How far, how fast, how much energy?

Because the source is moving through the air while it emits. Approaching, each successive wave crest starts closer to you than the last, so the crests arrive more often and the pitch is higher. Receding, they are stretched out and the pitch drops. The change happens most sharply at the moment of closest approach, not gradually along the road.

The physics rhymes but the formulas differ. Sound travels in a medium, so it matters whether the source or the listener is the one moving — the two cases give slightly different answers. Light has no medium, so only the relative velocity matters, and the relativistic formula includes time dilation, which has no acoustic counterpart.

E = hf, or equivalently hc/λ. A green photon at 550 nm carries 3.61 × 10⁻¹⁹ joules, which is 2.25 electronvolts. Both h and c have been exact defined constants since the 2019 SI revision, so this is one of the few calculations in physics with no measurement uncertainty in its constants.

Because damage depends on the energy of the individual photon, not on total brightness. Breaking a chemical bond takes a minimum quantum of energy, and red photons at about 1.8 eV simply do not have it however many arrive. UV photons at 4 eV and up do, so even a weak UV source causes damage a blazing red one cannot.

For nearby stars, by parallax: the star appears to shift slightly against the background as the Earth moves from one side of its orbit to the other, and the distance in parsecs is one divided by that shift in arcseconds. The angles are minute — under one arcsecond even for the closest star — which is why the first successful measurement came only in 1838.

A parsec is the distance at which one astronomical unit subtends one arcsecond, which the IAU fixed exactly in 2015 as 648,000/π AU. That works out to 3.2615638 light-years, or about 30.857 trillion kilometres. The commonly printed 3.26 is a rounding of an exactly defined value.

For small redshifts, v ≈ cz, so z = 0.001 means about 300 km/s. Beyond z ≈ 0.1 that overestimates, and the relativistic form v = c[(1+z)² − 1]/[(1+z)² + 1] takes over — which keeps the result under the speed of light no matter how large z gets.

Its recession velocity as we define it at cosmological scales can exceed c, and relativity is untroubled by that. Nothing is moving through space faster than light; space between us and the galaxy is expanding, and there is no speed limit on the expansion of space itself. We still see such galaxies because light emitted long ago was travelling through regions that were closer.

About 38% of your Earth weight on Mars and 16.6% on the Moon. Someone at 180 pounds here weighs 68 pounds on Mars and 30 on the Moon. Your mass has not changed by a gram — what changed is the force each body’s gravity exerts on it.

Mass is how much matter there is, in kilograms, and it is the same everywhere in the universe. Weight is the force gravity exerts on that mass, in newtons or pounds-force, and it changes with where you stand. Everyday language uses "weight" for both, which is harmless on Earth and immediately confusing anywhere else.

Each half-life halves what remains, so it is exponential rather than linear: 50% after one, 25% after two, 12.5% after three. It never quite reaches zero. The half-life is a property of a large population — individual nuclei decay at random, with no memory of how long they have already survived.

About a thousand times more. Each whole point on the magnitude scale multiplies the ground-motion amplitude by ten but the energy released by roughly 31.6, and two points therefore multiply the energy by around 1,000. That is why the gap between a 6 and an 8 is the difference between local damage and a regional catastrophe.

It is the extra energy retained by each square metre of the Earth per second, relative to a baseline, before the climate has adjusted. Two watts per square metre sounds trivial until you multiply it by the planet’s 510 trillion square metres and let it run for decades — the accumulated energy is what drives the temperature change.

Because the strongest absorption bands become saturated: once most of the outgoing radiation at those wavelengths is already being captured, added molecules can only work at the weaker edges of each band. Each doubling of concentration therefore contributes about the same 3.7 W/m², so 280 to 560 ppm has roughly the same effect as 560 to 1,120.