One of the boundary conditions required for the free surface flow computation is the kinematic condition, which is derived from the mass conservation of the fluid domain. Within a finite element framework that employs an interface-tracking approach classified as the arbitrary Lagrangian-Eulerian (ALE) method, the kinematic condition can be imposed as a Dirichlet condition by predefining the direction of the nodal velocity of the free surface. However, imposing this Dirichlet condition poses a challenge: the non-uniqueness of the nodal normal vector required for evaluating the condition. This issue becomes influential when the free surface domain exhibits high curvature, as observed in coating processes. One solution is the weak imposition of the Dirichlet condition, where the normal vector field is evaluated at quadrature points on the surface instead of at the nodes. In this research, the free surface flow problem, along with the elastic mesh update method coupled with incompressible Newtonian fluid flow, is computed transiently using in-house finite element software. Various formulations with different penalty (stabilization) parameters are investigated, including the penalty method and symmetric/asymmetric Nitsche's methods. The results suggest that Nitsche's method performs marginally better among the weak impositions. The current limitations of weak impositions are analyzed by examining the coupling process and the `Lagrangian' mesh update nature of the ALE approach.