Wait, a moment. Assume you are in a complete and perfect vacuum (i.e. a closed system), and there is a constant gravitational acceleration of 10m/s/s downwards. You have and object A on the ground (having energy only of its mass), and a single actor (an actor is any object that is capable of exerting force), that has enough energy to accelerate object A at 1m/s/s for 1s upward. Assuming the actor B the performs this action, the box will lift off the ground and at its highest point its gravitational potential energy would have reached its max (and thus it would be at its maximum energy). 
If you apply an acceleration of 1ms
^{2} in the opposite direction of a constant accelerating force of 10ms
^{2}, object A will not move as the resultant force will be 9ms
^{2} down.
I may be wrong here but as a sidenote, would the kinetic energy halfway through the drop be a quarter of the total? 
Consider an object B, with a mass of 100kg, in constant gravitational field of 10ms
^{2}. We drop it from a height of 100m.
Initial gravitational potential energy of B: 100 * 10 * 100 = 100000 joules = 100 KJ
t = 0s Height: 100m Potential: 100KJ
t = 1s Height: 90m Potential: 90KJ
t = 2s Height: 70m Potential: 70KJ
t = 3s Height: 40m Potential: 40KJ
t = 4s Height: 0m Potential: 0KJ

Edit:
Actually, those figures are horribly approximate as it should be 4.5s total time to reach 0... I wonder if there is somewhere online you can make a nice graph of this?
I couldn't find somewhere to do a neat graph, but I did take the time to do the exact calculation (sorry if anyone sees any arithmetic errors, I was too lazy to use a calculator). I'm dropping object B, which weights 100kg, in gravitational field 10ms
^{2} from 125 meters (125m because it made the maths in my head easy. ;)).
Better figues:
t = 0s Height: 125m Potential: 125KJ
t = 1s Height: 120m Potential: 120KJ
t = 2s Height: 105m Potential: 105KJ
t = 3s Height: 80m Potential: 80KJ
t = 4s Height: 55m Potential: 55KJ
t = 5s Height: 0m Potential: 0KJ

(Useless fact of the day: I went a little further with the maths and figured the gravitational potential energy of an object with mass 100kg suspended about 1.1km high is about the same as the total energy you're going to use up today.) < That's all very bad arithmetic and don't listen to any of it.