The faster the wire moves the higher the voltage generated. This voltage acts in the opposite direction to the current we are feeding in, making it harder and harder to force that current through the wire. For a given current, low speed equals low back voltage and hence low electrical power input. A high speed generates a high back voltage and thus requires a high electrical input power. Now the mechanical energy out is simply equal to the force exerted times the speed.
Low speed equals low output power and high speed equals high output power.
Computing the input and output powers The power in watts which we are feeding into the wire is the product of the current and the voltage. If the wire is moving at a constant velocity and is lifting a weight at so many meters per second. Then the mechanical power being generated is just the product of the velocity times the force on the wire. The force is proportional to the current and the back voltage is proportional to the velocity so the input power is proportional to the output power. When you trouble to multiply out all the units it turns out that the mechanical power out is exactly equal to the electrical power in.