[UPSC IFoS 2020] Conservation Laws (Q13, 10M)

A rocket of mass m_1 + m_2 is launched with a velocity whose horizontal and vertical components are u_x and u_y. At the highest point in its path, the rocket explodes into two parts of masses m_1 and m_2 that separate in a horizontal direction in the original plane of motion. Show that the fragments strike the ground at a distance apart given by D = \frac{u_y}{g} \sqrt{\frac{2(m_1 + m_2)K}{m_1 m_2}} where K is the kinetic energy produced by the explosion. (Neglect air resistance, the mass of explosive, any spinning motion of the fragments and assume that g is constant)