Exploding Projectile Problem: Finding Velocities and Mass Ratios

[Noob.com]

Fire at angle 45 degree.
Initial kinetic energy = Eo
At top of projectile explode with addtional energy Eo into 2 fragment.
1. Mass = m1, go straight down.
2. Mass = m2, direction: unknown.

Question:
1. Velocity (mag and dir) of m2
2. Velocity of m1
3. What's ratio m1/m2 when m1 is maximun?

Homework Equations

Conservation of momentum:
mo x vo = m1 x v1 + m2 x v2
Eo = 1/2 mo x vo^2
mo = m1 + m2
Center of Mass:
No clue

The Attempt at a Solution

How can I use center of mass correctly to get the direction of m2? I can set up the conservation of momentum formula but I need the second formula to solve for the 2 velocity!

 

[chrisk]

The center of mass of the system will continue in an undisturbed parabolic path. Evaluate magnitude and direction of m2 such that the center of mass of m1 and m2 follows the undisturbed parabolic path.

 

[Noob.com]

Can you give me more detail on how this parabolic path can be done? I have worked out the conservation of momentum but couldn't find a way to get the direction of m2.

Before the explode:

Eo = 1/2 (m1 + m2) (vo)^2
=> (vo)^2 = 2 Eo / (m1 + m2)

vy = 0
vx = vo * cos(45) = \sqrt{}Eo / (m1 + m2)

After the explode:

x-dir:
(m1 + m2) * vx = m2 * vx2
vx2: Velocity of m2 in x direction.
vx2 = v2 * cos (theta)

y-dir:
m1 * v1 = m2 * vy2
vy2 = v2 * sin (theta)

I have worked out this much but there are 3 unknowns and 2 equations. Can you help me out from here?
By the way, I couldn't understand what is the additional Eo right after the explode. Thanks!