What's the longest possible time for Earth to know of an impending meteor strike? Like, could the people on Earth realize that there's a fatalistic asteroid collision decades before it happens?
Orbital mechanics (the observation and prediction of orbital movement of bodies) is a pretty well established science. The math was developed initially by Johannes Kepler way back in 1615. Using the observation data acquired by Tycho Brahe (aka The Man With the Golden Nose), Kepler determined the actual paths that planets follow around the Sun (ellipses, not circles; around the Sun, not around the Earth), and gave us a toolkit (the Three Laws of Planetary Motion) that we can still use to predict the orbital path of almost any body we observe.Newton refined and generalized them, and Einsteinian Relativity explained the weird cases that popped up, but we can pretty accurately predict orbital paths of objects with surprising accuracy. Indeed, in 1801, Karl Friedrich Gauss figured out how to predict an orbit using only three observations (link has lots of math - be warned).
(A personal note: I had to use this method to plot the orbit of Mars in one of my early Astronomy classes. I then used that calculation to predict when the next time Mars would be observable from my location. All by hand - just like Gauss and Kepler would have done.To my eternal surprise, I was correct. SCIENCE!)
The real problem in predicting impacts on Earth is a much more mundane problem - can we see the approaching object in time?
If you can observe the object (the more observations the better; the more places you can see it from, the better), you can predict it’s orbit for a couple of hundred hundred years into the future. There are several objects that are being tracked by NASA that might impact us in the next couple of centuries.
The data as to the time of impact is given as a possible range, as well as the chance of the impact happening. We can calculate the orbital path of a rock very precisely only if we know everything out there that could affect it. If our rock moves close to another asteroid that we don’t know about, that gravitational nudge might be enough to swing its path right at us, and we’d never be able to predict it.
Earth has had a number of near misses in recent years that we didn’t see until the rock was upon us, or even after the asteroid flew by.
On March 18, 2004, a 30-meter asteroid, 2004 FH, passed by the Earth at a distance of 42,600 km - 10% the distance to the Moon, We only saw it on the day it passed by.
Sometimes, we get hit with something that we never saw coming.
Stuff like this happens all the time. Makes you feel so safe. :-)
There could be one or more rocks big enough to replicate the KT impact (the one that killed the dinosaurs) heading towards us right at this moment, and we might not have a clue. The bigger and better our space program is, the better our chance of seeing that Earth-killer while it’s still way far away, and the more time we have to call Sean Connery to save us.
Summary: Once a rock has been observed, you’ll know if it’s going to hit your planet very quickly (within days of detecting it if you have a lot of telescopes). Your warning before impact may be days, years, or a century or two. Pick whatever timeframe you need for your story, and you’ll be fine.