In a perfectly inelastic one-dimensional collision between two objects, what condition alone is necessary so that the final kinetic energy of the system is zero after the collision?

In all types of inelastic collision we know that two objects move together with same speed after collision.

Now we know that in this type of collision energy is lost due to its permanent change in shape or size

Now when two objects will collide then the total momentum of the system will remains conserved as there is no external force on the system

So here we can say that

now we know that after collision if system comes to rest

then we will have

so here final kinetic energy is zero in all types of inelastic collision when the two objects have initial momentum zero during their motion