Problem $2$: $[15$ points$]$ A rocket is projected straight up and explodes into two equally massive fragments just as it reaches the top of its flight. Assume the angle $$ between the two fragments at separation is $90.$ The fragments are observed to land at time $t,$ after the burst. Using the conservation of linear momentum and kinematics equations, find the height $h(t)$ at which the fragmentation occurred and the distance of the two fragments upon landing $d(t).$