Chapter 5: Time and Distance – Measuring Motion

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Welcome back to the Deep Dive.

Today we're really going back to basics.

We're tackling how physics even starts to measure reality,

specifically time and distance.

Right, and we're doing it through the lens of Richard Feynman, looking at how we actually pin down the where and the when of things in the universe.

It's about that shift, isn't it?

Moving from just thinking about stuff philosophically to actually measuring it quantitatively.

Exactly.

Physics depends on quantitative observation.

It's not just about arguments.

It's about data.

How does the world actually behave?

And Feynman points right back to all about 350 years ago, to Galileo.

Before him, you had Aristotle, whose ideas about motion sort of dominated.

Yeah, if it sounded reasonable, it was often just accepted.

But Galileo, he was skeptical, which, you know, you need that for science.

You have to question things.

He didn't just question.

He experimented that famous inclined trough experiment.

That's where it really kicks off.

It really is.

Clocks weren't accurate back then, so what did he use?

His own pulse.

He was counting along as a ball rolled down this gentle slope.

You can almost picture him.

One, two, three.

Marking where the ball was at each beat.

Uh -huh.

And the finding was just revolutionary for its time.

He saw that the distance the ball traveled wasn't just random.

It was proportional to the square of the time.

You know, dollar propto T two two.

That's the money shot right there.

Not just an observation, but a mathematical law pulled from an experiment.

Absolutely.

It showed motion followed these clean mathematical rules that basically turned physics from, let's say, descriptive philosophy into a predictive science.

It laid the groundwork for everything Newton did later.

OK, so let's let's dive into time first.

The when Feynman is pretty upfront that defining time itself is well, it's tricky.

Yeah, what is time?

He throws out ideas like what happens when nothing else happens or simply how long we wait.

But the crucial part isn't the definition, it's how we measure it.

For that we need something periodic, something that happens over and over again, reliably.

But that's the kicker, isn't it?

How do you know if your clock, your chosen periodic event is truly regular?

Like the day, right?

We use the Earth's rotation, but a summer day feels different from a winter day.

Are they really the same length?

Exactly.

So you need a comparison test.

Feynman uses the example of an hourglass.

If you just count sunrise to sunrise, the number of times you flip the hourglass changes.

Ah, but if you measure from noon to noon when the sun's highest, suddenly the count becomes much more consistent.

Right.

And it's not about proving the day is perfect or the hourglass is perfect.

It's about finding different phenomena whose regularities seem to mesh to fit together.

We sort of have to trust that consistency.

Because you can't really rule out some, I don't know, omnipotent being slowing everything down uniformly.

Philosophically, it's a dead end.

We just work with the interlocking

Okay, so we have a standard, like the day.

How do we get smaller units?

Galileo again?

Yep, the pendulum.

He noticed that for small swings, the time period is constant.

That gave away to subdivide time.

Eventually, that led to defining the second 186 ,400th of an average solar day.

But for modern physics, a second is an eternity.

We needed faster clocks.

Much faster.

So enter electronic oscillators.

Instead of a swinging weight, you have electrical current oscillating back and forth incredibly rapidly.

How rapid are we talking?

We can now measure intervals down to about 10, 12 seconds, picoseconds.

Wow.

And how do you even check something that fast?

How do you calibrate it?

You use something like an electron beam oscilloscope.

It lets you visualize two different frequencies at once.

You can essentially compare the super fast oscillator against a known slower one and see how they line up.

Think of it like watching for time.

Okay, 10, 12 seconds is fast.

But physics pushes even further, right?

To time so short, you can't use oscillators anymore.

That's where things get really interesting.

We have to stop trying to measure time directly with a repeating event and start inferring it.

Inferring it how?

From motion.

If you know how far something traveled and you know how fast it was going, ideally close to the speed of light, you can calculate the time it took.

Simple division.

T, TA is DV dollar.

Ah, like that D -up -U -Mulmesson example Feynman gives.

Exactly.

It lives for such a short time you can't clock it.

But physicists saw its track in photographic emulsions.

They could measure the distance it traveled before decaying maybe 10 to 7 meters.

And since it's moving near the speed of light, you just divide that tiny distance by C.

And you get its lifetime, which turns out to be incredibly short, around 10, 16 seconds.

From pulse speeds to 10, 16 seconds, that's quite a leap.

And even that's long compared to some things.

There are these strange resonances, particles that exist for maybe 10, 24 seconds.

10 to the minus 24th.

What does that even mean?

Well, it's roughly the time it takes light to cross the smallest thing we really know about, the nucleus of a hydrogen atom.

Okay, that puts it in perspective.

But it also raises that deep question, Feynman asks.

Does time even make sense below that scale?

If processes happen faster than light can even cross a nucleus, is our concept of time still valid?

It's a frontier question.

Absolutely mind -bending.

Now let's swing completely the other way, measuring really, really long times.

Geological time.

Right.

Counting days or tree rings only gets you so far.

For the deep past, we need a different kind of clock.

Radioactive decay.

This isn't periodic, is it?

It's decay.

Correct.

It's not repeating, but it's incredibly regular in a statistical sense.

The key concept is the half -life, symbol T.

It's the fixed time it takes for half of the radioactive atoms in a sample to decay.

So after one half -life, half is left.

After two half -lives, a quarter is left, then an eight, and so on.

Precisely.

The fraction remaining follows the formula, 12T, where T is the time elapsed.

If you can measure the fraction remaining and you know the half -life, you can calculate T.

And this is how we date things.

Yes.

For organic stuff, wood, bones, charcoal, we use carbon -14.

It has a half -life around 5 ,000 years.

That works well for dating things back, oh, maybe up to 50 ,000 or even 100 ,000 years.

You need to know how much C -14 was there to begin with, though, right?

That's crucial.

You need to know the initial ratio of C -14 to stable carbon in the atmosphere when the organism was alive.

Luckily, that's relatively stable and well understood.

And for rocks.

For the age of the earth itself.

You need something with a much longer half -life.

Uranium is perfect.

Some isotopes have half -lives measured in billions of years, around 199 years.

So you measure how much uranium is left in an old rock, and how much of its decay product, lead, has built up.

Exactly.

That ratio tells you how many half -lives have passed.

And doing that for the oldest rocks gives us the age of the earth, about 5 .5 billion years.

It's amazing, really.

These tiny atomic processes give us a clock for cosmic history.

It also aligns with dating meteorites, suggesting the whole solar system formed around then, and the universe itself, maybe 10 or 12 billion years ago.

Raising another fundamental question.

Did time even exist before the Big Bang?

What does that mean?

Heavy stuff.

Let's bring back to standards.

We started with the day, but you mentioned it's not perfect.

Yeah, precision measurements showed the earth's rotation actually fluctuates slightly.

Some days are a tiny bit longer or shorter than others.

Our fundamental standard wasn't truly constant.

So we needed something better.

Much better.

And we found it in atoms.

Atomic clocks are based on the incredibly stable, characteristic frequency of vibrations within certain atoms, like cesium or the hydrogen atom clock Professor Ramsey worked on.

How accurate are these?

Astonishingly accurate.

Like one part in 199, one part in a billion, or even better.

They are so good that the plan is, or perhaps already has been, to redefine the second based on a specific number of these atomic vibrations.

Making the second totally universal and reproducible, independent of our wobbly planet.

Exactly.

It's a much more fundamental standard.

It's fascinating how often times, especially the speed of light, pops up when we try to measure distance too, are they becoming intertwined?

In many ways, yes, especially at large scales.

But let's back up and think about distance, the where from the ground up.

Okay.

Basic measurement.

You take a stick, call it a unit, and you count how many times it fits.

Or you subdivide it.

Simple.

Simple on a human scale.

But for larger distances, say, the moon carrying a stick isn't practical, so we developed triangulation.

Using angles and a known baseline to calculate the distance to a faraway point.

Right.

It's essentially a new definition of distance based on geometry.

Luckily, where we can check it, it agrees perfectly with the stick method.

We used it for Sputnik, maybe $5 times 155 meters up.

And the moon, much further, $4 times 188 meters.

But Feynman says it didn't work for the sun.

Not directly from Earth with early instruments.

The angles were too small, the baseline too short.

So for the solar system, the approach was clever.

First, figure out the relative distances of planets using Kepler's laws and astronomical observations.

So you know Venus is x times closer than Mars, etc.

But you don't know the actual distance in meters yet.

Correct.

You need one absolute measurement to set the scale for the whole system.

Historically, they triangulated on the asteroid arrows when it came close.

More recently, radar.

Ah, using time again.

Precisely.

Bounce a radar signal off Venus.

Measure the round trip time.

Multiply by the speed of light, which we know very accurately.

That gives you the distance to Venus in meters.

And boom, the whole solar system scale clicks into place.

Distance measured by time.

Okay, so for stars,

triangulation again, but with a bigger baseline.

A much bigger baseline.

Earth's orbit.

We measure in nearby stars apparent position against the background stars in, say, summer, and then again in winter six months later.

So the baseline is the diameter of Earth's orbit around the sun.

That's huge.

Massive.

It lets us measure the parallax, the tiny angular shift, for nearby stars and calculate their distance.

But that only works for nearby stars, right?

What about ones further out?

For those, parallax is too small to measure.

We need another trick.

The color brightness relationship.

How does that work?

Astronomers studied nearby stars where they could measure distance via parallax.

They found a strong correlation.

A star's color tells you its intrinsic brightness.

How much light it's actually putting out.

Okay, so red stars have one typical brightness, blue stars another.

Generally, yes.

So for a distant star, you measure its color.

That tells you how bright it should be intrinsically.

Then you measure how bright it appears to us.

Since apparent brightness fades with the square of the distance, you can calculate how far away it must be to look that dim.

Clever.

Indirect, but it works.

It's crucial.

It lets us estimate distances across our galaxy.

We figured out the center of the Milky Way is about 10 meters away, using methods based on globular clusters, which utilize this principle.

And even further,

to other galaxies.

We start making assumptions.

We assume distant galaxies are roughly similar in size and brightness to our own Milky Way.

We measure the tiny angle they subtend in the sky.

And use that plus the assumed size to estimate the distance.

Almost like triangulation on a cosmic scale.

Kind of.

It's how we get to the truly mind -boggling distances, like the 10 meters to the furthest galaxies observed, say, by the Palomar Telescope mentioned in the text.

The edge of what we can currently see.

Wow.

Okay, 10 meters down to...

Let's go the other way.

Super small distances.

Right.

We can subdivide our meter stick down to maybe microns, 10, 6 meters.

But then we hit a fundamental limit.

The wavelength of light.

Exactly.

Visible light has wavelengths around half a micron, $5 x 10, 7 meters.

You just can't see details smaller than the wavelength of the light you're using to look.

It's like trying to measure a grain of sand with a thick marker.

So we need probes with smaller wavelengths.

Yep.

Enter the electron microscope.

Electrons behave like waves, but their wavelengths can be much shorter than visible light.

That gets us down to maybe 10 meters, enough to see large molecules, like viruses, perhaps.

And to see atoms themselves, about 10 denataries.

For that, we usually use x -ray diffraction.

X -rays have even shorter wavelengths.

You fire them at a crystal.

Which has atoms arranged in a nice regular grid.

And you look at the pattern the x -rays make after they scatter off those atoms.

It's like a very tiny complex form of triangulation.

The pattern reveals the spacing between the atoms.

Okay.

So atoms are 10 den meters.

But then Feynman highlights this huge jump.

Yeah.

The nucleus inside the atom is way smaller, down around 10, 15 meters.

There's this vast emptiness, relatively speaking, inside the atom.

A factor of 105 .55 difference in scale.

How on earth do you measure something that small?

10, 15 meters.

You can't see it directly.

You have to measure its effect.

We use the idea of an effective cross -section.

That sounds statistical.

It is.

Imagine you have a very, very thin sheet of material.

You fire high -energy particles, like protons or neutrons, straight through it.

Most will miss the tiny nuclei and pass right through the electron clouds.

But some will hit a nucleus dead on.

Right.

And get stopped or deflected.

You measure the fraction of particles that don't make it through cleanly.

That fraction represents the ratio of the total target area presented by all the nuclei, compared to the total area of the sheet you shot at.

Ah.

So if one in a million particles get stopped, the total area of the nuclei shadows is one millionth of the sheet's area.

Exactly.

And that total shadow area is the number of nuclei times the cross -sectional area of each one.

We call that cross -section sigma.

Assuming the nucleus is roughly spherical, its area is u pi r two two dollars.

So from the fraction stopped, you calculate the effective radius.

Precisely.

And doing this reveals that nuclear radii are tiny, ranging from about one to six times 10, 15 meters.

And that unit 10, 15 meters gets its own name.

Yes.

It's called the Fermi, in honor of Enrico Fermi.

So nuclei are a few fermis across.

Just like with time, the standard for length is also moved beyond physical objects, right?

No more meter bar in France.

Correct.

The modern standard of Langley are found based on the wavelength of a specific spectral line of a particular atom.

A certain number of wavelengths equals one meter.

It's invariant, reproducible anywhere.

So we've spanned this incredible range from the Fermi, 10, 15 meters, all the way out to the edge of the observable universe at 10 meters, 41 orders of magnitude.

That's staggering.

It truly is.

But Feynman also emphasizes the limits.

Even with these refined definitions and techniques, measurement itself isn't absolute.

First, there's relativity.

Right.

Einstein showed that measurements of both distance and time are relative.

They depend on the observer's frame of reference, specifically on their motion.

Two observers moving relative to each other will literally measure different lengths and different time intervals for the same events.

There's no single universal now or here.

And then even within a single frame, there are quantum limits, the uncertainty principles.

Yes.

Heisenberg's principle tells us you can't know both the position and the momentum, which is mass times velocity, of a particle with perfect accuracy simultaneously.

The more precisely you pin down its location, the less precisely you can know its momentum and vice versa.

There's a fundamental trade -off related to Planck's constant dollar dollars.

It stems from the wave nature of particles.

To locate something precisely, you need to hit it with something energetic, which inevitably disturbs its momentum.

And there's a similar trade -off for time and energy.

Exactly.

The uncertainty in measuring the energy of a system, and the uncertainty in knowing the time duration over which that energy is measured, are also linked by Planck's constant, delta E, delta T, GTT day.

So if you want to know when something happened extremely precisely, very small delta 2T, you inherently introduce a huge uncertainty about the energy involved in that event.

That's the essence of it.

To know when, with high precision, you often need a high energy interaction for your measurement, which fundamentally disturbs what the energy state you were trying to measure.

Nature itself imposes these limits.

You can't know everything perfectly.

It's built into the fabric of reality.

It seems so.

We're forced to accept these fundamental uncertainties.

Well, this has been quite a journey.

From Galileo counting his pulse to defining the atomic vibrations, and from a simple measuring stick to calculating the size of a nucleus using probabilities.

It really shows how physics progresses by constantly refining and sometimes completely rethinking how we measure the most fundamental things, space and time.

And Feynman leads us with that really provocative question, doesn't he?

We've pushed measurement down to the scale of the nucleus, the Fermi, and the corresponding time scale of 1024 seconds.

He asks,

does our understanding of space and time even hold up at scales smaller than that?

Could the deep mystery still surrounding nuclear forces, for instance, require us to modify our very notions of geometry and measurement at those infinitesimal distances?

Is there a below the Fermi where space, as we conceive it, breaks down?

That's the frontier.

A frontier for future physics and something for you, the listener, to really chew on.

A perfect place to pause.

Thank you for joining us on this deep dive into Feynman's take on time and distance.