Adding the air resistance: the form of the formula can be guessed once you got the conservation laws down.
Other ideas:
The biggest advantage of lens flares may be in serving as a spectrometer.
Thermodynamics is a piece of cake once you got kinematics down: step 1 is to realize energy is conserved, step 2 is to write the max entropy (aka least knowledge/most general) distribution given energy conservation, congratulations you have the ideal gas law. If you’ve got atomism on the level of Democritus you can write down the van der Waals equation. Unsure what this gets you that’s relevant, maybe it helps constrain constants. You can from here get equipartition and thus the ultraviolet catastrophe, giving a hint at quantum mechanics.
If the rock (or anything else in frame) is wet with visible droplets then you learn something about surface energies from the contact angle. I don’t know how well you can calculate that from first principles—if you can, you get some constraints on what the stuff is made of and on the fundamental constants. Likewise, refraction and reflection off the drop.
If there are heat ripples, you might get some info about constants—but idk what first principles prediction of temperature dependence of refractive index would look like (maybe you only need ideal gas law).
If the sky is blue/not white, you can relate the wavelength of light to the particle size of gas molecules via rayleigh scattering. With clouds being white you get another constraint (as that comes from mie scattering resonances that average out). To deduce the laws of scattering you need to know electromagnetism—however, you can get electromagnetism by assuming only coulomb’s law (which you can guess from geometry, or guessing inverse squares hold matter together after seeing gravity be inverse square) and lorentz invariance (if you guess some reasonable symmetries (laws same in all inertial/straight line frames, isotropy aka same in every direction) you’ll end up with either gailelean relativity or lorentz transforms).
If you assume at least one law works like the simplest form of attraction/repulsion (that is, central forces, which is really a symmetry that the law doesn’t give a preferred “angle”) then you’ll get something like a magnetic force, so you’ll have to deal with the absurdity of atomic orbits collapsing after radiating their energy away (though unsure if that still happens for what you’d get if the central force wasn’t inverse square). You might then guess Bohr’s ad hoc quantization, though maybe not. You may however just directly guess the schrodinger equation—the two hardest guesses there are relating energy to time frequency and momentum to space frequency, but you might guess that if you squint at the energy and momentum of electromagnetic waves. If you got Planck’s relation from earlier then you’ll have an easier time.
You can of course go from quantum mechanics to spin and quantum field theory. As an example, you could deduce that gravity (being attractive) should be mediated by an even integer spin boson, which also suggests that the source is probably either a scalar or a rank-2 tensor, which at the very least might help you get to general relativity (or at least reassure you you are on the right track).
Most of the semiempirical mass formula for nuclei binding energies can actually be derived theoretically once you know a couple fundamental constants (electric charge, hbar, coulomb’s constant, some way of estimating nuclear radius (idk how you’d do this)). If you can figure out the couple steps that I can’t, then you should be able to make good guesses about what nuclei are stable and what decays are possible. This would help you a lot when guessing what rocks are made of—you mostly need to look up to iron or a couple more.
I’m not sure how well brute forcing ab initio calculations of colors of stuff (and checking consistency with other objects) would work.
Generally the question “what chemicals is the stuff made of?” and “what are the fundamental constants” seem like the hardest ones to answer here.
And this is just what an undergrad came up without trying that hard :p
Adding the air resistance: the form of the formula can be guessed once you got the conservation laws down.
Other ideas: The biggest advantage of lens flares may be in serving as a spectrometer.
Thermodynamics is a piece of cake once you got kinematics down: step 1 is to realize energy is conserved, step 2 is to write the max entropy (aka least knowledge/most general) distribution given energy conservation, congratulations you have the ideal gas law. If you’ve got atomism on the level of Democritus you can write down the van der Waals equation. Unsure what this gets you that’s relevant, maybe it helps constrain constants. You can from here get equipartition and thus the ultraviolet catastrophe, giving a hint at quantum mechanics.
If the rock (or anything else in frame) is wet with visible droplets then you learn something about surface energies from the contact angle. I don’t know how well you can calculate that from first principles—if you can, you get some constraints on what the stuff is made of and on the fundamental constants. Likewise, refraction and reflection off the drop.
If there are heat ripples, you might get some info about constants—but idk what first principles prediction of temperature dependence of refractive index would look like (maybe you only need ideal gas law).
If the sky is blue/not white, you can relate the wavelength of light to the particle size of gas molecules via rayleigh scattering. With clouds being white you get another constraint (as that comes from mie scattering resonances that average out). To deduce the laws of scattering you need to know electromagnetism—however, you can get electromagnetism by assuming only coulomb’s law (which you can guess from geometry, or guessing inverse squares hold matter together after seeing gravity be inverse square) and lorentz invariance (if you guess some reasonable symmetries (laws same in all inertial/straight line frames, isotropy aka same in every direction) you’ll end up with either gailelean relativity or lorentz transforms).
If you assume at least one law works like the simplest form of attraction/repulsion (that is, central forces, which is really a symmetry that the law doesn’t give a preferred “angle”) then you’ll get something like a magnetic force, so you’ll have to deal with the absurdity of atomic orbits collapsing after radiating their energy away (though unsure if that still happens for what you’d get if the central force wasn’t inverse square). You might then guess Bohr’s ad hoc quantization, though maybe not. You may however just directly guess the schrodinger equation—the two hardest guesses there are relating energy to time frequency and momentum to space frequency, but you might guess that if you squint at the energy and momentum of electromagnetic waves. If you got Planck’s relation from earlier then you’ll have an easier time.
You can of course go from quantum mechanics to spin and quantum field theory. As an example, you could deduce that gravity (being attractive) should be mediated by an even integer spin boson, which also suggests that the source is probably either a scalar or a rank-2 tensor, which at the very least might help you get to general relativity (or at least reassure you you are on the right track).
Most of the semiempirical mass formula for nuclei binding energies can actually be derived theoretically once you know a couple fundamental constants (electric charge, hbar, coulomb’s constant, some way of estimating nuclear radius (idk how you’d do this)). If you can figure out the couple steps that I can’t, then you should be able to make good guesses about what nuclei are stable and what decays are possible. This would help you a lot when guessing what rocks are made of—you mostly need to look up to iron or a couple more.
I’m not sure how well brute forcing ab initio calculations of colors of stuff (and checking consistency with other objects) would work.
Generally the question “what chemicals is the stuff made of?” and “what are the fundamental constants” seem like the hardest ones to answer here.
And this is just what an undergrad came up without trying that hard :p