Satellite Orbit Trajectory and Velocity Doubling: Parabolic or Hyperbolic?

[physics148]

Homework Statement

A satellite moves in an elliptic orbit with eccentricity e=1/2 around a planet which it was launched. When it arrives at an apsis( a radial turning point), its velocity is suddenly doubled. Show that the new orbit will be either parabolic or hyperbolic depending on which of the turning points the velocity doubling occurs.

Homework Equations

if the E<0 and e<1, the orbit is an ellipse
if E>0 and e>1 the orbit is a hyperbola

The Attempt at a Solution

I tried to solve this question, but I got stuck and couldn't get anywhere cause I wasn't sure what to do.

 

[jbsteele]

I know that the eccentricity of an orbit determines its shape, but I'm not sure how to use that in this question. Any help would be much appreciated!