A space-time finite element method based on the Arbitrary Lagrangian-Eulerian description of the incompressible Navier-Stokes Equations is developed. The developed method is used for predicting the flows past bodies in relative motion. The governing equations are expressed in a fixed frame of reference, wherein the terms related to grid motion are included. Superparametric space-time elements are used in the discretization of the domain in which, finite elements are both allowed to move and deform in (i) simple and (ii) automatic manner. The equations for the rigid body motion are integrated to calculate the trajectory of the moving object under aerodynamic and gravitational forces. The code developed here is first tested on the flow about a drifting and failing sphere which starts to move at Reynolds number of 10(4) from a steady stale. To study the flows about the bodies in relative motion, store-separation from a wing problem is investigated as a turbulent flow for the Reynolds number of 2.5 x 10(6) based on the length of the store.