The biphasic adhesion-velocity relation is a universal observation in mesenchymal cell motility. It has been explained by adhesion-promoted forces pushing the front and resisting motion at the rear. Yet, there is little quantitative understanding of how these forces control cell velocity. We study motion of MDA-MB-231 cells on microlanes with fields of alternating Fibronectin densities to address this topic and derive a mathematical model from the leading-edge force balance and the force-dependent polymerization rate. It reproduces quantitatively our measured adhesion-velocity relation and results with keratocytes, PtK1 cells, and CHO cells. Our results confirm that the force pushing the leading-edge membrane drives lamellipodial retrograde flow. Forces resisting motion originate along the whole cell length. All motion-related forces are controlled by adhesion and velocity, which allows motion, even with higher Fibronectin density at the rear than at the front. We find the pathway from Fibronectin density to adhesion structures to involve strong positive feedbacks. Suppressing myosin activity reduces the positive feedback. At transitions between different Fibronectin densities, steady motion is perturbed and leads to changes of cell length and front and rear velocity. Cells exhibit an intrinsic length set by adhesion strength, which, together with the length dynamics, suggests a spring-like front-rear interaction force. We provide a quantitative mechanistic picture of the adhesion-velocity relation and cell response to adhesion changes integrating force-dependent polymerization, retrograde flow, positive feedback from integrin to adhesion structures, and spring-like front-rear interaction.