Question

An internal explosion breaks an object, initially at rest, into two pieces, one of which has 2 times the mass of the other. If 22500 J is released in the explosion, how much kinetic energy does each piece acquire?

Answer 1

The lesser mass will have twice the velocity.

KE= 1/2mv^2 +1/2 2M (v/2)^2=22500
1/2mv^2+ 1/4mv^2=
so the smaller piece has twice as much energy than the larger
so smaller has 2/3 22500 Joules

Answer 2

It is not necessary that all of the energy of the explosion be converted to kinetic energy of the two pieces. There will usually also be heating of the pieces, and the release of gas.

IF 22500 J is released in the explosion as kinetic energy of the only two pieces, then the approach of BobPursley will yield the answer.