Primary & Middle School Physics · Grades 1–9

61The Speed of Light; Distances in the Universe

For seven years this book has whispered that light is “as good as instant” — the flash before the thunder, the seen strike starting the stopwatch. This chapter keeps the family’s oldest promise: light’s speed is not infinite, humans measured it against all odds, and the number is so vast that it becomes, in the same breath, astronomy’s ruler and its time machine.

61.1 The impossible measurement

Example 61.1 (Lanterns on two hills)

Four centuries ago, the great pioneer of experiments set two observers on distant hills at night, each with a shuttered lantern: uncover yours on seeing mine, and the round-trip delay should betray light’s speed. The delay they found was only human reflexes; between the hills, light’s true crossing took a few millionths of a second. The experiment failed magnificently — proving, at least, that light is faster than anything a hilltop could clock, and daring astronomy to find longer race-tracks.

Example 61.2 (The moon that ran late)

The race-track came courtesy of Jupiter. Its innermost great moon eclipses into Jupiter’s shadow on a superbly regular schedule — a clock in the sky. Yet a patient astronomer found the clock running late by many minutes in the months when the Earth’s orbit had carried us to the far side of the Sun from Jupiter, and early again as we swung near. His leap: the moon is punctual — its light is delayed, needing extra time to cross the extra width of the Earth’s orbit. From the lateness and the orbit’s size came the first estimate of light’s speed — breathtakingly close, for the sixteen-hundreds, to the truth.

The moon that ran late: eclipse news from Jupiter reaches the far-side Earth many minutes behind schedule — the delay is light, caught commuting.

Proposition 61.3 (The speed of light)

In vacuum — and in air, to excellent approximation — light travels at

c = 300\,000\,\mathrm{km}/\mathrm{s} = 3 \times 10^{8}\ \mathrm{m}/\mathrm{s},

three hundred thousand kilometres each second: seven and a half laps of the Earth between two heartbeats. Every color of the blend travels at this same $c$ in vacuum; nothing that carries matter or news has ever been clocked faster. Later, Earth-bound ingenuity confirmed the astronomers — a beam chopped by a spinning toothed wheel, racing eight kilometres to a mirror and back through the next tooth-gap — and modern instruments have polished $c$ to a certainty so complete that the metre itself is now defined from it.

Example 61.4 (Travel times in the neighborhood)

With $t = d/c$ , the Solar System acquires a timetable. The Moon, $384\,000\,\mathrm{km}$ : $384000 \div 300000 \approx 1.3\,\mathrm{s}$ — Earth–Moon radio conversations carry that awkward pause. The Sun, $150$ million km: $150000000 \div 300000 = 500\,\mathrm{s}$ — eight minutes twenty: sunshine is always eight-minute-old news. Mars, at a typical $2 \times 10^{8}$ km: about eleven minutes — rover drivers send the day’s plan and wait; no joystick can steer across a twenty-minute round trip.

A spiral galaxy: light from its stars crosses millions of years of empty space before reaching any telescope.

61.2 The light-year

Definition 61.5 (Light-year)

A light-year is a distance: the stretch light covers in one year — about $9.5 \times 10^{12}$ km, nearly ten million million kilometres. The name misleads the hasty (it is no more a time than a “footstep” is a foot); astronomers use it because kilometres collapse under the sky’s true scale.

Example 61.6 (The neighborhood in light-years)

The nearest star beyond the Sun: about $4$ light-years — tonight’s photons left it while you were in your first physics chapters. The bright stars of the winter sky: tens to hundreds of light-years. The Milky Way, our city of stars: about $100000$ light-years across. The nearest great neighboring galaxy: some $2.5$ million light-years — its faint smudge, visible to a dark-adapted naked eye, is the oldest light a human can see unaided.

Proposition 61.7 (Telescopes are time machines)

Because light’s news travels at $c$ and the universe is vast, looking far means looking back. The star at $4$ light-years shows itself as it was $4$ years ago; the neighbor galaxy, as it was when our ancestors first chipped stone tools. The g5 stargazer’s suspicion is now a theorem of arithmetic: every telescope aimed outward is aimed pastward, and astronomy is history read by starlight.

Method 61.8 (Converting distance and lookback)

distance in light-years $\to$ lookback in years: the same number — that is the unit’s whole genius;
distance in kilometres $\to$ light-time: divide by $300\,000\,\mathrm{km}/\mathrm{s}$ , then tame the seconds into minutes, hours or years;
light-years $\to$ kilometres (rarely worth it): multiply by $9.5 \times 10^{12}$ — and remember why astronomers seldom do.

Remark 61.9 (What “now” means out there)

Is the four-light-year star still shining right now? Almost surely — stars live long — but no faster answer than light’s own can ever reach us: the freshest possible news is four years stale, always. For the deep sky, the question softens into strangeness: some of the faintest smudges in great telescopes are galaxies whose light left before the Earth existed. Physics will sharpen what “now at a distance” can even mean in the university years; for this year, carry the honest version: the sky is not a scene — it is an archive.

61.3 Exercises

Exercise 61.1 ★

Give $c$ in $\mathrm{km}/\mathrm{s}$ and in $\mathrm{m}/\mathrm{s}$ (powers-of-ten form). What failed on the two hills, and why?

Solution

Solution of Exercise 61.1.

$c = 300\,000\,\mathrm{km}/\mathrm{s} = 3 \times 10^{8}\ \mathrm{m}/\mathrm{s}$ . The hilltop lanterns measured only human reflexes: light crossed the hills in millionths of a second, far beneath any hand’s timing.

Exercise 61.2 ★

Retell the late-moon argument in three sentences: what was punctual, what was late, and what the lateness measured.

Solution

Solution of Exercise 61.2.

Jupiter’s moon eclipsed on a punctual schedule — a sky clock. The news of each eclipse arrived late whenever the Earth stood far from Jupiter, early when near. The lateness measured light’s travel time across the width of the Earth’s orbit — and so light’s speed.

Exercise 61.3 ★

Compute light’s travel time: Moon ( $384\,000\,\mathrm{km}$ ); Sun ( $150$ million km). Why is sunshine “old news”?

Solution

Solution of Exercise 61.3.

Moon: $384000 \div 300000 \approx 1.3\,\mathrm{s}$ . Sun: $150000000 \div 300000 = 500\,\mathrm{s}$ — eight minutes twenty. Sunshine always shows the Sun as it was eight minutes ago: old news by construction.

Exercise 61.4 ★

A light-year — time or distance? Define it, and give its size in kilometres (powers-of-ten form).

Solution

Solution of Exercise 61.4.

A distance: the stretch light covers in one year — about $9.5 \times 10^{12}$ km.

Exercise 61.5 ★

The nearest star sits about $4$ light-years away. What is tonight’s lookback — and what were you doing when its light departed?

Solution

Solution of Exercise 61.5.

Four years. Answers vary — four years ago you were several grades back down this very book.

Exercise 61.6 ★

Why can no one joystick a Mars rover from Earth? Use the timetable example’s numbers.

Solution

Solution of Exercise 61.6.

At eleven-ish minutes one way, a steering correction answers a picture at least twenty-two minutes stale — the rover would be in the crevasse before the joystick’s twitch arrived. Hence plans, not joysticks.

Exercise 61.7 ★

Order by lookback: the Moon; the Sun; the nearest star; the neighbor galaxy — with each one’s rough figure.

Solution

Solution of Exercise 61.7.

Moon: about $1.3\,\mathrm{s}$ . Sun: eight minutes. Nearest star: four years. Neighbor galaxy: two and a half million years.

Exercise 61.8 ★

“The metre is now defined from $c$ .” What does this say about which of the two — ruler or light — physicists trust more deeply?

Solution

Solution of Exercise 61.8.

Light. The speed $c$ proved steadier than any metal bar or national ruler — so the ruler is now derived from the light, not the light measured by the ruler.

Exercise 61.9 ★★

Radar ranging: a radio pulse (traveling at $c$ ) bounced off the Moon returns in about $2.6\,\mathrm{s}$ . Recover the Moon’s distance from this round trip — and name the old sonar rule that carries over.

Solution

Solution of Exercise 61.9.

Round trip: $300000 \times 2.6 = 780000$ km; the Moon stands at half — $390\,000\,\mathrm{km}$ . The sonar rule carries over whole: every echo pays for the round trip, halve before believing.

Exercise 61.10 ★★

Sort by the archive’s depth: light in your room from a lamp ( $3\,\mathrm{m}$ ); the Sun; a star at $100$ light-years; the neighbor galaxy. For the lamp, estimate the lookback in billionths of a second ( $3 \div (3 \times 10^{8})$ — tame it in words).

Solution

Solution of Exercise 61.10.

Lamp: $3 \div (3 \times 10^{8}) = 10^{-8}$ s — a hundredth of a millionth of a second: even the room is an archive, too shallow to notice. Then the Sun (eight minutes), the star (a century), the galaxy (two and a half million years) — the deeper you look, the older the page.

Exercise 61.11 ★★

A news report says “astronomers watched a star explode last Tuesday, $8000$ light-years away.” Rewrite the sentence with honest tenses: when did the explosion happen, and what happened last Tuesday?

Solution

Solution of Exercise 61.11.

“Last Tuesday, the light of a star’s explosion reached us — the explosion itself happened some $8000$ years ago, and its news has been traveling ever since.”

Exercise 61.12 ★★★

The late-moon lateness peaks at about $1000$ seconds — light’s time to cross the full width of the Earth’s orbit, about $3 \times 10^{8}$ km. Run the seventeenth-century division yourself and compare with the modern $c$ : how honest was the sky’s first answer?

Solution

Solution of Exercise 61.12.

$3 \times 10^{8}\ \text{km} \div 1000\ \text{s} = 3 \times 10^{5}$ $\mathrm{km}/\mathrm{s}$ — the modern value on the nose (the historical estimate, with rougher orbit figures, landed within about a quarter of it: for the sixteen-hundreds, an astonishing first answer).

61.4 Problem: Mission Control

Problem 61.1

Weekend problem — a shift at deep-space mission control; light-lag conversations from the Moon to the stars; the operations manual

Tonight you direct the communications desk: every message travels at $c = 300\,000\,\mathrm{km}/\mathrm{s}$ , and the light-lag rules every decision.

Part I — Near operations.

The lunar base ( $384\,000\,\mathrm{km}$ ) requests confirmation of a supply drop. How long after your “confirmed” does the base hear it, and what is the soonest you can hear their acknowledgment?
A satellite in high orbit at $36\,000\,\mathrm{km}$ relays a television feed. Compute its one-way lag — and say why intercontinental phone calls via such satellites carry a just-noticeable hesitation (round trip up and down).
The solar observatory reports a great flare “as it happens”. Correct the operator’s phrase: when did the flare truly erupt?
Protocol asks all clocks to allow for lag. Which near-Earth link — Moon or high satellite — needs the larger allowance, and by roughly what factor?

Part II — The Mars window. Mars stands tonight at $1.8 \times 10^{8}$ km.

Compute the one-way lag in seconds, then in minutes.
The rover meets an unexpected crevasse and its cameras show it approaching at walking pace. Explain, with the round-trip figure, why a joystick rescue is hopeless — and what the rover must therefore carry on board.
The evening’s plan upload takes $3\,\mathrm{min}$ of transmission plus the lag. When must the upload start to be fully received by the rover at 21:00 control time?
Six months hence, Mars will stand at $3.8 \times 10^{8}$ km. What becomes of the lag, and what does this teach about “the” distance to a planet?

Part III — The far desk.

An interstellar probe has reached one full light-day from Earth. Express that distance in kilometres (powers of ten; a day holds $86400$ seconds), and state the round-trip conversation time with the probe.
Dreamers at the back desk draft a “message to the nearest star”. Draft the schedule instead: sent tonight, when is the soonest any reply arrives?
The archive wall shows tonight’s deep image of the neighbor galaxy, $2.5$ million light-years out. Add the caption’s honest tense — what era of Earth’s past does this snapshot broadcast back, were anyone there to look?
End-of-shift log: write the desk’s three rules — one about instantaneous (that nothing is), one about joysticks and lag, one about telescopes and the archive.

Solution

Solution of Problem 61.1.

1. About $1.3\,\mathrm{s}$ after you speak; the soonest acknowledgment returns after the round trip, about $2.6\,\mathrm{s}$ . 2. $36000 \div 300000 = 0.12\,\mathrm{s}$ one way; a question-and-answer rides up and down twice — near half a second of hesitation, just at the ear’s threshold. 3. “The flare erupted eight minutes twenty before our screens lit — we watch it as it was.” 4. The Moon link: $1.3\,\mathrm{s}$ against $0.12\,\mathrm{s}$ — roughly ten times the allowance. 5. $1.8 \times 10^{8} \div (3 \times 10^{5}) = 600$ s: ten minutes one way. 6. The round trip is twenty minutes: the rescue picture is ten minutes old and the rescue command ten minutes late — the crevasse wins by nineteen minutes. The rover must carry its own reflexes: on-board hazard-stopping. 7. Last bit must arrive by 21:00; it leaves Earth at $21{:}00 - 10$ min $= 20{:}50$ ; transmission began three minutes earlier: start at 20:47. 8. The lag doubles to about twenty-one minutes one way: planets have no “the” distance — both worlds orbit, and every conversation is scheduled against a moving target. 9. $300000 \times 86400 \approx 2.6 \times 10^{10}$ km — twenty-six thousand million kilometres; a question-and-answer takes two full days. 10. Light reaches the star in about four years; the soonest reply lands about eight years after tonight’s transmission. 11. “This light left its galaxy two and a half million years ago: the snapshot broadcasts the Earth of the earliest stone tools — our deepest naked-eye page of the archive.” 12. For example: “Nothing is instantaneous — every signal, ours included, travels at best at $c$ . Joysticks lose to lag beyond the Moon: distant machines must think for themselves. And every telescope is an archive reader — the farther the object, the older the news, with no fresher edition printable.”

Terms defined in this chapter

Light-year

See all 393 terms in the glossary