Calculating Maximum Range of a Catapulted Rock

[Gank]

Homework Statement

Assuming that the rock launched by a catapult is released from ground level, show that the theoretical maximum range is:
R=2Mh/m, where M = mass of counterweight
m = mass of rock
h = height counterweight falls

Homework Equations

Ep=mgh
Ek=(mv^2)/2

The Attempt at a Solution

I equated the two expressions and then split the velocity into its components so this splits the Ek up into horizontal and vertical but I can't progress from here

 

[voko]

At what angle should a projectile be launched to obtain a maximum range?

 

[Gank]

45degrees

 

[voko]

What is the range when the angle is 45 degrees and the magnitude of velocity is some v?