This seems to be THE classical problem of Newtonian Dynamics.
With r^2 = x^2 +y^2 (change to polar coordinates as Michael proposed) my (old) textbook gives
t - t0 = Integrate[1/(Sqrt[2/m (Energy - U[r] - C^2/(2 m r^2))]), r]
and
phi = Integrate[ C / ( m r[ tt ]^2 ), {tt, 0 ,t }]
Where Energy is the total Energy of the body and C its angular momentum.
As far as I remember at least one of the integrals is an ellipitical integral without closed solution, so NIntegrate would be appropriate.
But then of course one could directly use NDSolve.
