[Physics] How to find the minimum height to open a parachute for a safe landing

Question

What concepts/formulas can be used to find the optimum/minimum release height of a parachute?

Answer 1

I'm going to talk concepts, because I think the formulas are not terribly applicable - you'll see why.

The answer depends a LOT on the purpose of the parachute and a lot less on fundamental physics.

For cargo chutes, where the rate of deceleration isn't a major issue, the answer is pretty much how long the lines are times about two. The chute pulls out, typically from a cloth bag pulled off by a much smaller drogue chute, and once it hits the air it opens very rapidly and decelerates to its terminal almost instantly. This still takes a little time, but much less time than it takes for the cargo to leave the plane. If you watch films of this on YouTube, you'll see the cargo is at terminal and descending vertically within about 1/2 the length of the plane and the cargo is perhaps 50 feet below it. This can be further lowered basically to zero if one opens the chute with the cargo still in the aircraft, the LAPES concept, at which point the parachute is fully (or almost) deployed before the cargo moves.

We punny humans are not built for such punishment,so our chutes are equipped with systems that deliberately slow down the opening so the peak deceleration is much slower. A typical skydiver falls at about 120 mph and once the chute is open we're going perhaps 20 mph. That is the equivalent of a very very bad car accident. So we fold them up so they don't catch the air, cover them with a "slider" to further slow things down, then jam the whole thing into a surprisingly small bag held shut by rubber bands. Now it takes several seconds to open. In that case the minimum altitude is distance you travel in (say) 5 seconds (to be safe) at 120 mph, which is a little under 900 feet. In practice we always pull above that in case the main doesn't open, so we have a couple of seconds to pull the reserve. Unlike the main, the reserve is deliberately packed so it does not slow down the opening, and opens fully in a few hundred feet. I've pulled a reserve only once, and I can tell you my whole body hurt for a day. In any event, in this case the rate of opening is defined entirely by physiology, not physics.

BASE rigs are also designed to open almost instantly. They are packed without a sleeve or bag, they simply fold them up accordion style and put them in a velcro-closed backpack. According to one story that made sure I would never BASE, an acquaintance knows from direct experience that they can open fully from the balcony of a 13-story apartment building (which they were not at the top of, IIRC).

So... that's why it's less physics and more concept. The basic answer is "zero, if you want that, but likely don't".

All of that said, there is an interesting formula in all this. A six-foot snowdrift will safely stop you in freefall. That's easy to calculate, and there are several real-world examples of this happening.