Classical problem: projectile exploding into two parts with one part going down

[knightpraetor]

So my problem is: a mass M that is fired at 45 degrees with KE E_0, at the top of the trajectory, projectile explodes with additional energy E_0 into two parts, the first fragment travels straight down , what is the velocity of the the first and the velocity and direction of the second part.

Also, what is the ratio of m1/m2 when m2 is maximized.


so far I've played with it for awhile, but basically using conservation of energy and conservation of momentum in the X and Y directions..gives me 3 equations for 3 unknowns (theta [angle the second particle fires off at], and k1 and k2, where

v1 = -k1*v, and v2 = k2*v

, that I'm to write in terms of other unknowns that it is assumed i will know when calculating for actual values i guess ( ie mass of the first and second particles, and initial velocity of the big mass M and the starting energy)


Anyways, I'm wondering if I'm doing this right, i start with a bunch of equations

$$4E_0 = m_1 (k_1 v)^2 + m_2 (k_2 v)^2$$

$$-m_1 k_1 v = m_2 k_2 v sin(\theta)$$

$$(m_1+m_2) v = m_2 k_2 v cos(\theta)$$

are these the right equations? ...and so to solve these qquation i wish to solve this system of equations for equations involving k1, k2, and theta...i assume it's easiest to solve for k2, plug it into k1, then plug both those equations into the energy one and solve for theta...is taht right? and i have no clue how you would even begin to go about the second part

 

[Astronuc]

What if one determines the initial energy/momentum of both at the time of the explosion. At the top of the arc, the vertical velocity is zero, but each mass has a horizontal velocity. Determine horizontal and vertical velocity in terms of E₀. Neglecting air resistance, the combined mass would be moving in a parabolic trajectory.

Then determine the initial velocities of each mass at the explosion (with respect to CM). Note that one accelerates upward and the other downward, then there is gravity acting on each mass.

If the bottom mass goes straight down, then the horizontal component of it's velocity must be equal to it's initial horizontal velocity.