.
If the impact is "perfectly plastic", it means that after the collision, the two mass joint (clue) in one mass 2 + 2 = 4 kg.
This term also means that the momentum conservation law is held.
The mechanical energy conservation law is not held in this case: part of the mechanical energy transforms in the heat energy,
so the mechanical energy conservation law IS NOT included to the model in this case.
Only the momentum conservation law works, and it is just enought.
So, your equation, based on the momentum conservation law, is
= 2*4 + 2*0.
The left side is the momentum after the collision; the right side is the momentum before the collision.
The value of "v" in the left side is the resulting speed of the joint mass .
Substitute the values of masses into the last equation, simplfy and solve for "v", the only unknown.
(2 + 2)*v = 2*4 + 2*0
4v = 8
v = 8/4 = 2 m/s.
ANSWER. The velocity of the joint mass after the impact is 2 m/s.
Solved.
--------------
Notice that the question to the problem is formulated INCORRECTLY.
The correct formulation is THIS :
"Determine the velocity of the joint mass after impact, if the impact is perfectly plastic."
Also note, that instead of the term "the impact is perfectly plastic", more often the term "the impact is perfectly inelastic" is used.