Energy and Conservation Laws

The Currency of Physics

Energy is a conserved quantity that flows through every physical process — the universal currency of physics. Work is the transfer of energy by a force acting over a distance: $W = \int \vec{F} \cdot d\vec{x}$, reducing to $W = Fd$ for a constant force along the motion (and $W = Fd\cos\theta$ in general, 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, $KE = \tfrac{1}{2}mv^2$. Note the square: doubling the speed quadruples the kinetic energy.
• Potential energy — stored energy of configuration. Gravitational PE near Earth is $PE = mgh$; a spring stores $\tfrac{1}{2}kx^2$.

The work–energy theorem ties work to motion: the net work on an object equals its change in kinetic energy. The crown jewel is conservation of energy: in an isolated system with only conservative forces (no friction), total mechanical energy $KE + PE$ is constant — energy converts between forms but is never created or destroyed. A falling ball trades potential for kinetic energy; a pendulum swaps them back and forth. Friction converts mechanical energy to heat, which is why real systems appear to "lose" energy — the energy is not gone, just transformed.

Power is the rate of doing work, $P = \frac{dW}{dt}$, in watts — distinct from energy (the total). Conservation laws are among the most powerful tools in physics: they let you relate initial and final states without tracking the detailed motion in between, turning hard dynamics problems into simple bookkeeping. Momentum and angular momentum are conserved too, each tied (by Noether's theorem) to a symmetry of nature.

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. And $KE = \frac{1}{2}mv^2$ depends on $v$ squared: doubling the speed quadruples the kinetic energy, not doubles it (which is why stopping distances grow so sharply with speed).

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.

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