Energy & Conservation

The one rule the universe never breaks

Junior level — plain language, no maths

Of all the laws in physics, one stands out for never, ever being broken: energy cannot be created or destroyed. You can move it around and change what form it takes, but you can never make a single drop of it from nothing, and you can never make any of it vanish. This is the conservation of energy, and it holds everywhere from a firefly to a supernova.

Energy just wears different costumes. A ball held up high has potential energy — stored energy of position, waiting to be used. Let it go and that stored energy turns into kinetic energy, the energy of motion, as the ball speeds up on the way down. At the bottom it's all motion; at the top it was all stored. The total never changes — the ball simply keeps trading one kind for the other.

You can watch this trade everywhere: a skateboarder pumping up and down a ramp, a swing arcing back and forth, a roller coaster cresting a hill and plunging down the other side. So why does everything eventually stop? Because a third costume, friction, keeps quietly stealing a little energy each cycle and turning it into heat. The energy isn't gone — it's just spread out as warmth, too scattered to use. In the simulation below, drop a ball into a bowl and watch height become speed become height.

Kinetic energy, potential energy and mechanical conservation

Student level — the core equations

Two quantities do most of the work in mechanics. Kinetic energy, the energy of motion, is \(KE = \tfrac{1}{2}mv^2\) — note the \(v^2\): doubling your speed quadruples your energy, which is exactly why stopping distances balloon at high speed. Gravitational potential energy, the energy of height, is \(PE = mgh\). In any system where only gravity does work, their sum is fixed: \(\tfrac{1}{2}mv^2 + mgh = \text{constant}\).

That single equation solves a huge range of problems without ever tracking the messy details of the motion. Drop the ball from height \(h\) and all its \(mgh\) converts to \(\tfrac{1}{2}mv^2\), giving an impact speed \(v = \sqrt{2gh}\) — independent of mass, the same result Galileo found. A pendulum, a roller coaster, a ski jump: each is just potential energy and kinetic energy trading places while their total holds steady.

Energy also connects to force through work, \(W = Fd\), and to time through power, \(P = W/t\) — the rate of energy transfer, measured in watts. Real systems, though, are never perfectly conservative: friction and air resistance do negative work and convert mechanical energy into heat. The energy is still conserved overall — it has just left the tidy mechanical ledger for the disordered thermal one, which is why real pendulums wind down.

The work-energy theorem, potentials and why energy is conserved

Scholar level — full mathematical depth

01Work as the bridge from force to energy

Energy is not a separate postulate bolted onto Newton — it follows from his second law. Integrate \(F = ma\) along a path and you get the work-energy theorem, \(W_{\text{net}} = \int \vec{F}\cdot d\vec{r} = \Delta KE\): the net work done on a body equals its change in kinetic energy. Kinetic energy \(\tfrac{1}{2}mv^2\) is not a definition plucked from the air; it is precisely the quantity that this integral of force-through-distance changes.

02Conservative forces and the birth of potential energy

Some forces — gravity, springs, electrostatics — do work that depends only on start and end points, never the route taken. Such conservative forces have \(\nabla\times\vec{F} = 0\), which lets us define a potential energy \(U\) with \(\vec{F} = -\nabla U\). Potential energy is bookkeeping for the work a conservative force would do, and for exactly this class of forces the total \(E = KE + U\) is constant. Friction fails the test — its work depends on path length — so it has no potential and drains mechanical energy.

03The deepest reason: symmetry

Why is energy conserved at all? The profound answer is Noether's theorem (1918): every continuous symmetry of a system's laws yields a conserved quantity, and the symmetry behind energy is time-translation invariance — the fact that the laws of physics are the same today as tomorrow. Energy conservation is not an accident of mechanics; it is the shadow cast by the uniformity of time itself. Momentum conservation follows identically from the uniformity of space.

04When energy seems to go missing

Apparent violations are always incomplete accounting. A ball dropped in honey does not conserve mechanical energy, but track the heat and the ledger balances — the first law of thermodynamics, \(\Delta U = Q - W\), simply widens the books to include internal energy. Historically each "loss" forced the concept to grow: chemical energy, then the equivalence of heat and work (Joule), then \(E = mc^2\), which revealed mass itself as concentrated energy and closed the last apparent gap in nuclear reactions.

05The subtle case of an expanding universe

There is one place the tidy law gets slippery. In general relativity, energy conservation is strictly local; globally, in an expanding universe, there is no time-translation symmetry, so total energy is not straightforwardly conserved. The photons of the cosmic microwave background lose energy as space stretches (their wavelengths redden) with nothing obvious catching it. Far from breaking Noether's theorem, this is its fine print: no time symmetry, no global conservation law.

06Free energy: the currency that actually flows

For real processes, raw energy is the wrong bookkeeping. The second law says usable energy degrades: what matters is free energy, \(G = H - TS\), the portion actually available to do work once entropy is paid. Every engine, battery, muscle and living cell trades in free energy, not energy, and its inexorable decline — energy conserved but ever more disordered — is what gives time its direction and ultimately points toward the heat death of the universe.