Conservation of momentum and explosions I am working on a problem involving a projectile launched at an angle from the ground at some initial velocity and then explodes into 3 parts after reaching its maximum height and they want us to find the velocity of the piece that is moving upwards. I know that in an explosion momentum is conserved, so I used $p_i=p_f$, where I used the initial velocity in the $y$-direction to find the initial momentum in the $y$-direction. Two of the pieces after the explosion move in the $y$-direction, so I used those in my my final momentum to find the final velocity. Unfortunately when I did this I got the wrong answer because in the solution you are supposed to use the momentum right before the explosion when it's at its maximum height. My question is why do I have to use the momentum right before the explosion, why can't I use the momentum at the very beginning, if momentum is conserved?
 A: 
why do I have to use the momentum right before the explosion, why can't I use the momentum at the very beginning, if momentum is conserved?

Momentum is conserved only in the absence of external forces, i.e. when the net (external) force acting one the system is zero. The impulse-momentum theorem is defined as
$$\vec{p}_i + \vec{J} = \vec{p}_f$$
where $\vec{p}_i$ and $\vec{p}_f$ are initial and final momentums of the system, and $\vec{J}$ is the impulse that acts on the system. Emphasis is on the word system which can include more than one object. Internal forces between the objects are by third Newton's law of motion equal in magnitude and opposite in direction. When you calculate the total impulse of the system of two or more objects, all internal forces cancel and their contribution to the total impulse is zero. Momentum is conserved only when the total impulse is zero, i.e. $\Delta \vec{p} = \vec{p}_f - \vec{p}_i = \vec{0}$.

While the projectile is moving towards the maximum height, there is a force of gravity that shows in the total impulse $\vec{J}$. This means that the system momentum in vertical axis is not conserved. If you neglect the drag, momentum in horizontal axis will be preserved since there is no external force acting in the horizontal axis.
However, when the explosion happens, the internal forces are much larger in magnitude than the gravitational force. Although theoretically the momentum just before and after the explosion is not conserved, for all practical purposes we can say the momentum is conserved because explosion happens for a (very) short period of time during which gravitational force does not contribute to the impulse much.
