Horizontal M*V doesn't change:
Initial horisontal M*v = 100 * cos(60) * 80 = 4000
M*v after crasch = 60 * 80 - 40 * x = 4000 => x = 20
We can find the time for max height by noting that the y component of the velocity of the rocket
is equal to 0 at that time.
Plugging that time into the y equation gives the max height.
For the explosion stage of the problem, note that the explosion is internal, meaning that the momentum of the rocket and its pieces is conserved. Thus:
( is the velocity at max height, which will simply be the x component of the velocity.)
Solve for v.
For the last question, you know v and you know the height. So you can set up your x and y equations again and find out where Q lands. (You will need both the x and y coordinates of the max height to start you off on this.)