Energy, work and power — common mistakes

Always draw a clear diagram and mark on all forces. When using the energy principle, clearly state your initial and final points and your chosen zero level for GPE. Be meticulous with your energy accounting: Initial Energy + Work In = Final Energy + Work Out.

Kinetic energy ($KE = \frac{1}{2}mv^2$) can never be negative, as mass and $v^2$ are always non-negative. Gravitational potential energy ($GPE = mgh$) can be negative if the object is below the chosen zero-level (e.g., in a ditch, if ground level is $h=0$). A change in energy, $\Delta E$, can certainly be negative, which signifies that the system has lost energy.

They have the same magnitude but opposite signs. 'Work done by friction' is negative because the frictional force opposes the direction of motion; it represents energy removed from the system. 'Work done against friction' is positive, representing the energy that must be supplied by another force to overcome the frictional loss. In energy balance equations, you either subtract the 'work done by friction' from the initial energy, or you add the 'work done against friction' to the final energy side of the equation. For example: $E_i = E_f + W_{against friction}$.

It is the instantaneous speed. The formula gives the power being delivered at the specific moment in time when the object has speed $v$ and is being pushed by a driving force $F$. If the driving force and velocity are constant, then the power is also constant.