I recently encountered a question that made me think of this "paradox":
There are two forces: Force 1 and Force 2. They accelerate a mass $m$ from the same initial velocity to the same final velocity in the same amount of time, but as you can see from the graph, when compared at the same instants in time, the two processes have different accelerations.
How does the work done by each process compare?
It is easy to see that the work done by both forces must be the same because the change in kinetic energy is equal, which is correct.
However, work can also be written as $Fd$. Now Newton's Second Law says $F=ma$, so $W=mad$.
From the graph, it seems that the average accelerations are equal for both forces, as they result in the same change in velocity over the same interval of time. But it's also clear that Force 1 results in greater displacement (area under a velocity-time graph). Therefore, shouldn't Force 1 do more work?
I have a vague sense as to why the first reasoning was correct and the second incorrect, which has to do with average accelerations vs. instantaneous accelerations, but can anybody give me a definite explanation as to why the work done for both processes are the same?