newtonian mechanics - How can we move an object with zero velocity?

Consider there is a box of mass $m$ at rest on the floor. Most books give an example that we need to do a work of $mgh$ to lift the box $h$ upward.

If we analyze this work done, the external force acting on the box by us should be equal to the weight of the box. Therefore the net force is zero which in turn there is no acceleration. If there is no acceleration and the initial velocity of the box is also zero, how can the box move upward?

Answer

In introductory problems about work you're normally taught that it's force times distance:

$$ W = F \times x $$

and you treat the force as constant. If you look at the problem this way then you're quite correct that if the force is $F = mg$ then the box can't accelerate so it can't move. However a more complete way to define the work is:

$$ W = \int^{x_f}_{x_i} F(x) dx $$

The force $F(x)$ can be a function of $x$, and to get the work we integrate this force from the starting point $x_i$ to the final point $x_f$. Because $F(x)$ can vary we can make $F > mg$ at the beginning to accelerate the box then make $F < mg$ towards the end so the box slows to a halt again.

DavePhD comments that work is not a state function, and in general this is true. However in this case the work done is equal to the change in potential energy so as long as the box starts at $x_i$ at rest and ends at $x_f$ at rest we'll get the same work done regardless of the exact form of $F(x)$.

If you're really determined to have $F$ constant then start with $F > mg$ at the beginning and $F < mg$ at the end, then gradually reduce the initial value of $F$ and increase the final value to make the force more constant. This will cause the time taken to move the box from $x_i$ to $x_f$ to increase. The limit of this process is a completely constant value for $F$, in which case it takes an infinite time to move the box.