I am trying to make sense of the meaning of the definition of work. The original definition of work was also known as "the weight lifted through a height." I was hoping that our mathematical definition of work would correspond to this notion in a well-defined way but it seems like it doesn't.
According to our mathematical definition of work, if I "lift the weight through a height" at two different constant accelerations, say, I get two different values for the amount of work I have done.
Why do we make the definition like this? Why don't we make the definition such that the work done by "lifting a weight through a height" doesn't depend on the velocity/acceleration with which it is lifted. It seems the definition was originally intended for this and it got changed along the way somewhere.
Consider a path $x(t): x(0) = 0, x'(0) = 0$ taking values on the real line and $x''(t) = a = F$. Then if $T_a = \frac{\sqrt{2}}{\sqrt{a}}$ then $x(T_a) = 1$. Also $W = \int_0^{T_a} F \cdot x'(t) dt = \int_0^{T_a} a \cdot at dt = a$.