A roller coaster never climbs higher than its first hill. A stretched spring remembers exactly how hard you pulled. Rub your hands and the motion doesn't vanish — it warms.
Energy is never created or destroyed, only traded between forms. Let's follow the trades — and launch, roll and lift them yourself.
Push something and you do work: W = F·d. That work doesn't disappear — it becomes energy stored somewhere, and it can always be traced. Lift a book and the work is banked as height. Compress a spring and it's stored in the coils. Speed a car up and it's carried as motion.
Every process in this chapter is a trade between those accounts. The total never changes; it just moves. That single accounting rule is arguably the most useful idea in all of physics.
Energy banked by position — height in a gravity field (mgh) or squeeze in a spring (½kx²). Silent, waiting, ready to be spent.
Energy carried as motion, ½mv². Notice the square: double the speed and you carry four times the energy — which is why crashes get so much worse so fast.
Everything below is hands-on. No sliders for the physics — you set the coaster's first hill, compress a spring launcher, drop a skater into a half-pipe, haul a crate against a stopwatch, and light a bulb from a dam.
Every coaster ever built obeys one iron rule: after the lift hill, no later hill may be as tall. The reason had to be settled by an argument that raged for fifty years. Newton's followers insisted a moving body's "quantity of motion" went as mv. Leibniz argued for mv². In the 1740s Émilie du Châtelet ended it — dropping brass balls into soft clay and showing that a ball at twice the speed sank four times as deep. Energy goes as the square of speed.
So the trade is exact: mgh = ½mv². Height converts to speed, speed converts back to height, and the sum stays fixed. The car cannot buy a hill it never paid for on the way up.
Set the first hill, release the car, and watch the two energy bars trade back and forth without ever changing their total.
Kinetic energy scales as the square of speed, so a car at 60 km/h carries four times the energy of one at 30 — and needs four times the braking distance. Du Châtelet's clay experiment is quietly built into every crash barrier and airbag on the road.
In 1676 Robert Hooke published a meaningless string of letters — ceiiinosssttuv — to stake his claim without revealing it. Two years later he unscrambled it: ut tensio, sic vis — "as the extension, so the force." Pull a spring twice as far and it pulls back twice as hard: F = kx.
Because the force grows steadily from zero, the energy you store isn't F×x but the area under that straight line: E = ½kx². Squared again — so compressing twice as far doesn't double the launch, it quadruples the stored energy. Release it and every joule pours into speed, then into height.
Haul the block back against the spring and let go. Watch the green store empty into gold, then into blue.
A pole-vaulter's fibreglass pole stores the energy of their sprint and hands it back as height. Car suspensions, mousetraps, watch mainsprings, bows, and the tendons in your own legs all bank energy the same way — and give it back on demand.
A real skater in a real half-pipe comes back a little lower every time, and eventually stops. For centuries that looked like energy being destroyed. Then a brewer's son named James Joule spent years measuring, with fanatical precision, exactly how much heat a falling weight produced when it churned a paddle in water. He even lugged a thermometer on his honeymoon to measure a waterfall's temperature at top and bottom.
He proved that the missing mechanical energy turns up precisely as heat. Nothing is lost; it is degraded — spread into the disordered jiggling of atoms, from which it will never spontaneously reassemble. Friction does negative work, W = −μmg·d, and that exact amount appears in the thermal account.
Drop the skater in and watch the total bar refuse to shrink — even as the red thermal slice eats the ride.
Joule found that 4.2 joules of mechanical work always produces one calorie of heat — the mechanical equivalent of heat. It welded mechanics and thermodynamics into one science, and the SI unit of energy now carries his name.
Doing work is one thing. Doing it quickly is another, and it is what people actually pay for. James Watt knew this: to sell steam engines to mine owners who thought in horses, he measured how fast a strong horse could haul coal and defined one horsepower as 550 foot-pounds per second — about 746 watts. He was, deliberately, a little generous, so his engines would always outperform the promise.
Power is simply the rate of energy transfer: P = W/t = F·v. Lift the same crate to the same height slowly or quickly and you do identical work — but the fast lift demands far more power. That distinction is the difference between a bicycle and a superbike.
Three challenges. Do the work, then do it fast, then out-power a horse.
Watt's horsepower was a sales tool — and it outlived the steam engine, the horse, and Watt himself. Your kettle draws about 2,000 W, a family car around 100,000 W, and a trained cyclist can hold roughly 300 W for an hour. One horsepower, sustained, is beyond almost any human.
The same trade you felt on the coaster — height for speed — can be sold as electricity. In 1895 the Niagara Falls Power Company opened the first large commercial hydroelectric plant: water dropped through penstocks, spun turbines, and generators pushed current to factories miles away. George Westinghouse and Tesla's AC system won the “war of the currents” by making that long-distance delivery practical.
Ideal hydraulic power is the energy of falling water, delivered every second: P = η·ρ·g·h·Q — efficiency × density × gravity × head × flow rate. Raise the reservoir (more head) or open the gate (more flow) and the power climbs. That power is the load: a filament that glows only when joules arrive fast enough.
Open the dam gate, spin the turbine, and light the bulb — PE → KE → electrical → light.
Every extra metre of head multiplies power. A tall reservoir is a PE battery; the penstock converts it to speed; the turbine and generator hand the joules to the grid. Your bulb is just the last trade — electrical energy into light and heat — the same conservation ledger as every experiment above.
Everything you just did by hand — the car that could never out-climb its first hill, the spring that quadrupled its store, the skater whose ride turned into warmth, the crate you hauled against the clock, the dam that lit a bulb — is one law seen five ways.
Work done is energy transferred; stored as height or squeeze.
The total never changes — it only moves between accounts.
The rate of the trade. Same work, less time, more power.
Once you see the ledger you see it everywhere: food becoming muscle motion, sunlight becoming wind, a phone battery draining into light and sound, a dam trading height for electricity, and every joule you have ever spent still out there somewhere, quietly warmer.
Energy is the universe's currency, and no one has ever caught it being counterfeited. That's the whole chapter.