sensing models the evolution of the image frames of a video sequence as a linear dynamical system (LDS). This reduces the video recovery problem to first estimating the model parameters of the LDS from compressive measurements, from which the image frames are then reconstructed. We exploit the low-dimensional dynamic parameters (state sequence) and high-dimensional static parameters (observation matrix) of the LDS to devise a novel compressive measurement strategy that measures only the dynamic part of the scene at each instant and accumulates measurements over time to estimate the static parameters. This enables us to lower the compressive measurement rate considerably yet obtain video recovery at a high frame rate that is in fact inversely proportional to the length of the video sequence. This property makes our framework well-suited for high-speed video capture and other applications. We validate our approach with a range of experiments including classification experiments that highlight the purposive nature of our framework.