Time and Distance

The first bench has a 4 m track, tilted just enough for the ball to gain 0.5 m/s of speed every second, and no clock at all. The only way to know when the ball passes each point is to use one's own heart as the clock and scratch the track, wherever the ball happens to be, on every beat. The example comes from Feynman, who rebuilds Galileo's first inclined-plane experiments in this way, and the numbers are ours.

With a 0.8 s pulse, the marks land 0.16 m, 0.64 m, 1.44 m and 2.56 m from the start, and the fifth falls on the end of the track, at 4 m. Divided by the first, these distances give 1, 4, 9, 16 and 25, the squares of the beat number. The gaps between neighbouring marks grow as 1, 3, 5, 7, so that on each beat the ball covers a little more track than on the one before. Today we write this as \(d \propto t^2\), and the missing constant is half the acceleration.

Nobody's heart keeps metronome time, and the bench lets the pulse vary from one beat to the next. The marks scatter, and the first suffers most, because it rests on a single beat and serves as the yardstick for all the others. Even so, release after release, the mean ratio hovers around 1, and the law of squares is still there, only buried in the noise.

On the bench, distance is the easy part, since a tape measure laid along the track will do. Time is the hard part, because the ball takes 4 s to come down and the mechanical clocks of the period drifted by minutes over a day. In Feynman's account, Galileo's way out was his own body, and this improvised stopwatch is enough to replace treatises in the Aristotelian mould with a table of numbers.

It is measurement that turns talk about motion into physics, since a relation such as 1, 4, 9, 16 only shows up once there are numbers. From here on, describing a motion will mean answering two questions, which Feynman reduces to where and when, and the rest of the chapter is about answering them with ever greater precision.

In Feynman: §5-1 Motion ↗

The two graphs on the bench cover a whole year in São Paulo, at 23° south. The upper one shows by how much a day counted from one sunrise to the next exceeds 86,400 s, and the lower one does the same sum from one noon to the next. By noon we mean the Sun crossing its highest point, not twelve o'clock on the clock face. In late December both sums give about 29 s extra, and in mid-September the first drops to 63 s short, while the second stays at 22 s short.

The attentive reader will spot a cheat, because speaking of 86,400 s presupposes a clock, and a clock is precisely what we do not yet have. To break the circle, time comes to be defined by the way we measure it, by picking something that repeats and counting the repetitions. The day is the natural candidate, and it is the day that the bench puts to the test.

For that, the bench relies on a second clock that does not depend on the Sun, a one-hour hourglass, the same device Feynman proposes for the job. What it counts are the turns accumulated between two events of the Sun, which assumes someone keeping watch over the glass without a break.

Counted from sunrise to sunrise, the number of turns does not hold steady. In São Paulo it ranges from 23.98 in September to 24.01 in January, and at 50° of latitude the range widens to 23.96 to 24.03. Which of the two clocks is wrong? With only these two, there is no way to tell. What changes at sunrise is the length of daylight, which grows or shrinks with the seasons and brings sunrise forward or back from one morning to the next. At the equator daylight lasts practically 12 hours all year round, and the two curves on the bench become one.

From noon to noon, the count stays between 23.99 and 24.01 at any latitude, a swing smaller than the error of a real hourglass, which would read it as a round 24 turns. This does not prove that the sand always runs at the same rate, nor that the Sun returns to its highest point at identical intervals, and Feynman takes care to point it out. What the bench shows is that the two clocks agree to within ±0.01 turn a day, and it is this agreement, not a proof, that we call regular time. With better clocks, which Zoom on noon imitates, noon itself starts to wander, by up to about 30 s from one day to the next. This is the equation of time. It has two causes, the Earth's elliptical orbit, travelled fastest in early January, and the tilt of the axis, and, built up over the months, it leaves a sundial up to about a quarter of an hour fast or slow.

In Feynman: §5-2 Time ↗

The bench releases two identical 0.994 m pendulums at the same moment, the left one pulled out to 5° and the right one to 60°. The first takes 2.001 s to go out and come back, the second takes 2.146 s, and the 0.145 s difference piles up with every oscillation. After some seven round trips, one is heading right while the other heads left.

The time of one full oscillation, the period, therefore depends on the size of the swing. For small swings the dependence is weak, and \(T \approx T_0(1 + \theta_0^2/16)\) holds, but it grows quickly with the amplitude. At 90° the period is already 18% longer than \(T_0\), and the approximation, which gives 15%, begins to lag behind. The exact calculation uses the arithmetic–geometric mean, which converges in a few iterations and draws the solid curve on the graph.

Between 5° and 10°, on the other hand, the period changes by less than 0.15%. In the limit of small swings it tends to \(T_0 = 2\pi\sqrt{L/g}\), which depends only on the length of the string and the acceleration due to gravity, and it is this limit that makes the pendulum a clock.

In Feynman's account, it was Galileo who turned this limit into an instrument, treating the short-swinging pendulum as a marker of equal intervals. The hourglass from the previous step lets us check the idea. The 0.994 m pendulum completes about 1,800 oscillations in each hour of sand, and trading a 3° swing for an 8° one changes that count by about two oscillations, probably less than a real hourglass can resolve. Since the day has already passed the test in step 2, the pendulum lets us split the day far more finely than into hours.

The second is the 86,400th part of the mean day, \(24 \times 60 \times 60\) pieces, and the 0.994 m pendulum is called a seconds pendulum because each one-way swing, from one extreme to the other, lasts 1 s. The period, which counts there and back, is 2 s, and it is easy to confuse the two.

In a pendulum clock, an escapement gives a small push on each swing and restores what friction takes away, so that the amplitude stays the same. The dependence on amplitude is still there, as a fixed factor that the clockmaker compensates for by adjusting the length. What would spoil the rate is a change of amplitude, because going from 5° to 10° slows the pendulum by about 5 s an hour, or 2 minutes a day.

In Feynman: §5-3 Short times ↗

The grid on the bench holds 400 nuclei of a single radioactive isotope, each lit for as long as it has not decayed. None of them has an appointed hour, and at every instant the bench draws lots to decide which go out. Even so, the count passes close to 200 after one half-life, \(T\), close to 100 after \(2T\) and close to 50 after \(3T\), always hugging the theoretical exponential. The fluctuations around the curve grow, in relative terms, as fewer nuclei remain.

Every nucleus on the grid has the same chance of decaying per unit time, whatever its age, and so the sample as a whole loses, in each half-life, half of what it still has. Read backwards, the fraction remaining gives the time elapsed. With an eighth of the grid lit, three half-lives have gone by, with a thousandth, close to ten, and in general the ratio \(B/A\) between what is left and what there was at the start equals \(\left(\tfrac12\right)^{t/T}\), which is enough to read off the age \(t\).

This clock has no tick. There is nothing to count, only a fraction to measure, and it keeps running long after any human counter has gone home, which makes it useful over thousands or billions of years.

The practical difficulty is knowing how much there was at the start, and for carbon the atmosphere supplies the answer. Cosmic rays make carbon-14 out of nitrogen high in the atmosphere, and decay removes it, so the fraction of carbon-14 in the carbon dioxide of the air tends to stay roughly constant. A living tree exchanges carbon with the air and keeps that fraction, and once it dies the exchange stops and the clock starts to run. The half-life is 5,730 years, a value measured with more care than Feynman's round 5000, and the small variations of the fraction in the air over the millennia are corrected with calibration curves. These curves come from natural archives in which every year leaves its own trace, such as tree rings and the yearly layers of sediment at the bottom of certain lakes, counted one by one and also dated by carbon-14.

What limits the dating is the count. Each decay is a click in the detector, \(N\) clicks carry a statistical uncertainty of about \(\sqrt N\), and that turns into an age uncertainty of \(T/(\ln 2\,\sqrt N)\). A 12,000-year-old sample keeps 23% of its original carbon-14, and a detector that would register 10,000 decays from a fresh sample registers 2,342 in the same time, which gives ±171 years. At 60,000 years only 0.07% is left, the same measurement counts some 7 decays and is off by more than 3,000 years. In practice, background radiation and contamination by recent carbon put the limit at around 50,000 years, and longer times call for isotopes with longer half-lives, such as uranium-238, which dates rocks and the Earth itself.

In Feynman: §5-4 Long times ↗

In the Star scenario, the bench starts from a parallax of 0.77″, the value measured for Proxima Centauri, the smallest of the three stars in the Alpha Centauri system and the closest to the Sun. Seen six months apart, it shifts slightly against the far more distant stars of the background, because the Earth has moved to the other side of its orbit. The base of the triangle is the diameter of the orbit, 2 AU, about 300 million km, and the parallax is half the angle at the apex. With 0.77″, the star lies at 1.30 pc, or 4.2 light-years, some 270 thousand times the distance from the Earth to the Sun.

The parsec is the distance at which one astronomical unit, set perpendicular to the line of sight, spans one arcsecond, and it was chosen so that the distance in parsecs is the inverse of the parallax in arcseconds. What this simple sum hides is the error. A telescope measures angles with a more or less fixed error, whatever the distance of the star, and the relative error of the distance is that error divided by the parallax. With 0.01″ of error, Proxima comes out with 1.3%, a star at 10 pc with 10%, and at 100 pc, where the parallax is as large as the error itself, the measurement no longer says anything about the distance.

From the ground, through the atmosphere, getting down to 0.01″ is already hard work. The Hipparcos satellite, in the 1990s, measured parallaxes with errors near 0.001″. Gaia, which observed the sky from 2014 to 2025, reaches a few hundredths of a thousandth of an arcsecond on the brightest stars, which takes triangulation out to thousands of parsecs, still within our galaxy.

In the Satellite scenario, 1° of error in each sighting becomes a 3.8% error in the height, some 19 km out of 500. The same triangle shrunk to the size of a courtyard, with a 10 m base and a point 5 m up, can be checked with a tape measure, and the two readings coincide within those same 3.8%, or within whatever the tape and the protractor allow. This is what, for us, makes triangulation a measurement of distance, and not merely a calculation with angles: wherever a tape can be laid alongside the triangle, the two readings match. That match is a result of measurement, and it tells us that the angles of the triangles we are able to measure add up to 180°, a point Feynman also stresses.

The Satellite and Moon scenarios use two observers on the Earth's surface. A satellite 500 km up, seen from two stations 1,000 km apart, appears at wide angles, and even 1° of error in each sighting costs less than 4% in the height. For the Moon, at 384,000 km, a 10,000 km base leaves an apex angle of only 1.5°. An error of 0.1° already costs 14%, and from about 0.75° upwards the triangle may not even close.

The Sun stays off the bench because of a number. With the Earth's diameter as the base, the apex angle would be some 18″, on a disc half a degree wide with no sharp edges or fixed features to aim at. Today the distance to it comes from a clock rather than an angle: a radar pulse sent to Venus returns within a few minutes, and the round-trip time, measured with the oscillators and atomic clocks of steps 4 to 6 and multiplied by the speed of light, gives the distance in kilometres. That one distance is enough, because the positions of the planets and Kepler's laws already fix the proportions of the whole system, and before radar the same role fell to triangulating nearby asteroids, such as Eros, the two examples Feynman cites.

Beyond the reach of parallax, the way forward is comparison. If a star has the same colour and the same spectrum as others whose distance parallax has already given, it seems reasonable to assume that it emits the same power, and the flux reaching us falls with the square of the distance. The Standard candle scenario does this with a star just like the Sun. The error now comes from how closely the star resembles the model, and a 20% uncertainty in the power becomes 10% in the distance, near or far. With ever brighter candles, the same idea reaches the galaxies, and the most distant ones yet observed lie at distances of order \(10^{26}\) m.

In Feynman: §5-6 Large distances ↗

The bench shows a carbon plate face on, as a particle in the beam would meet it, with each nucleus drawn as a disc. The electrons are left out, since they are too light to stop a very fast particle, and what removes a particle from the beam is a collision with a nucleus. At true scale, the nuclei in 1 cm of carbon block only 2.5% of the area and would barely show. That is why the figure multiplies this area by a factor, ×20 with the initial values, which amounts to looking at a 20 cm plate. Each shot lands at a random point, and the counts refer to this equivalent plate, with the factor taken out when the radius is worked out.

The area a target blocks off, as the beam sees it, is called its cross section, \(\sigma\). In a thin layer, of thickness \(dx\), with \(n\) nuclei per unit volume, the discs cover a fraction \(n\sigma\,dx\) of the area, and that is the fraction of the beam the layer removes. Layer upon layer, the fraction that makes it through a thickness \(x\) falls as \(e^{-n\sigma x}\), the same exponential as in step 5, with thickness in place of time.

For carbon, \(n \approx 1.1\cdot10^{23}\) nuclei per cm³. If 97.5% of the beam gets through 1 cm, then \(n\sigma x = \ln(1/0.975) \approx 0.025\), \(\sigma \approx 2.3\cdot10^{-25}\) cm² and, treating the nucleus as a sphere, \(r = \sqrt{\sigma/\pi} \approx 2.7\) fm. One fm, or fermi, is \(10^{-15}\) m, and the empirical rule \(r \approx 1.2\,A^{1/3}\) fm, where \(A\) is the number of protons and neutrons, gives practically the same value for the 12 of carbon.

One might expect that, in a whole centimetre of carbon, the nuclei would be hiding one another, and that is not what happens. With 2.5% of the area covered, a nucleus on the back face has at most some 2.5% chance of having another in front of it, and on average, across the thickness, half that, so the simple sum, stopped fraction equal to \(n\sigma x\), is only slightly off. In 10 cm the summed coverage goes past 25%, the shadows begin to overlap, and the simple sum, which gives 25% of particles stopped, already departs from the 22% of the exponential.

Like the ladder of oscillators in step 4, the measurement of distance comes down in rungs, and each stretch of the ruler under the bench has its own instrument, the tape measure down to the millimetre, the optical microscope to about a micrometre, the electron microscope down to atoms, X-rays for the spacing in crystals, close to \(10^{-10}\) m, and, at the scale of the nucleus, only the particle counting of this bench.

From atom to nucleus there is still a jump of \(10^5\), the same that separates a 7 m room from a hair, and no image spans that jump. The size of the atom had an independent check, the volume of a mole of crystal divided by Avogadro's number, whereas for the nucleus what we measure is how much it gets in the way of a beam. That is why the bench counts particles rather than looking.

The ruler under the bench is logarithmic, like the time ruler in step 4, and it runs from the nucleus to the most distant galaxy yet observed, from \(10^{-15}\) m to \(10^{26}\) m. That makes some 41 orders of magnitude, nearly as many as on the time ruler, and each stretch of it needed its own recipe for distance, one we accept only because it matches the previous recipe where the two overlap.

In Feynman: §5-7 Short distances ↗