What is the minimum height re how high you have to be be for a parachute to be able to deploy and carry a 200 lb man safely to earth?
Depends very much on the system. BASE jumpers (who go off of fixed objects) use equipment that is modified to open very quickly at low airspeeds; a few hundred feet is common and I’ve seen one tower jump that was about 100 feet.
For typical jumps out of an aircraft, the minimum recommended opening altitude for licensed skydivers is 2000-2500 feet (depending on experience level) which allows time to deal with an emergency situation. If you don’t have a landable canopy over your head at about 1600 feet that is the “hard deck decision altitude” where you initiate emergency procedures immediately (the joke is “You’ve got the rest of your life to fix it”).
Automatic activation devices (AAD) are a little gadget that detect whether you are descending very fast at a very low altitude and attempt to open the reserve canopy for you (typically by cutting the closing loop of the reserve pack). The most common model when I was active was the CYPRES which would trigger at 750 feet AGL if descent rate exceeded 78mph.
Reserve canopies are built to open as quickly as possible without injuring you…somewhere down in the high-mid “hundreds of feet” AGL as the CYPRES indicates.
If you cutaway from an open (but not landable) main canopy there’s a mechanical backup on most rigs called a reserve static line (RSL or Stevens Lanyard) that basically links the reserve to the main canopy - when you pull the main cutaway handle and start falling the RSL deploys the reserve canopy very quickly (it was designed to avoid fatalities where someone pulls the cutaway and then blanks, figuring that since they pulled a handle everything is OK and they forget to pull the reserve ripcord). The RSL deployment is very, very fast - I have seen video of a BASE rig with a reserve canopy that had the backup fully open within about 30 feet of cutting away the main.
And of course airborne troops do static line jumps from a few hundred feet all the time (their goal is to get safely to the ground in as short a time as possible, hanging around under canopy is a great way to get shot at).
Finally, ejection seats are meant to function in the “zero/zero” situation - zero altitude, zero airspeed. There’s a famous video from the Paris airshow years back where a fighter jet has an engine flameout, the pilot stayed with the plane to steer it away from the crowd and as it dives into the ground he ejects at a shockingly low altitude and he lived to tell about it.
When I was in sport parachuting (admittedly a while back) a rule of thumb I saw mentioned was that if you are deploying the canopy at 1000’ above ground, your statistical chances are pretty good - but they erode meaningfully below that. At 500’ you might be looking at an 80% chance, and at 300’ possibly 30%. (These are obviously very approximate.)
The minimum possible altitude might be as low as 150’ (possibly less for a reserve canopy which has a simpler deployment system).
The ejection system on the Ka-50 helicopter will supposedly work and land a pilot safely even when it is on the ground. The rocket probably doesn’t shoot the pilot that high up into the air, 150’ sounds about right, after that the parachute supposedly deploys almost instantly. I sure would hate to be the first guy to test that feature out though.