This is a problem from the book 'Problems in General Physics' by I E Irodov. Two blocks of masses $m_1$ and $m_2$ are placed on a rough horizontal surface, connected by a light spring.
Find the minimum constant force that has to be applied on the block with mass $m_1$ so that the other block just begins to slide.
Consider the limiting case:
Suppose the spring has compression $x$ by that time. Block $m_2$ is just about to move.
We can write, $$kx=\mu m_2 g$$
And, by the work energy theorem
$$Fx-0.5kx^2-\mu m_1 gx- \mu m_2gx= 0.5m_1 v^2$$ For the system of $m_1$, $m_2$ and the spring.
This is where I am stuck. According to the answer, the force required is minimum when $v=0$.
Why is the block $m_1$ at rest in the limiting case? I cannot find any suitable justification for this, neither an intuitive explanation. Also, is something wrong with my Work Energy theorem expression?