[Potironette]

Ohh, that makes sense. Because that's what velocity is ><'.

Does the force * time thing have anything to do with e=mc^2 ? And rocket fuel is using up m for e..? Or is e=mc^2 referring to something else?


Impulse is a word I rarely hear, so it sounds really odd. Thanks for the clarification!

[Coda]

No, E=mc^2 is mostly used for nuclear reactions, not chemical ones. You CAN theoretically use it empirically to determine the energy of a chemical bond by breaking that bond and measuring the change in the mass of the system -- I worked the math for this in an assignment in high school but I don't know if it actually gets used this way in practice.

Rather, rocket thrust works based on the principle of conservation of momentum. When you throw something off the back of the rocket, the equal-and-opposite-reaction pushes the rocket forward.

A simple example that you should be able to work given the knowledge you already have:

A drone weighing 250 grams when empty is armed with a pellet gun. Pellets weigh 1 gram each, and the gun can fire them at 200 meters per second. If you load the drone with a single pellet, how fast will it be going due to the recoil after it shoots?

(I haven't actually WORKED the answer.)

To make this more like a rocket, load the drone up with 10 shots and fire one per second. This takes more work -- you haven't studied the math that would make this easier, so you'd have to work out the math one shot at a time -- but you can see that it starts off at 259 grams going one way and 1 gram going the other way for the first shot, but 250 grams going one way and 1 gram going the other way for the last one -- the same impulse results in a different change in velocity.

[Potironette]

Ohh, okay. I remember there being problems about using E=mc^2 for coal in some previous unit at school, but in real life it makes sense that it isn't useful.

(200 gm/s) / 251 g = 0.7968 m/s about.

200/251 + 200/252 + 200/253 + 200/254 + 200/255 + 200/256 + 200/257 + 200/258 + 200/259 + 200/260 = 7.8288 m/s about.

I'm glad it was only ten x'D.
Thanks for the drones problem! I didn't notice that the pellet added mass to the drone until later :x

[Coda]

Oh, no, no, E=mc^2 IS useful, just not for the kinds of chemistry you can do in a basic lab. Nuclear chemistry is a totally different beast.

Pellet doesn't add to the mass of the one-shot example, because it's already fired the shot by the time you calculate it. 10-shot example you should have gone from 250 to 259 instead of 251 to 260.

I'll check your math later, I've gotta run right now.

[Coda]

Okay, so your math is right if you fix the mistake I already mentioned.

As for a real-world example of E=mc^2... Photons don't have mass, because they're pure energy. They travel at the speed of light, never any faster or any slower. But we know that photons can transfer kinetic energy to other objects -- which means that photons must have momentum.

Except momentum is mass times velocity, and photons have no mass!

That's where E=mc^2 comes in. We can say that a photon's effective mass, for momentum purposes, is equal to its energy, divided by the speed of light.

[Potironette]

woops. So it's actually

(200 gm/s) / 250 g

And

200/250 + 200/251 + 200/252 + 200/253 + 200/254 + 200/255 + 200/256 + 200/257 + 200/258 + 200/259

Oh wow, that's interesting--but also confusing. A photon doesn't have a mass but it does have an "effective" mass? Also, if E=mc^2 holds, then without a mass there's no energy..?

[Coda]

Now we're getting into relativity. Fun stuff!

E=mc^2 is actually a simplification of the full formula, which is E^2 = p^2*c^2 + (m*c^2)^2, where p is momentum and m is rest mass, by setting momentum equal to zero (that is, considering an object at rest). If you instead set mass to zero, like for a photon, then you get E = p*c, which is to say, the momentum of a photon times its speed (obviously, the speed of light) equals its total energy. Mass-energy equivalence means that a particle's mass while in motion is greater than its mass when at rest.

When it comes down to it, energy is energy, and energy is the thing that actually does stuff. So it doesn't matter if the energy comes from rest mass or momentum; you can combine the two into into a single measurement and treat them as interchangeable.

It should be noted that a photon's relativistic mass is very very small. It's possible to set up an experiment to observe the effect of photon momentum on Earth, but it requires sensitive tools, because friction (including drag from the air) will easily overwhelm it. However, a spacecraft was launched in 2010 that uses a solar sail for propulsion -- the only thing making it move is the light from the sun pushing on its sail like wind in a kite. And it's still accelerating today!

To give you an idea of just how small the forces involved are: A solar sail 800m x 800m square can catch enough sunlight to exhibit a force of only around 5 newtons.

[Potironette]

Now we're getting into relativity. Fun stuff!

How is this relativity..?
Granted, I don't know what relativity is, and everything I've seen that explained it shot out of my head like gibberish x'D

E=mc^2 is actually a simplification of the full formula, which is E^2 = p^2*c^2 + (m*c^2)^2, where p is momentum and m is rest mass, by setting momentum equal to zero (that is, considering an object at rest).

The full equation looks a lot like the Pythagorean formula :o. Why is it simplified though? Is it just so that I, as a student, can understand the formula (since we learned energy before momentum)?

...What about p = mv though? If the mass is zero, the p turns zero too


Oh wow, solar sails are like the stuff of fantasies xD. Makes me wonder though, if E makes up that equation, where things like thermal energy and the rest come in--though I haven't yet learned about any of those :x

[Coda]

This is relativity because energy-mass equivalence is something that only really matters at relativistic scales.

At normal scales, you can just use p=mv and just ignore the contribution that motion adds to the rest mass. I mean, look at that full equation -- the mass term has c^4, while the momentum term only has c^2. And c is a BIG number. You'd need a momentum in the neighborhood of 10^17 (or a mass on the order of 10^-17 kg) in your equation for those terms to be on the same order of magnitude.

p = mv can be true for massive objects with no velocity. That's why the simplification is used -- it lets you reason about the energy equivalence of the mass itself at scales where the relativistic contribution of its velocity is negligible.

Macroscopic measurements of different types of energy come from the collective behavior of huge numbers of particles. At subatomic scales -- a proton has a mass of less than 10^-30 kg -- many things like "heat" don't have meaning. Temperature, for example, is a measure of the average kinetic energy of the particles in the thing being observed. There are field forces (like gravity and magnetism) that still come into play, so potential energy is a thing that still comes into the picture, but that doesn't participate in energy-mass equivalence as far as I can tell in my research. (I actually had to go look this up because I didn't know.)

Solar sails DO totally seem like sci-fi stuff, don't they?

[Potironette]

Beyond the idea of actually moving from sunlight, how simple they look is really amazing! Reminds me of reading books about space-ships XD. But sunlight pushing stuff is really cool :o

Thanks for all the answers!

[Potironette]

What is scattering of light? In class we were talking about how "Rayleigh scattering" was gas molecules absorbing and re-emitting light in all directions, "with the proportion of light scattered related to its wavelength, yielding blue sky and red sunsets." And then: 'Because the Sun’s output begins to drop off at violet wavelengths, most of what reaches us due to scattering is blue and green, yielding familiar “sky blue.”'

So, lights are made of photons at different energies, and when they hit stuff like gas molecules they get "absorbed" then "re-emitted" (whatever those mean) and then the "proportion" (amount?) of light "scattered" (seen?) is based on the "wavelength" (of the light before it hit the gas?) and sunlight has photon particles with energies that make blue-light wavelengths which is why we see blue?


I end up nodding off a lot in physics class since it's later in the day. Hence the many holes in my learning :/

[Coda]

In your defense, Rayleigh scattering is usually taught very poorly at the high school level.

So before I start getting into the technicalities you're asking about... I'm just going to make one big statement: The sky is blue because the Sun is blue.

Now I'm going to roll it back to the beginning and build back up to that statement.

Though first, one quick definition, since I'm going to use it in my description and you were asking about it: A proportion is just a fraction -- a description of some part of a whole that has a certain property. If two things are proportional, then there's a relationship between them such that they go up and down together at related rates.

Part 1: Wavelengths of Light

As you know, what we perceive as "white" light is made up of a combination of colors. No individual photon of light is "white." (For that matter, no individual photon of light is magenta, either. Magenta doesn't exist except in your head.)

The color of light is determined by its wavelength, and the wavelength of light is directly determined by its energy. (For ANY wave, if you know two of wavelength, velocity, and energy, you can determine the third one. We know that light's velocity is c, so that means its wavelength and its energy have a directly inverse relationship with each other. When one goes up, the other goes down.) So whenever you see "color" or "wavelength" or "energy" you know it's all talking about the same thing.

The Sun is a fairly good approximation of a blackbody radiator. That means it emits photons in a continuous range of energies (this is called an "emission spectrum"), with the most of them at an energy level determined by the temperature of the blackbody (this is known as its "spectral peak") and then you get less of them the farther away from that energy level you go. The Sun burns at around 6000K, so it emits blue-green light the brightest, a little less violet light and yellow light, a little less near-ultraviolet light and red light, a little less far-ultraviolet light and infrared light, and so on. There's no gaps; every color in between is represented.

(Incandescent light bulbs are ALSO blackbody radiators, but since they're much cooler than the Sun, their spectral peak is in the infrared region of the spectrum, so most of the light coming from an incandescent bulb is heat, and most of the visible light is on the red end of the spectrum. Fluorescent and LED lights aren't blackbody radiators, so their emission spectra aren't smooth and continuous.)

Upshot: When your book talks about sunlight dropping off in the violet region, it's referring to the shape of this emission spectrum.

Part 2: Reflection

Throw a ball at a big flat surface. It bounces off at a predictable angle, right? Light does the same thing: shoot a beam of light at a big flat perfect surface (like a mirror) and it'll bounce off predictably.

Throw a ball at an uneven surface. Still bounces, but it isn't perfectly predictable; if you throw a bunch of balls, each one might bounce a different direction. You can estimate where they're likely to end up, but you can't predict any individual throw. Again, light does the same thing: shoot a beam of light at a sheet of paper, and while you know it's going to bounce, it's not going to bounce perfectly. It scatters, because each of the photons gets bounced in a little bit different direction.

Throw a ball at another ball. You can't even be sure if it's going to bounce, this time, because that's an awfully small target, and it's pretty likely that you're going to miss. You throw a bunch of balls, and most of them are going to go right on by without even so much as deflecting. And even if you hit, you can't be sure which way it's going to bounce because both balls are round, and you can't even be sure how hard it's going to bounce because the other ball is going to get knocked away too.

Now we're getting closer to the idea of Rayleigh scattering, but we're still not there yet.

Part 3: Crazy Physical Analogies That Actually Work!

Forgive me for that silly outburst of clickbait.

Blue light has a wavelength of 475 nanometers. A nitrogen molecule, like what the vast majority of our atmosphere is made up of, has a scattering diameter of a little under 1 femtometer -- around 600 million times smaller! That's like trying to hit a speck of dust with a beach ball, except the speck of dust weighs a trillion kilograms. (I actually calculated that number; I didn't just pick a big number arbitrarily -- I used E=mc^2 here!)

Now, fortunately for sanity, Newton's third law does in fact still apply at this scale. The principle of equal and opposite reaction still applies. (The first and second laws get fuzzy when you start approaching the speed of light. We already discussed the changes to the second law -- F = m*a is really an approximation of E^2 = m^2*c^4 + p^2*c^2 -- and the first law requires that you can construct a coordinate system with the object in question at relative rest, and you can't do this for photons because they're ALWAYS traveling at c no matter how you look at them.)

So given the third law, you can treat this situation like shooting said (indestructible, perfectly rigid) beach ball with said speck of dust traveling at the speed of light.

Surprisingly, the math actually does work out that way.

There's some fiddly electromagnetic fudge factors you have to take into account in order to define how big the balls are, but once you've established the effective scattering cross-section, Rayleigh scattering turns out to be a fairly clean formula that describes what you get when you average out all of the probabilities over a volume of space.

If you shoot the beach ball dead center, it's just going to go zipping directly in the opposite direction. If you hit it with a glancing shot on the side, both objects are going to ricochet in different directions.

Or, to drop the metaphor and go back to the actual question at hand: When a photon hits a nitrogen molecule, it'll bounce off in a direction based on how it hits and the relative sizes, and you can average out how far it's likely to deflect from its original straight-line path. And since different colors of light have different wavelengths and therefore different sizes, they have different statistics.

The Rayleigh scattering formula collects those statistics into a very convenient number: Given a beam of a given color of light shining through a unit volume of space, how intense is the light coming from the sides instead of passing straight through?

Part 4: Why Blue?

Now that we understand what Rayleigh scattering is, the next question is... Why blue?

Well... that's where those electromagnetic shenanigans I mentioned come into play. It turns out that the relevant cross-sectional radius has an inverse relationship with the wavelength, so that means the cross-sectional area has an inverse square relationship with the wavelength -- that is, counterintuitively, longer wavelengths mean a SMALLER target. (Actually, it's that higher frequencies make it less likely to have a collision, and higher frequencies mean lower wavelengths.) And then since the light can scatter in any direction, the amount of light coming in any one direction follows the inverse square law (brief description: a ray passing through the center of a sphere must pass through a point on the surface of that sphere, and surface area is 4pi*r^2), so Rayleigh scattering ends up being inversely proportional to the 4th power of the wavelength.

The important part of that last paragraph is that shorter wavelength -> more scattering.

For a ray of light that WOULDN'T otherwise pass in a straight line from the Sun to your eye, for you to be able to see it at all it would have to be scattered.

Remember that violet has the shortest wavelength of visible light, and red has the longest. So that means the red light from the sun is more likely to keep going instead of bouncing off of the air in order to come back and hit your eye.

Why isn't the sky purple, then? Because referring back to part 1, the Sun emits more blue light than it emits violet light, so even though a greater PROPORTION of purple light is getting scattered back to your eye, there's just not as much of it in the first place, so you see more blue.

Part 5: But You Said...!

Yep. I said the Sun is blue.

Sure doesn't look blue, does it? Looks yellow!

The light that IS coming directly from the Sun to your eye is ALSO getting scattered. The blue light is getting deflected off to the side more instead of reaching you. That leaves the light that gets scattered less -- the yellows and the oranges and the reds.

Why isn't it red, then, if red gets scattered least?

Same reason the sky isn't purple: the Sun emits more yellow light than red.

From outer space, the Sun looks white, and space looks black. There's no air in the way, so all of the colors of light mix together in our eye and we perceive white that might be ever so faintly tinted blue.

Part 6: Absorbing? Re-Emitting?

I didn't actually talk about this stuff above!

This is one of the reasons -- though by no means the ONLY reason -- I think Rayleigh scattering is taught poorly. The whole nitty-gritty details about absorption and re-emission is diving into too much detail on the WRONG PART of the phenomenon. It suggests that it's relevant to the macro-scale things we observe, when in fact you can understand how Rayleigh scattering works without ever needing to talk about it.

But to actually discuss it...

Photons don't actually BOUNCE off of things. Despite our fun little relativity discussion, they don't have mass; they're pure energy. So when a photon hits something, it transfers its energy into whatever it hits. But atoms can't hold arbitrary amounts of energy; they want to be at the lowest stable energy state they can, so if nothing happens to make the higher-energy state stable they quickly shed that extra energy in the form of another photon. And since the energy of the photon determines its wavelength, the photon that comes out is exactly the same color as the one that went in. (Obviously there are exceptions -- those exceptions are where chemistry happens.)

Photons also aren't just particles. They're also waves, and no macro-scale physical model can fully imitate ALL of the weirdness that implies. It's the wave nature of a photon that causes the counterintuitive behavior where longer wavelengths mean shorter cross-sections for collisions.

But you don't NEED to worry about that for a high-school level description.

[Potironette]

Oh wow sections of information! Thanks for taking the time to answer with all that!

So the sky is blue because the sun is mostly blue and the blue wavelengths hit nitrogen more (and nitrogen makes up most of the air) because it's a higher frequency and shorter wavelength (I was confused by "Actually, it's that higher frequencies make it less likely to have a collision, and higher frequencies mean lower wavelengths." Isn't blue light a higher frequency than red light..?).

But why is it that sometimes the sky is pink or orange? Surely both the sunlight and the nitrogen in the sky stay the same? At night is it dark blue just because there's less light?

Does the sun having more energy cause it to have lots of blue wavelengths? Similar to how the blue bit of the fire is hotter--is that even related?

Why does Newton's first law require a coordinate system to be constructed? And what does "relative rest" mean? Why does it matter that the photon doesn't rest? Or does it mean that for some reason the first law requires there to be a moving photon to be compared to a not moving photon--which doesn't happen? And since photons are always moving at c...waves of light are basically how the photons are moving?

[Coda]

Originally Posted by Potironette

(I was confused by "Actually, it's that higher frequencies make it less likely to have a collision, and higher frequencies mean lower wavelengths." Isn't blue light a higher frequency than red light..?)

I think I actually meant to say "more likely" there and I messed up during one of my edits to the text.

Another way to think about it is that shorter wavelengths means there's less of a gap between the crests of the waves, so there's a greater probability that one of those crests will intersect with the target.

But why is it that sometimes the sky is pink or orange? Surely both the sunlight and the nitrogen in the sky stay the same?

Nope, there's more sky between you and the sun as the sun approaches the horizon -- think about two concentric circles and a line passing between a point on the inner circle and a distant point. That means the blue light is getting scattered even further, leaving the redder shades of light to pass through to where you can see them.

At night is it dark blue just because there's less light?

Yup, got it in one.

Does the sun having more energy cause it to have lots of blue wavelengths? Similar to how the blue bit of the fire is hotter--is that even related?

Bingo, got it again! This goes back to the discussion of blackbody radiation. Objects that emit light because of how hot they are follow a very predictable pattern.

Why does Newton's first law require a coordinate system to be constructed? And what does "relative rest" mean?

The first law is inertia, that is, an object won't change velocities unless a force acts upon it. A consequence of this fact is that -- in the absence of external forces -- you can pick any object you want, call it the center of your universe, define its velocity as zero, and judge the velocity of everything else relative to it.

The classic example is riding in a car. From your perspective inside the car, the seat, the steering wheel, the radio knobs, and the doors aren't moving -- they have a velocity of zero. You look out the window, and everything is moving backwards at 90mph. From the perspective of the cop on the side of the road, the street has a velocity of zero, and you're breaking the law. Newton's first law says that both of these descriptions are equally valid.

Why does it matter that the photon doesn't rest? Or does it mean that for some reason the first law requires there to be a moving photon to be compared to a not moving photon--which doesn't happen? And since photons are always moving at c...waves of light are basically how the photons are moving?

The second guess is closer.

If you pass a slowpoke going 70, then you, the slowpoke, and the cop agree that there's a 20mph difference in your velocities. (The cop still thinks you're breaking the law.)

The cop has a radar gun. After all, that's how he knows you're going 90mph faster than he is, and how he knows the slowpoke is going 70mph faster than he is. If he took off in his cruiser and got up to 100mph so he could catch up with you, the radar gun would say that you were going -10mph, and the slowpoke was going -30mph.

So far, so good. But here's where things get weird...

No matter who's measuring it, EVERYONE -- you, the slowpoke, the cop -- agrees that the radar beam coming out of that gun is going at the speed of light, c.

Not c - 70. Not c - 90. Just c. All the time. Always.

Even the alien spacecraft zipping by the planet at 0.3c sees the beam going c, not 0.7c. (The cop isn't going to try to write the alien a ticket. It gets off with a warning.)

So because photons always travel at c no matter how you look at them, you can't measure the velocity of things relative to the photon. Oh, sure, you can try, and you can get a number, and that number might even seem reasonable -- if the alien were in a drag race with a laser beam, the judge at the finish line would say that the laser beam was going 0.7c faster. But unlike you and the slowpoke, where everyone agreed you were going 20mph faster, the alien thinks the laser is going 1.0c faster than it.

The space police don't write speeding tickets; the interstellar speed limit is c, and you can't break it. On the other hand, drag racing a laser beam is reckless, so the cop comes up to the race at 0.5c. He thinks he's going roughly 0.2c faster than the alien (not EXACTLY 0.2c, but that's a lecture for another lesson), and he thinks the laser is going 1.0c faster than him, so the cop thinks that the laser is outrunning the alien by 1.2c. The cop doesn't have to write a speeding ticket for the laser, though, for the same reason it's not illegal to pass someone going 140mph relative to them on the highway.

So the judge, the alien, and the space cop disagree on how much faster the laser is than the alien.

That's why Newton's first law breaks down at relativistic speeds: when you're going that fast, then things look like they're going varying speeds relative to each other even if there AREN'T forces acting on them.

[Potironette]

Oh wow, light is strange! I wonder why it happens that regardless of relativity, everyone sees it as 1.0c faster than themselves?
Uh, but if lightbeamA were racing lightbeamB would lightbeamA see lightbeamB as going 1.0c faster. And why Newton's first law doesn't work is because even without another force acting on it, the light travels 1.0c faster?

Err, so the Earth rotates around every day and around the time the skies turn pink/orange at the place, the place is further away from the sun. And because it is further away the sunlight has to go through more air stuffs to reach the person's eyes at that place. And because there is more sky, the blue's already been scattered a lot further away, now what's left to be scattered to reach the person is the red/orange light? And so those reach the person's eyes? But if that's so then why is it that during nighttime the sky is more bluish than reddish?

EDIT: Attempted to make it more coherent.

[Coda]

Originally Posted by Potironette

Oh wow, light is strange! I wonder why it happens that regardless of relativity, everyone sees it as 1.0c faster than themselves?

That's Einsteinian relativity instead of Newtonian relativity. The shortest description I can give you is that the speed of light is really better described as the speed of time, that is, a measurement of the relationship between time and space. If it sounds confusing that's because it is and I don't know if I can make it intuitive.

Uh, but if lightbeamA were racing lightbeamB would lightbeamA see lightbeamB as going 1.0c faster.

Answering this fully gets pretty deep into relativity and I'm not sure I can satisfactorily answer it without going overboard -- and then THAT risks being too confusing.

At the core of it, though, is that you can't consider things from a photon's perspective. The math breaks down; trying to set up that frame of reference ends up quite literally dividing by zero.

You might have heard about time dilation -- the faster you go, the slower time appears to pass for everyone else. When you hit the speed of light, the universe stops experiencing time at all from your perspective; if you COULD look through a photon's eyes, you'd see yourself crossing the entire universe without any time having passed at all. You couldn't see another photon moving because there would be no time for the photon to move in: your velocity formula of (change in position) / (change in time) ends up with a zero on the bottom.

And why Newton's first law doesn't work is because even without another force acting on it, the light travels 1.0c faster?

It's not so much that Newton's first law doesn't work so much as it is that it doesn't apply. The law has a big "if" in it. At normal scales you can simplify that condition to "no net force" but the more formal statement of the law is "in an inertial reference frame" and light doesn't have one of those, for the above reason.

Err, so the Earth rotates around every day and around the time the skies turn pink/orange at the place, the place is further away from the sun. And because it is further away the sunlight has to go through more air stuffs to reach the person's eyes at that place. And because there is more sky, the blue's already been scattered a lot further away, now what's left to be scattered to reach the person is the red/orange light? And so those reach the person's eyes?

You got it.

But if that's so then why is it that during nighttime the sky is more bluish than reddish?

Because the Earth itself is in the way. Only the light that gets scattered the most can reach you, because anything taking a more direct path is busy shining on the daytime side of the planet.