If it takes 10 energy units to move a 10kg weight 10 meters, does the energy expended go up as the time used decreases?
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Movement, in and of itself, does not take energy.
Movement *against a force*, like friction or gravity, takes energy, at least from the perspective of just you and the moving object (energy is still conserved overall, but you + the moving object might gain or lose it when taken in isolation).
If you’re pushing an object along the ground, the main force resisting movement is friction. And for everyday objects at everyday speeds, friction (with a surface) is independent of how fast the object moves: it’s just the weight of the object (or more properly, the normal force, but the two are equal for an object lying on a surface) times a number called the coefficient of (kinetic) friction that describes how “slippery” the object is. In other words, regardless of your speed, for any fixed object, the frictional force is ~the same.
The energy required to move an object against a force (or the energy gained by letting it move with a force) is called *work*. And the formula for work is W = F * d, where F is the force you’re moving with or against and d is the distance you move. Since in this case both the force from friction, and the distance moved, are the same, the amount of work required is the same, too.
Note that we’re making a lot of simplifying assumptions here, though. Like most engineering/mechanics questions, there are details that can matter, like the exact surface structure of the object we’re moving.
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