Work, Energy, and Power

The Currency of Physics

Energy is the universal currency of physics — the capacity to do work, conserved through every process. Work is the transfer of energy by a force: $W = F d$ for a force acting over a distance in its direction (more precisely $W = Fd\cos\theta$, so a force perpendicular to the motion does no work). Work is measured in joules.

Two central forms of mechanical energy:

• Kinetic energy — energy of motion, $KE = \tfrac{1}{2} m v^2$. Note the square: doubling the speed quadruples the kinetic energy (which is why stopping distances grow so fast with speed).
• Potential energy — stored energy of position. Near Earth, gravitational PE is $PE = mgh$ (mass times gravity times height).

The work-energy theorem ties them together: the net work done on an object equals its change in kinetic energy. And the crown jewel is conservation of energy: in an isolated system with no friction, the total mechanical energy $KE + PE$ stays constant — energy merely converts between forms. A falling ball trades potential for kinetic energy; a pendulum swaps them back and forth. Energy is never created or destroyed, only transformed (friction converts it to heat, which is why real systems seem to "lose" energy).

Power is the rate of doing work: $P = \frac{W}{t}$, measured in watts. Two engines can do the same total work; the more powerful one does it faster. This distinction — energy (total) versus power (rate) — is exactly the one that matters for computing hardware, where a chip's energy per operation and its power (heat) budget are separate, critical constraints.

Common pitfall: confusing energy with power, and forgetting the square in kinetic energy. Power is the rate of energy transfer ($P = W/t$), not the energy itself — a low-power device can still deliver lots of energy over time. And $KE = \frac{1}{2}mv^2$ depends on $v$ squared: doubling the speed quadruples the kinetic energy, not doubles it.

A ball dropping: its accent potential-energy bar shrinks as the kinetic-energy bar grows by an equal amount, their sum a constant total — energy converting form while conserved.

$$KE + PE = \text{constant}$$