As you are trying to move a heavy box of mass m, you realize that it is too heavy for you to lift by yourself. There is no one around to help, so you attach an ideal pulley to the box and a massless rope to the ceiling, which you wrap around the pulley. You pull up on the rope to lift the box.

Use g for the magnitude of the acceleration due to gravity and neglect friction forces.

Once you have pulled hard enough to start the box moving upward, what is the magnitude F of the upward force you must apply to the rope to start raising the box with constant velocity?

As you are trying to move a heavy box of mass m, you realize that it is too heavy for you to lift by yourself. There is no one around to help, so you attach an ideal pulley to the box and a massless rope to the ceiling, which you wrap around the pulley. You pull up on the rope to lift the box.

Once you have pulled hard enough to start the box moving upward, what is the magnitude F of the upward force you must apply to the rope to start raising the box with constant velocity is given below

Explanation:

When you pull with a force upward so that the box is just about to rise, the sum of the two tensions (one from the ceiling and one from your hands pulling) equal the weight of the box.  Thus C + H = W; where C is the ceiling rope tension and H is the hand rope tension.  W = mg, the weight of the box.

Because there is no mechanical advantage in the single pulley, C = H, the two tensions are equal.  So, we can write 2H = W = mg; and thus, H = W/2 = mg/2 when the box is just about to rise.  To raise the box, you must have 2H>W; so that H > W/2 which means you need to yank upward on the rope in your hand with a force H (your F) just a bit over half the weight of the box.

Now here's the tricky part.  Once you've gotten the box to accelerate (start moving), you can relax the force H a bit so that it exactly equals H = W/2.  Why?  Because by keeping the net force f = ma = W - 2H = 0, acceleration settles in at a = 0 and the box is neither speeding up nor slowing down.  It is then at constant velocity, which is what you specified.

Lesson here is that it takes a bit more force to get something to start moving, but once it starts to move we need to slack off a bit to maintain a constant velocity (too much force and it continues to speed up, too little and it slows down).

PS: Your question starts off with "Energy" then you talk and ask about force.  They are not the same thing.  I recommend you read up on the differences between energy and force.  That will avoid a lot of confusion in future physics studies.