Fiona Robin Hood collision

In the movie Shrek, Fiona has recently been rescued from a dragon by Shrek.  As Shrek is bringing Fiona to Lord Farkwad, Fiona is abducted by Robin Hood.  He swings from the trees and scoops Fiona up in a completely inelastic collision.  (They move together with the same velocity after the collision).  

Click on the analysis below for a close up look of the equations that describe this event.

For a challenge, try to predict their final height as a fraction of Robin Hood's initial height before you look at the solution.

Step 1 (Swing Down) - Robin Hood's potential energy (PE) at the start of his swing is converted to kinetic energy (KE) at the bottom of his swing.  Applying conservation of energy, PEtop = KEbottom.

Step 2 (Collision) - Robin Hood collides with Fiona in a completely inelastic collision.  Applying conservation of momentum, m1v1 = 2mv3.  For simplicity, we'll assume Fiona and Robin Hood have equal masses.

Step 3 (Swing Up) - Their kinetic energy (KE) together after the collision converts to potential energy (PE) at the top of the swing.  Applying conservation of energy, KEbottom = PEtop.