Module 07
Energy
Two identical balls. One rests at the top of a hill, the other rests in the valley below. Walk around them with every instrument this course has built. Velocity: zero and zero. Momentum: zero and zero. Mass, the same. State for state, the dials agree completely.
Now nudge each ball an inch. The valley ball dribbles and stops. The hilltop ball leaves, gathers speed all the way down, and arrives at the bottom moving fast enough to hurt.
That motion came pouring out of a ball whose every reading was zero. It was not in the velocity; the velocity was nothing. It was somewhere the instruments of modules 01 through 06 cannot point. Where, exactly, is motion hiding while it is not happening?
The takeaway
Energy is the second conserved currency. Motion holds it as kinetic energy, position holds it as potential energy, and forces move it between the two as work. In a frictionless closed world the total is untouchable, and whole journeys collapse to a single subtraction.
The second set of books
Module 06 taught the move: find a quantity the world can only relocate, never create or destroy, and entire calculations fall away. Momentum was the first such currency. This module opens the second, and it is the one that answers the hilltop.
Start with the transaction. When a force pushes a body through a distance, the force does work: force times distance moved along it. Push a cart for ten meters and you have deposited something in it; the deposit shows up as speed. The account it lands in is kinetic energy, motion's holdings, and the math section will show the balance is exactly .
The hilltop ball's secret is the other account. Lifting the ball there took work, deposited against gravity, and gravity is an honest banker: it holds the deposit as long as the ball holds the height, and pays it all back, with nothing skimmed, on the way down. Holdings that live in position rather than motion are potential energy. The hilltop ball and the valley ball had identical checking accounts, both empty, and wildly different savings.
So run the two-account analogy in full: kinetic is checking, visible and spendable; potential is savings, quiet and positional; work is the transfer between them; and the combined balance, in a closed frictionless world, never moves. Here is where the analogy breaks: real ledgers leak. Friction skims every transaction and wires the take to trillions of accounts too small to audit, the jiggling of atoms, and classical mechanics at this altitude writes that money off as gone. The total is still conserved in the universe's books; it has left ours. The frictionless idealization is the price of a clean theorem, and this course pays it knowingly.
What the theorem buys is a different kind of shortcut from momentum's. Momentum skipped the inside of an instant, the crash too fast to follow. Energy skips the length of a journey. Ask how fast the ball arrives at the bottom and conservation answers without touching the path: total at the top equals total at the bottom, solve, done. The slope's shape, the twists, the time it all took, never enter. Two currencies, two kinds of silence, and whole regions of every problem stop needing to exist.
The math
Take a constant force pushing a mass through distance along its motion. Define the work done:
Module 04 solved constant acceleration. Take its two results, eliminate time between them. From , time is . Substitute into the position formula and simplify:
Multiply both sides by and use :
Read it right to left: work in, and the quantity rises by exactly that much. That quantity is the kinetic energy , and this is the work-energy theorem.
Now let the force be gravity, with up positive. Raising a body by height means gravity does of work, so the body's account fell by on the way up and recovers it on the way down. Define the potential energy and the total:
Check that cannot move. Differentiate along the motion, using and :
Zero, identically, at every instant of any flight. Work the drop: fall from height starting at rest, then gives . From 4.9 meters: meters per second, and the mass cancelled again on its way out.
Play with it
A marble in a landscape you own. Drag the five posts to reshape the terrain, drop the marble in, and watch the two accounts trade under a total that refuses to move. Build a wall it cannot climb; the bars will tell you why it cannot.
Check yourself
Problem 1. A ball drops from rest at 4.9 m, with g = 9.8. How fast is it moving at the ground?
Reveal the solution
mgh = ½mv², mass cancels, v = √(2 × 9.8 × 4.9) = 9.8 m/s.
Problem 2. A skater starts from rest 5 m up a frictionless ramp of completely unknown shape, g = 10. How fast at the bottom, and why does the shape not matter?
Reveal the solution
v = √(2 × 10 × 5) = 10 m/s.
Conservation compares the two ends only: the total at the top equals the total at the bottom, whatever happened in between. The ramp's shape lives in the skipped middle.
Problem 3. A 2 kg cart at 6 m/s crosses a rough patch that removes 16 J of energy. How fast does it leave the patch?
Reveal the solution
Kinetic energy in: ½ × 2 × 36 = 36 J. The patch skims 16, leaving 20 J: ½ × 2 × v² = 20 gives v = √20, about 4.5 m/s. The 16 J is not destroyed; it moved to accounts too small for this ledger.
What this did to the demon
The demon now runs two sets of books, and between them entire computations go quiet: crashes settle by momentum, journeys settle by energy, and only the in-between still needs the old tick-by-tick grind. What it does not have is a reason. Why these two currencies and not others? Why does nature keep books at all? Module 08 stops asking what happens next, and asks instead what the whole path is up to.
Least action