Work, Energy, and Power

Translational kinetic energy, $K = \frac{1}{2}mv^2$, is the energy an object has because it is moving. It's a scalar, so direction doesn't matter and it is never negative. Observers in different reference frames measure different speeds, so they can measure different kinetic energies for the same object.

Work is energy transferred into or out of a system by a force acting over a displacement: $W = \int_a^b\vec{F}\cdot d\vec{r}$. It equals the area under a graph of the force component along the motion versus position. Only that parallel component does work, so work can be positive, negative or zero. By the work–energy theorem, the net work on an object equals its change in kinetic energy. Work done by a conservative force like gravity doesn't depend on the path taken, but work done by friction does.

Potential energy belongs to a system of objects that interact through conservative forces, and it depends on their positions: $\Delta U = -\int\vec{F}\cdot d\vec{r}$ and $F_x = -\frac{dU}{dx}$. On a U(x) graph, minimums are stable equilibrium points and maximums are unstable ones. Key cases are a spring, $U_s = \frac{1}{2}k(\Delta x)^2$, gravity near Earth, $\Delta U_g = mg\Delta y$, and gravity between spherical bodies, $U_G = -\frac{Gm_1m_2}{r}$.

Mechanical energy is the sum of a system's kinetic and potential energies. If no external work is done on the system and nothing like friction acts inside it, its mechanical energy stays constant; otherwise energy is transferred between the system and its surroundings or turned into thermal energy and sound. Choosing the system carefully decides which energies you track and whether the total stays the same.

Power is how fast energy is transferred or changed from one form to another: average power is $P_{avg} = \frac{\Delta E}{\Delta t}$ and instantaneous power is $P = \frac{dW}{dt}$. For a force on a moving object, $P = \vec{F}\cdot\vec{v}$, so only the force component along the velocity delivers power. Power is measured in watts, or joules per second.