Why is potential energy and why is change in potential energy equal to negative of work done.
prove this formula.
\[\delta\] u=-w

Potential Energy can be negative or positive because it is all relative. calling U = mgh (for approximations close to the surface) you could make U=0 the floor or U=0 the ceiling, it wouldn't matter, since the change would be the same anyway. The change in energy can be positive or negative. Kinetic energy can never be negative, since energy is a scalar quantity, but the change in Kinetic Energy can be positive or negative. Work can be positive or negative, depending on the direction of the force and displacement. However, Work is equal to the CHANGE in energy, not the actual quantity of energy itself.
Do you know the work-energy theorem(Change in Kinetic energy = Total work done)? According to conservation of energy, K1+U1=K2+U2 ΔU=U2−U1=−(K2−K1) By the work-energy theorem, ΔW=K2−K1 Thus, ΔU=−ΔW