Say that I am a force which wants to act to keep an object which is in motion at the same velocity it is already in motion.

D is a set of forces which want to act to bring the object to a complete stop.

D must expend at least as much energy as is in the object to bring it to a stop as the object actually has. If it is a mass which weighs 10kg and is traveling 1m per second, then my object has a total of 10 joules at its disposal and 10 joules worth of work has to be done in order to bring the object to a standstill.

Now say that it would take D 10 feet to bring the object to a standstill. We can say then that D is expending 1 joule of energy per foot, on average.

In my understanding, the law of inertia says that things don’t like to lose their energy. I don’t need to expend 1 joule of energy every foot to counteract D because I have inertia on my side. To pick a random number, we’ll say that I only have to expend 0.5 joules of work per foot in order to keep the object moving at a constant velocity.

If the object starts at rest, however, then I have the same job to do as D does, except in reverse. I have to expend at least 10 joules to bring the object up to speed. In this case, I don’t have inertia on my side, so if I bring it up to speed over a space of 10 feet, then I also have to match the 10 joules that D attempts to apply in the reverse direction. For the first 10 feet of my trip, I expend 20 joules. For the remaining 90 feet, I expend 45 joules, for a total of 65 joules of work for the 100 foot walk.

If I bring it up to 1 m/s then let it drift to a stop, bring it back up to 1 m/s let it drift to a stop, etc. then my math becomes 20 joules per every 20 feet. Over the course of my trip, I end up expending 100 joules instead of the 65 I could have done.