Let's say there is a 1kg lump of rock floating in space.
Through some means I apply an acceleration of 3 m/s² to the rock for one hour.
How much energy did I spend, assuming none was wasted?
Here is what I'm trying:
Force = mass * acceleration. So F = 1kg * 3m/s² = 3 newtons.
This must mean I'm applying 3N for 1 hour.
Joules = newtons * metres.
What? This doesn't make sense. I thought we'd already accounted for all derivatives of position. Suppose I stopped for half an hour in the middle of the exercise. I would have spent the same amount of total energy, but the rock would have travelled further.