The following rules about gravity appear in D&D 5th edition:
- You take 1d6 damage per 10 feet fallen, to a maximum of 20d6 (Player's Handbook).
- If you're flying, and you stop, you fall (Player's Handbook).
- You fall 500 feet per turn (Xanathar's Guide to Everything , as per this question).
- The average man can make a long jump of 10 feet, or 3 feet up, with a run-up; a highly exceptional human can easily long jump 20 feet, or 8 feet up. (Player's Handbook). By comparison, the world record long jump is 29 feet, and high jump is 8 feet.
- The Ethereal Plane ignores gravity (Dungeon Master's Guide).
We can reasonably infer that gravity exists, and that its effect is approximate to that of our Earth, in that a five foot fall will not injure you but a ten or twenty foot drop will, and a 200 foot drop will almost certainly kill you; and the most exceptional human can jump eight feet straight up. Were gravity much higher or lower, that eight foot jump would be either impossible or so easy that everyone can do it, and people would either float safely down from rooves or crawl cautiously over deadly stairs.
However, we can also discern that D&D's rules are an approximation of gravity for fast gameplay. Realistically, terminal velocity for a human is 53m/s or 1,043 feet per 6-second round, and you take 12 seconds or two rounds to reach that velocity, at which point you have fallen 1,500 feet. Terminal velocity also varies by creature, so a cat has different terminal velocity than a human.
If you ignore that D&D rules are an approximation, the laws of physics don't behave as we understand them. The gods simply will all creatures to fall at the same speed, hitting terminal velocity of about 25m/s after 200 feet. In reality, it's an artefact of the rules being a simplification for gameplay purposes.
According to lore, the planets of Abeir-Toril (Forgotten Realms), Oerth (Greyhawk) and Eberron are all spherical, and approximately the same size as our planet. Where gravity is substantially higher or lower than our Earth, such as on another plane, the rules always describe it as such. We can assume that gravity works approximately the same way as our Earth, and that the rules serve adequately to adjudicate any gravity-based situation that occurs in normal play.
The Rules are Unclear
Your question is reasonable, because Feather Fall's casting time might lead you to conclude that it can only be used at the start of a fall:
Casting Time: 1 reaction, which you take when you or a creature within 60 feet of you falls.
The issue at hand is that "fall" has two relevant definitions (as a verb):
1. move downward, typically rapidly and freely without control, from a higher to a lower level.
2. (of a person) lose one's balance and collapse. "she fell down at school today"
(Source: Oxford Dictionaries)
If we use use the second definition, then the "fall" only happens "when" the creature begins descending. But if we use the first definition, then someone "falls" whenever they are still descending under the influence of gravity (e.g. "during this round, the creature falls").
The spell has considerably reduced utility if it can only be cast at the start of a fall. For example, since it only lasts a minute, it could not save a creature from a fall of more than 600 feet. And since it has a range of 60 feet, it also could not be used to save a creature that falls from higher than that above a spellcaster. It seems likely that a DM would rule that it can be used during a creature's descent, to stop it from being an extremely niche use spell, but it is up to the individual DM.
Using once you are 60 feet above the ground
Let's assume, for the moment, that your DM permits you to cast Feather Fall during a fall, not just at its start. Your example (casting it when you are within 60 feet of the ground) still may not work as you intended.
If you want to cast the spell when you are within 60 feet of the ground after falling normally for an extended period, you are essentially asking to cast it when you will spend exactly one round falling gently, and then still be able to use your full movement on the ground (an understandable goal). But while you fall, you do not have an accurate readout of their current height at your disposal. And if you have been falling for 500 feet, you are moving very fast (realistically, you will have less than half a second to cast the spell while you are within 60 feet of the ground: and if you are moving 500 feet in 6 seconds, the timing is similar). You might cast the spell too soon, and be more than 60 feet above the ground, or too late (and splat).
A DM could call for any type of check (Perception, for example) with whatever DC they wished if you wanted to attempt a HALO style jump with the Feather Fall spell. Although "rule of cool" could certainly grant you some leeway, you might want to check with your DM before you attempted this tactic.
Best Answer
If this is important at your table, make your own rulings
The rules as written to not simulate falling with any degree of accuracy because a) long falls don’t happen that much, b) when they do, the rules are mostly ok most of the time.
However, in the real world, when you fall, you fall really, really fast - 1,000 feet in the first 10 seconds as you speed up to belly down terminal velocity and then 5 seconds for every 1,000 feet. So the TCoE rule gives a falling speed less than half what actually happens.
By changing your attitude, you can change your velocity both vertical and horizontal. You can increase the rate of descent by between 20 and 45% by doing this. A 1 second (round) head start will take 3-4 seconds (rounds) to make up.
So if you know what you’re doing, react quickly enough and have enough air (time) to work with you can catch up to your companion.
Now, go and work out the rulings for what you want to achieve.