Cell motility is an important biological action in the creation, operation and maintenance of our bodies. One of the best studied types of motility is the lamellipodial motility on flat, hard and sticky surface. To advance our understanding of this essential biological process, it is necessary to develop a mathematical model explaining cell motility. We have considered the turning behavior and dependence of the motile behavior on the model parameters and boundary conditions of 2D square shaped cells. The model consists of force-balance and myosin transport equations which solved numerically by using finite difference method in MATLAB after nondimensionalized the governing equations along with initial and boundary conditions. The model analysis shows that initiation of motility critically depends on four dimensionless parameter combinations which represent the myosin contractility, characteristic viscosity-adhesion length, effective velocity and local boundary velocity. By simulating numerically a minimal free-boundary model we observed that cells are stationary when contractility is weak and motile behavior of a cell is sensitive to conditions of force balance at the cell boundary.