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So the book I'm using derived this connection between work and potential energy when a partikle is moved from point 1 to point 2

[tex]W_{1-2} = V_1 - V_2 [/tex]

Now imagine i lift an object a distance h i would get

\begin{cases}V_1 = 0 \\ V_2 = mgh\end{cases}

and I end up with a negative work

[tex]W_{1-2} = -mgh [/tex]

Is the work supported to be negative when i lift something? Also if i did this work over a certain time ##\Delta t## i get

[tex]P = \frac{-mgh}{\Delta t}[/tex] a negative power. Which gives me a sign error on every exercise i try while in the book's examples it calculate the work to be negative.

Then to make things even more confusing for me theres this second law of kinetic energy

[tex]W_{1-2} = T_2 - T_1[/tex]

So before i start lifting something it got a velocity of 0 just sitting there at a certain height while after I'm done with the lift and just hold it a bit higher up the velocity is again 0. so i get

[tex]T_2 = T_1 = 0 \Longrightarrow W_{1-2} = 0[/tex]

how is this consistant, where am i going wrong?