Illustrated Physics
physics, drawn
Concept

Momentum conservation — where it appears

The rule that the total momentum of an isolated system is unchanged by anything happening inside it, which makes a collision's details irrelevant. It is what allows the outcome of a collision to be computed without knowing anything about the forces during it.

Named by 19 essays across 8 fields — each of them below, with the objects they name alongside it.

A collision with restitution 0.6. Two bodies before and after a head-on collision. Momentum is the same on both rows by construction; kinetic energy is only preserved when the collision is elastic.

Collisions are easier than forces, and momentum is the reason

Nobody knows what happens inside a collision. Momentum conservation makes that ignorance irrelevant, which is the whole trick — and energy, deliberately, is not conserved.

mechanics · Momentum Pressure, counted as momentum arriving at a wall. Molecules with speeds drawn from the Maxwell–Boltzmann distribution and directions drawn uniformly in the plane of the figure. Those moving toward the wall will bounce off it, reversing the component perpendicular to it and delivering twice that momentum each — and pressure is nothing but the rate at which that momentum arrives. Because the directions here lie in a plane rather than in space, the perpendicular component carries half the energy rather than the third it carries in a real gas.

Pressure is a rate of arrival, and the gas law falls out of counting

Nothing in a gas is pushing on the walls. Molecules arrive, bounce, and leave, and pressure is the momentum they deliver per second — from which the ideal gas law follows with no thermodynamics in it at all.

thermodynamics · Kinetic theory Total energy against speed, in units of the rest energy. The total energy of a moving body divided by its rest energy, against speed as a fraction of the speed of light. The Newtonian answer, one plus half v squared over c squared, is drawn beside it: the two agree to 0.004 per cent at a tenth of light speed and disagree by 39 per cent at nine-tenths. The relativistic curve has a vertical asymptote at c, which is why nothing with mass reaches it.

Mass is a form of energy, which is not the same as a source of it

The famous equation is usually read as a promise that mass can be turned into energy. It says something stricter and stranger — that a mass is an energy already, sitting there, whether or not anything ever releases it.

relativity · Mass-energy The wavelength shift against scattering angle. How much longer a scattered photon's wavelength is, against the angle it scattered through. The shift runs from nothing at 0 degrees to 4.853 picometres straight back, passing through the electron's Compton wavelength of 2.4263 picometres at 90 degrees. Nothing about the incident light or the target material appears anywhere on this axis.

A photon with a momentum, and a collision that proves it

X-rays bouncing off electrons come back with a longer wavelength. How much longer depends on the angle they turned through, and on nothing else — not the incident wavelength, not the target material, not the intensity.

quantum · Photon An electron's wavelength against the voltage that accelerated it. The de Broglie wavelength of an electron after falling through a potential difference, in picometres. At 100 volts it is 122.6 picometres, at 400 volts it is 61.3 picometres, at 900 volts it is 40.9 picometres. The wavelength goes as the inverse square root of the voltage, so quadrupling the voltage halves it. The calculation is non-relativistic; at a kilovolt that costs a tenth of a per cent. The dashed line is the 215 picometre spacing between atomic planes in nickel, which is what makes an electron beam diffract off a crystal at all.

Everything has a wavelength, and almost nothing shows it

If light with a momentum can behave like a particle, a particle with a momentum can behave like a wave. The wavelength is Planck's constant over the momentum, which for anything larger than a molecule is a number too small to have consequences.

quantum · Matter waves A collision with restitution 0.4. Two bodies before and after a head-on collision. Momentum is the same on both rows by construction; kinetic energy is only preserved when the collision is elastic.

The point that keeps moving as if nothing had happened

Newton's third law makes every internal force cancel against its own partner, which leaves the external sum governing a single mass-weighted average of positions. In the collision below the total momentum stays at 4.00 kg·m/s while 63 per cent of the kinetic energy leaves, and the average travels at 1.00 m/s throughout, before and after.

mechanics · Momentum What puts a scale on a tilted axis. A spacetime diagram with a second observer's axes at β = 0.6. The curves are the sets of events at a fixed interval from the origin — c²t² − x² = s², one branch each for s = 0.5, s = 1, s = 1.5 — and the whole point of them is where they cross. A unit of the moving observer's time is wherever the s = 1 curve meets the tilted time axis, and on the page that point is 1.250 times as far from the origin as the stationary observer's own unit. Without the hyperbolae the tilted axes carry no scale at all, and every argument about which of two clocks is behind is unreadable off the diagram. Each drawn crossing reads back as its own interval to 0.0e+0.

The invariant that survives a boost

Energy and momentum are both answers to the question "how fast is it going, and according to whom". One combination of them is not, and that combination is the mass — which is why two photons of 511 keV can be a thing of mass 1.022 MeV or a thing of no mass at all, depending only on the angle between them.

relativity · Mass-energy 5 balls whose contact force goes as the overlap to the three halves, touching. The velocity of every ball in a line of 5, against time in units of one binary contact, with the first arriving at unit speed. Nothing about collisions is assumed: neighbours push on each other with k times their overlap raised to the power 1.5, and the equations of motion are integrated. With the balls touching there is no such separation — several overlaps are non-zero at once and the disturbance crosses the line as a single compression wave. The far ball leaves at 0.989 of the striking speed and the others keep 0.011 between them, which is why a real cradle's balls do not quite come to rest. Momentum and energy are conserved to 4.4e-16 and 4.1e-8, so the difference between the two cases is the contact law and not the bookkeeping.

Five balls, and the law that does not choose

The usual account of a Newton's cradle says that momentum and energy conservation force one ball out at the striking speed. For three balls or more they do no such thing: the two laws leave a whole curve of possible outcomes, and what picks one is the shape of the force between two touching spheres.

mechanics · Momentum A loop at rest carrying 2.2e-12 kg m/s. A square loop of area 100 cm² carrying 20 amps, sitting still in a uniform electric field of 1.00 megavolts a metre. The carriers going up the field on one side cross 100 kilovolts on the way, so the ones in the top wire are less energetic than those in the bottom by that much per unit charge. The current is the same all the way round, so the same number of carriers pass per second in each wire — but they carry different energy, and momentum is energy times velocity over c². The two wires therefore contribute unequally, and the difference is 2.23e-12 kilogram metres a second, pointing across both the field and the dipole. Nothing in the picture is moving as a whole. The electromagnetic field round the loop carries exactly that momentum the other way.

The momentum of something that is not moving

A current loop sitting still in an electric field has momentum in the space around it. Nothing is moving, so something must be carrying an equal and opposite amount — and it is the loop, whose carriers on the high-potential side are more energetic than those on the low.

electromagnetism · Field energy A line is what two bodies give. The electron energy spectrum of tritium, against the vertical line a two-body decay would produce. A nucleus emitting one particle has no choice about how to share the energy: momentum conservation fixes it, and every electron comes out at the same energy. The observed spectrum is a continuum running from zero to the full available energy, with a mean at 0.31 of the endpoint. The shape is a count of the ways the energy can be divided between an electron and something else, and the something else was proposed for no other reason than that the shape requires one.

The energy that did not all arrive

A nucleus emitting one particle has no choice about the energy it comes out with — momentum conservation fixes it, and the spectrum is a line. Beta decay gives a continuum instead, from zero to the full available energy, and the shape of that continuum is a count of the ways the energy can be shared. Counting it required a third body nobody had seen, and the way the count approaches its endpoint is still the best place to weigh one.

quantum · Decay A beam of sound has a weight. The force a fully absorbed acoustic beam exerts, against its power, for 3 media. It is the power divided by the speed of sound and nothing else — a watt in water gives 675 micronewtons, which is the weight of 68 milligrams, and a watt in air gives 2.9 millinewtons because the sound is slower there. This is not an analogy with light: it is the same statement, that a wave carrying energy carries momentum, with a much smaller speed in the denominator. The consequence is that acoustic power is measured by weighing. A radiation-force balance — an absorbing target on a laboratory balance, with the transducer beneath it — is the primary standard for ultrasonic output, and every therapeutic and diagnostic transducer is calibrated against one.

Where the loudness goes

An absorption coefficient removes energy from a wave, and energy removed has to appear somewhere. It appears twice, from the same coefficient: as heat, and as momentum. So a beam of sound has a weight — a watt absorbed in water weighs sixty-eight milligrams — and acoustic power is measured by putting an absorber on a balance. The ratio of the force to the heating contains no intensity at all.

waves · Attenuation Snell's law with space and time exchanged. Two constructions on the same diagram of frequency against wavenumber, with the light lines of a medium of index 1 and of index 1.5. On the left, a boundary in space: the wave crosses a still surface, the frequency is conserved, and the horizontal line at the incident frequency meets the new medium's line at a wavenumber 1.5 times larger — the familiar shortening of the wavelength. On the right, a boundary in time: the whole medium changes at once, the wavenumber is conserved, and the vertical line at the incident wavenumber meets the new medium's line at a frequency 0.667 times the old one. The vertical line also meets the new line's negative-frequency branch, which is a wave running backwards: a reflection in time. A spatial boundary reflects into the same frequency and a temporal one into the same wavelength.

The reflection that needs no surface

Change the refractive index of a whole medium at one instant and a wave already travelling through it splits in two, one part running on and one running back, though there is no surface anywhere for it to reflect from. A boundary in time is Snell's law with space and time exchanged: the wavelength is kept and the frequency changes, momentum is conserved and energy is not.

optics · Refraction What a beam's energy buys, three ways. The speed reached against the energy intercepted, measured in the body's own rest energy, for a perfect mirror pushed by a beam, a perfect absorber pushed by the same beam, and a photon rocket that carries the same energy as fuel and throws it out behind. The mirror's curve is γ(1 + β) = 1 + 2E/mc², a rapidity of ln(1 + 2E/mc²); the dots integrate the reflected beam's force, (2P/c)(1 − β)/(1 + β), and agree with it to 10⁻¹⁵. To reach 0.2c the mirror needs 0.1124 of its rest energy, the absorber 0.2500 and the photon rocket 0.2247 — for a 1 g sail, 10.1 terajoules against 20.2. The rocket's rapidity is ln(1 + E/mc²), so the mirror is the rocket with its fuel left at home and each joule used twice, once arriving and once leaving. The absorber does worst, because the energy it keeps becomes rest mass it then has to carry.

The rocket that leaves its fuel at home

A mirror pushed by a beam from the ground carries no propellant, and relativity gives its speed in closed form: its rapidity is ln(1 + 2E/mc²), the photon rocket's equation with the fuel left behind and every joule used twice. What stops it is not the energy, which can be stored for days, but diffraction, which fixes the distance over which the energy can be handed over — and so demands an acceleration of tens of thousands of g.

relativity · Mass-energy A repeat in space gaps the frequencies; a repeat in time gaps the wavenumbers. Two media with the same modulation depth, 0.2, computed the same way. Left: permittivity repeating in space, a stack of layers. For each frequency the wave equation is integrated across one spatial period, and the Bloch wavenumber drawn against frequency; between frequencies 0.478 and 0.526 (in units of c over the period) there is no real wavenumber, so light of those frequencies cannot travel and is reflected. Right: permittivity repeating in time. For each wavenumber the equation is integrated across one period, and the frequency drawn against wavenumber; between wavenumbers 0.474 and 0.518 there is no real frequency. The axes of the two panels are swapped, and so is everything else: the spatial gap is a band of frequencies that decays in space, the temporal gap is a band of wavenumbers that grows in time.

The crystal made of moments

A stack of layers that repeats in space refuses a band of frequencies and reflects them. A medium that repeats in time — its refractive index swung up and down everywhere at once — refuses a band of wavenumbers instead, and a wave with a wavenumber in that band does not reflect. It grows, exponentially, drawing on whatever is swinging the index. The construction is exact, the gap is computable from one period of the modulation, and the obstacle to building one for light is how fast a material would have to change.

optics · Refraction Helium's excitations, and the line that decides how fast it can flow. The energy of an elementary excitation in superfluid helium-4 against its momentum, as a smooth fit through published neutron-scattering values: a phonon branch rising at the speed of sound, 238 m/s, a maximum near 13.9 K, and the roton minimum of 8.62 K at 1.92 Å⁻¹. A straight line from the origin has a slope that is a velocity. The steepest line that just touches the spectrum is Landau's critical velocity, 57 m/s, and it touches at 1.98 Å⁻¹, just past the roton minimum. A body moving slower than that cannot make any excitation at all and so cannot lose energy to the liquid; the dashed line is the speed of sound, far above it.

The speed below which nothing can be made

An object moving through an ordinary liquid always feels drag, because it can always pass some of its energy to the liquid. In a superfluid there may be nothing it can pass it to. Landau's argument turns the question into geometry: an excitation of momentum p and energy ε can be made only by a body moving faster than ε/p, so a liquid flows without friction below the smallest value of that ratio. For helium the smallest value is not the speed of sound but 57 m/s, set by the dip in its spectrum called the roton — and an ideal gas, with no dip and no sound, has no such speed at all.

quantum · Superfluidity Equal masses, one at rest, struck head on at four energies. A particle striking an identical particle at rest and scattering elastically, with the two leaving at equal angles either side of the incoming line, at four incident kinetic energies measured in units of the particle's rest energy. At 0.001 mc² the two tracks open at 90.0°. At 0.5 mc² the two tracks open at 83.6°. At 2 mc² the two tracks open at 70.5°. At 10 mc² the two tracks open at 44.4°. Slowly, the angle between them is a right angle, as every billiards player knows. Fast, it closes, because the incoming particle's energy is part of its inertia and the pair is carried forward. A proton at 2 mc² — about 1.9 GeV — and an electron at 1 MeV are both well inside the narrowing.

The right angle a fast collision closes

When one billiard ball strikes an identical one at rest, the two always leave at right angles — a fact every snooker player uses without knowing its proof. The proof is Newton's, and it fails for fast particles. Two protons leaving a collision at 1.9 GeV open at 70 degrees, not 90, and the narrowing follows one law, tan θ₁ tan θ₂ = 2/(γ + 1), in which the right angle is the slow limit. The circle that forced the right angle has been squashed into an ellipse by the energy the incoming particle carries as inertia.

relativity · Relativistic dynamics The block has to move, and which way decides the momentum. A pulse of light crosses a free block of glass of index 1.5 and length L, with time in units of L/c. The pulse falls behind where it would have been in vacuum by 0.50 L while it is inside. The block's displacement, in units of EL/Mc², is obtained from momentum conservation alone: if the light inside carries Abraham's E/nc the block takes up the rest of E/c, moves forward, and ends 0.500 units along — Balazs's (n − 1). The sum of the pulse's lag and the block's shift, which is the centre of energy's departure from uniform motion, is then zero to within 0.0004 units at every moment. If instead the light carried Minkowski's nE/c, the block would have to recoil backwards and the centre of energy would end 1.25 units behind a straight line, which no isolated system can do. For a one-joule pulse through a kilogram of glass ten centimetres long the real shift is 5.6 × 10⁻¹⁹ m.

The momentum light carries into glass

A pulse of light in glass has two momenta, nE/c and E/nc, and for a century the argument was about which one is right. Both are measured. A free block that the light crosses must move forward, which only Abraham's value explains; a Cherenkov cone exists at all, which only Minkowski's allows. They answer different questions, and the difference between them is momentum carried by the glass.

electromagnetism · Field energy Two pulls that should be equal and opposite. Two bodies of equal passive mass, one of an ordinary material and one of a material whose active mass is 30 per cent larger than its passive mass (the difference exaggerated to be seen). Each pulls the other with its active mass and is pulled on its passive mass. The pull the unusual body exerts is therefore larger than the pull it receives, the two arrows are not equal and opposite, and the pair, with nothing outside it, accelerates towards the ordinary body. Newton's third law and the conservation of momentum both require the ratio of active to passive mass to be the same for every material, and that is a separate claim from the one torsion balances test, that passive and inertial mass agree.

The pull that has to equal the pull back

Every test of the equivalence principle drops two things and asks whether gravity pulls them equally. None of them asks whether they pull equally. A body's mass enters gravity twice — once as what is pulled and once as what does the pulling — and if the two ratios differed from one material to another, Newton's third law would fail and a lopsided body could push itself through space. The first test of this used a plastic cylinder floating in a liquid. The best uses the Moon, whose light crust is thicker on the side that never faces the Earth.

astrophysics · Equivalence principle The spring falls from the top down. Snapshots of the falling spring, every fifth coil, at 0.00, 0.05, 0.09, 0.14, 0.18, 0.23, 0.26, 0.29 seconds after release, for a spring that hangs 1 m; the dashed line is the height of the bottom coil before release. The coils collapse onto one another from the top, and the collapsed block falls on the coils below; everything below the block hangs exactly as it did before the spring was let go, the bottom coil included. Only when the block reaches the bottom, 0.261 s after release, does the bottom begin to fall, and then the whole collapsed spring falls together.

The spring whose bottom does not know it has been dropped

Hang a soft spring from its top, let go, and its bottom end stays exactly where it was — not falling slowly, not falling at all — while the top races down to meet it. A spring that hangs a metre takes about a quarter of a second to collapse, and only then does the bottom start to fall. Nothing holds it up. The centre of mass falls freely from the first instant, as it must, and the bottom simply has not been told: the news that the top has been released travels down the spring as a front of collapsing coils, and nothing below the front can feel anything until the front arrives.

mechanics · Momentum

Named alongside it

The objects these essays reach for when they reach for this one.

Energy conservationCentre of massElastic collisionConservation lawsE = mc²The Lorentz factorMomentumPhotonDispersion relationDissipationFour-momentumImpulse

All concepts