物理 · Physics

Energy & Momentum

Day 3 · 2026 · Phase A — The Skeleton of Classical Mechanics
Instead of chasing how the force changes at every instant, watch the few numbers that never budge — they are the cheapest, and the deepest, structure in all of physics.
Once you've met force and motion, a smarter question appears: stop tracking the force frame by frame, and watch instead the quantities that refuse to change no matter what you do. A thrown ball, a collision, a planet orbiting the Sun — the process shifts wildly, yet a few numbers sit perfectly still: momentum and energy. This isn't a lucky shortcut; it's the deepest scaffolding in physics — conserved quantities are the shadow cast by symmetry (the seed planted back in "What Is a Physical Law" — here it lands). Four cards: conservation of momentum, the forms of energy, where energy goes in a collision, and why conservation runs deeper than force.

Momentum: the "Amount" of Motion Momentum & Its Conservation

Classical mechanics · conservation
Intuition A gun kicks back when you fire it; you push off the diving board and it launches you upward; toss a wrench in space and you drift off the opposite way. These are the same event: shove "amount of motion" one way, and you get an equal amount of motion the other way. That "amount of motion" = mass × velocity, and it's called momentum.
Mechanism Momentum p = mv (mass times velocity, with direction). Newton's second law, in its original form, isn't F=ma but
F = dpdt
Read it as "force equals the rate of change of momentum." d means "a tiny change": dp is a small change in momentum, dt a short interval of time, and their ratio is "how fast momentum changes." For an isolated system (no outside force), the internal pushes come in pairs, and Newton's third law makes each pair equal and opposite — they sum to zero, so total momentum is conserved: total Δp = 0 (that 0 is the number zero).
Before Total momentum = 0 After −M v +m v Total = 0 (unchanged)
Recoil isn't an accident — whatever momentum the bullet carries off, the gun takes back the opposite; the books stay balanced at zero.
The counterintuitive point Recoil isn't a "flaw in the gun" or a stray side effect — it's forced by the conservation law: whatever momentum the bullet leaves with, the gun must reclaim in the opposite direction. A rocket flies in vacuum, with no air to push against, for exactly this reason — it hurls exhaust backward and is pushed forward. Conservation of momentum is a rocket's only means of propulsion; it is not "pushing against the air behind it."
Cross-read · AI Deep learning's momentum method (SGD with Momentum) borrows both the word and the intuition directly: it lets a parameter update behave like a ball with mass, carrying inertia so it isn't jerked around by every noisy gradient but rolls through small dips along the accumulated direction. Physical momentum lets motion remember its direction; optimizer momentum lets descent remember its direction — one idea, working in two places.
In a sentence: you can't gain motion from nothing — only trade it with something else.
Ponder: When you push off a wall to move backward, the wall gains an equal momentum — so why doesn't it visibly budge?
Because the wall (bolted to Earth) has enormous mass. Conservation gives equal-and-opposite mv: your small m means a large recoil v, while the wall+Earth's huge M gives v≈mv/M≈0, too tiny to measure. The momentum really did transfer — Earth's share of speed is just imperceptible.

Energy: One Ledger, Many Forms Energy: One Ledger, Many Forms

Classical mechanics · kinetic / potential
Intuition A roller coaster barely moves at the top of the hill and is fastest at the bottom; a pendulum freezes for an instant at each end and races through the lowest point. Height buys speed, and speed buys height back. It's like money shuffled between two pockets — whatever the left loses, the right gains, and the total never changes. That total is energy.
Mechanism Energy hides in two places: the part stored in speed is kinetic energy; the part stored in height/configuration is potential energy.
Ek = 12mv2 , Ep = mgh
Ek is kinetic energy: 12 is one-half and v2 is speed squared (so doubling the speed makes kinetic energy four times larger). Ep is gravitational potential energy: g is the acceleration of gravity (about 9.8 metres per second per second at the surface), h the height. With no friction, their sum is constant: Ek + Ep = constant. As a ball falls, h shrinks and v grows — traded off exactly, not a scrap lost.
Total energy E constant (top line fixed) Highest Midway Lowest Potential Kinetic
As it falls, potential energy (blue) turns steadily into kinetic (amber), but the two stacked together always reach the same top line — the total never moves.
The counterintuitive point Energy isn't a "thing" — you can't cut off a piece of it and hold it. It's a bookkeeping number. Feynman's famous image: energy is like a child's building blocks — count as you may, the total stays fixed, yet you never see "the blocks themselves," only that they've changed form: kinetic, heat, light, chemical bonds. What's conserved is that number, not some substance. When "friction dissipates energy," the energy has merely slipped from the ledger you were watching into one you weren't (heat).
Cross-read · AI Training a neural network is letting parameters roll downhill across a "potential-energy landscape" (the loss surface): loss = potential energy, gradient = slope, gradient descent = gravity. A momentum-based optimizer will even overshoot small dips and oscillate in a basin, just like a real ball. "Rolling to the bottom" in physics and "converging to a minimum" in training are the same geometric picture — no wonder optimization is full of terrain words like "landscape" and "basin."
In a sentence: energy is neither created nor destroyed — only redressed in a new form.
Ponder: Does friction make energy "vanish"? Or does it move to a ledger you simply weren't keeping?
It didn't vanish — it became heat. Friction turns macroscopic motion into random molecular motion (thermal energy), plus a little sound and wear. Energy conservation never breaks — “disappearance” always means a ledger you didn't count, usually heat. Counting heat in is exactly what gave us the first law of thermodynamics.

Collisions: Where the Energy Goes Collisions: Where the Energy Goes

Classical mechanics · elastic / inelastic
Intuition In a Newton's cradle (a row of steel balls), lift one and let it strike, and a ball flies off the far end — back and forth, almost forever. But two lumps of clay smashing head-on go "splat," stick, and stop. Same collision, yet one loses almost nothing while the other's "kinetic energy" seems to lose most of itself out of thin air — has a conservation law been broken?
Mechanism No. The key: momentum is always conserved (as long as there's no outside force), while kinetic energy is not always. An elastic collision conserves both momentum and kinetic energy (steel balls and billiards come close). An inelastic collision still conserves momentum, but part of the kinetic energy becomes heat, sound, deformation (clay warms slightly after impact). The extreme is a perfectly inelastic collision: the objects stick and move as one. Confusing "conservation of energy" with "conservation of kinetic energy" is the most common trap here.
Before After Elastic Momentum ✓　Kinetic energy ✓ Inelastic heat / deformation Momentum ✓　Kinetic energy ↓ (→ heat)
Momentum is conserved in both; the difference is kinetic energy — an elastic collision keeps it, an inelastic one turns part of it into heat and deformation.
The counterintuitive point In a car crash, you actually want kinetic energy not to be conserved — better it all become crumpled sheet metal and heat than stay, intact, in the passengers. That's the physics of the crumple zone: the front is deliberately made "soft," using a stretch of controlled inelastic deformation to eat the kinetic energy, lengthen the collision, and shave down the shock passed to people. A car so "rigid" it doesn't deform on impact is, paradoxically, more dangerous.
Cross-read · Engineering Safety design is full of "deliberate inelasticity": the foam in a helmet, highway crash barriers, packing foam in a parcel — all use deformation to turn an impact's kinetic energy into harmless heat and plastic deformation. What an engineer computes is often not "how to avoid the crash," but "once it happens, where should the kinetic energy be routed."
In a sentence: momentum is always conserved; kinetic energy, it depends.
Ponder: A perfectly inelastic collision loses the most kinetic energy — so why isn't a scrap of its momentum lost?
Because two different laws govern them. Momentum has no dissipation channel — internal collision forces cancel in pairs, so total momentum must be conserved; kinetic energy can convert to heat and deformation and isn't conserved. So sticking together loses the most kinetic energy yet keeps every bit of momentum.

Why Conservation Runs Deeper Than Force Why Conservation Runs Deeper Than Force

Symmetry · Noether, landed
Intuition Computing the force at every instant of a real collision is all but impossible — the deformation at the contact surface and the microscopic interactions are hopelessly tangled. But you needn't solve any of it: just write "total momentum before = total momentum after," and the answer is pinned down. A conservation law lets you skip the tangle in the middle and connect the two ends directly.
Messy interior deformation · friction · unknown forces ? Total momentum (in) Total momentum (out) in = out　·　no need to untangle the middle
A conservation law is a constraint laid across the whole process: whatever happens in the box, the total momentum going in must equal the total coming out.
Mechanism These conservations aren't separate empirical rules — they all spring from symmetry (the theorem was stated in "What Is a Physical Law"; here it lands): time-translation symmetry (the same experiment today or tomorrow) → conservation of energy; space-translation symmetry (the same here or there) → conservation of momentum; rotational symmetry (the same whichever way you point it) → conservation of angular momentum. These are the three most famous landing points of Noether's theorem (for the rigorous proof, see the mathematics page "Noether's Theorem").
Why it matters The power of a conservation law lies precisely in not caring about details: however filthy the collision's interior, however complex the forces, as long as space is uniform, total momentum is necessarily conserved. So conservation laws are often more reliable and more useful than the equations of motion — they are constraints guaranteed by symmetry, laid across every process. Facing an unfamiliar system, a physicist's first move is often not to write down the forces but to ask: what symmetries live here, and therefore what quantities are conserved?
Cross-read · AI A conserved quantity = a system's invariant, the first key to reading any dynamical system — and this intuition is welded into AI too: a Hamiltonian Neural Network is built to learn a conserved "energy," so the long-horizon trajectories it predicts automatically don't leak or blow up. Baking the conservation constraint into the network is cheaper, steadier, and more accurate than forcing it to memorize the rule from raw data.
In a sentence: ask about symmetry first, conservation second, and force only last.
Ponder: If some quantity were one day found to quietly fail to conserve on a cosmic scale, which broken symmetry should you suspect first?
Suspect the matching symmetry. Energy non-conservation ↔ broken time-translation symmetry — and cosmology is exactly this: expansion makes spacetime change with time, so “total energy” isn't strictly conserved cosmologically. A loosened conservation law is always a loosening symmetry (Noether in reverse).

Going Deeper

Both momentum and kinetic energy carry velocity — one is mv, the other ½mv². Why do we need two? Isn't one enough?
Because they are conserved under different conditions and govern different things. Momentum is a vector (it has direction) and is conserved by vector addition in any collision (elastic or not); it governs "how motion is shared out among objects." Kinetic energy is a scalar (magnitude only) and is conserved only when things are elastic/lossless; it governs "how violent the motion is, how much work it can do." Vector vs scalar, first power vs square — that lets them each rule their own kind of problem: use momentum for recoil and flying debris; use energy for how high something climbs or how hard it hits. The two are independent, and you need both.
Does "conservation of energy" still hold in an expanding universe?
An honest trap. Locally, in the lab, energy conservation is exact to an exquisite degree. But pushed to the whole universe, the answer turns subtle: energy conservation corresponds to time-translation symmetry, and the expanding cosmic background itself changes with time — it lacks that symmetry. So on cosmological scales "total energy" has no clean, conserved definition — light crossing the expanding universe has its wavelength stretched (redshift), its energy seemingly "lost," with nowhere to point to where it went. This isn't the law failing; it's a reminder that a conservation law is bound to a symmetry, and once the symmetry is gone, conservation is no longer automatic.
Where is potential energy actually stored? When you lift a rock, is the extra energy inside the rock?
Not in the rock — this is exactly where "potential energy is stored energy" most easily misleads. The extra gravitational potential energy of a lifted rock belongs to the whole system of rock plus Earth (it lives in the gravitational field / relative configuration between them), not the rock alone. Spring potential energy lives in the electromagnetic interactions inside the compressed spring. So potential energy is always energy "between a pair of interacting things"; the potential energy of a single object is meaningless — you must specify "relative to what" and "where you set the zero." That's also why potential energy can be negative: it measures a relationship, not a stash of something sitting somewhere.
Why can't a perpetual-motion machine be built? Which conservation law blocks it?
This means a perpetual-motion machine of the first kind: one that does useful work continuously without consuming energy. It runs straight into conservation of energy — output work must have an energy source, and producing work from nothing amounts to letting that conserved bookkeeping number grow on its own. Countless ingenious designs have failed at the same spot: somewhere, one part is quietly drawing on energy you left off the books (some height, elasticity, a magnet, or simply the hand that started it). Conservation of energy doesn't forbid any clever mechanism; it forbids exactly one thing — the total growing out of nothing. (There is also a "second-kind" perpetual-motion machine that doesn't violate energy conservation yet still can't be built; that involves entropy — for the topic of entropy and the arrow of time, later.)
If everything is conserved, isn't the universe just "nothing happening"? Why is there change at all?
What's conserved is the total, not the distribution. Total energy stays fixed, but it can turn from potential to kinetic, from ordered mechanical energy into diffuse heat — the form and the distribution shift constantly, and it's exactly this redistribution that makes up everything we call "happening." A conservation law doesn't say "nothing may move"; it says "however you move, the books must balance." What actually decides which direction change runs (why heat flows from hot to cold, why a broken mirror won't reassemble) isn't conservation but a subtler law — the growth of entropy. That's left for the topic of entropy and the arrow of time.

Further Reading