For a planet of mass m in orbit around a sum of mass M >> m, the properties of the orbit are given by Kepler's Laws.
Let's examine the orbits corresponding to several initial velocities:
Case 1: Circular orbit. We choose the radius so that this initial velocity v = 1.
Case 2: v < 1. The mass follows an elliptical orbit. The starting point is the aphelion, the point furthest from the Sun.
Case 3: v = 0. The object gets sucked in to the Sun along a straight line.
Case 4: √ 2 > v > 1 but total energy is still negative. The orbit is again elliptical, but this time the starting point is the perihelion - the point closest to the Sun.
Case 5: v = √ 2. This is the escape speed where the total energy equals zero. The orbit is parabolic and never returns.
Case 6: v > √ 2, so the total energy
is positive. The orbit is hyperbolic -
straighter than the parabola.