A finite element method (FEM) based on a level set method and a monolithic formulation was investigated for the analysis of the entry problem of an elastic body. Both the free surface tracking and the interaction of an entry body with fluid flow should be considered in an entry problem. Free surface tracking was achieved using a level set method, in which advection and reinitialization equations for the level set variable are discretized using the least-squares FEM (LSFEM). The coupling of the motion of the elastic body with fluid flow was conducted using a monolithic mixed finite element formulation. The arbitrary Lagrangian Eulerian method was used for the movement of a grid because the downward motion of the body is dominant in the entry problem. Since a mixed finite element was used, a linear basis function that belongs to the $H^1$ space was used for velocity and level set variables on each sub-element ($T_h$), and a linearly interpolated pressure variable that belongs to the $L^2$ space was adopted on each element ($T_{2h}$). The level set method based on the LSFEM was verified by solving some free surface tracking problems. Then, the finite element formulation for the entry problem was validated by comparing it with existing experimental results. Finally, the entry of an elastic body was compared with that of a rigid body.