OK, here’s a quick back-of-the-envelope calculation for you. Stokes’ Law states that the drag force on a sphere of radius a traveling through a medium of viscosity η at velocity v is:
Now, for the assumptions:
- We’re going to initially ignore the buoyant force of the air on the sphere, which we will find to be an unsatisfactory approximation.
- we’re going to ignore the fact that the viscosity of air changes with height, and approximate it using the value at ~15 degrees C
- We’ll assume the restraints are solid enough to keep a human alive through the course of an 80mph collision with a stationary object (the earth). thus, we let
and solve for a:
We’ll assume the sphere is light, and weighs about the same as the human, so the mass is twice the mass of an average 150lb human. Thus the values we plug in are:
η=1.810^-5 Ns/m^2 (viscosity at 15C)
This gives us a radius of a=1.09*10^5 m = 109 km. This seems unreasonably large. The first thing we note is that at this size, the buoyant force is almost certainly not negligible. In fact, the buoyant force Fb would be
using the value of the density of air (d) at sea level (1.29 kg/m^3), this results in a buoyant force of 6.9*10^16 N, which is huge compared to the drag force (1331 N).
So, to revise our model, we include the buoyant force in the equation:
plugging in the values given above and solving for a in Maple yields the value a=2.93 m, a much more reasonable number, though it seems a bit small. It’s worth noting that since d will decrease with increasing height, this value of a is bound to be a low estimate. However, it is conceivable to make a sphere 10x larger than this (~30 m), though it would be difficult to hoist into the air. Furthermore, this is assuming the harness structure is designed properly, so that it reduces the impulse (G’s) applied to the human by enough to keep them alive.
I’d like it if someone wanted to check my calculations – while i trust Maple, i don’t trust my ability to enter everything into it correctly.