BOUND AND UNBOUND TRAJECTORIES

Suppose that a ball is projected from a point 'P' that is at a distance r0  from the center of earth with speed V as shown in the figure above.The speed required for the ball to escape from the earth's gravitational field is called escape speed.
Let us consider the locus of paths for various values of v.
When v=0 ball will fall on the earth straight down.
When v is small , it follows an elliptical path and falls on earth
When v= Orbital speed the path is circle with E=1
When orbital speed< v < escape speed , ball starts moving in eliptical path around earth with earth as  focus and 'P' as perigee.
When v = Escape speed ,ball leaves the gravitational region of earth following a parabolic path
And For  v > escape speed ,ball follows a hyperbolic trajectory and leaves the gravitational field...