What Thickness of Cotton Can Safely Stop a 60 kg Person Falling from 10 km?

I saw an interesting question on Zhihu and compiled the problem and several answers.

1 Question

Assume a 60 kg person free‑falls from 10 km altitude, ignoring low‑temperature and other landing factors. At the instant of landing the person spreads arms and legs to maximize contact area with cotton/feather‑down. Is it possible to avoid injury by providing a sufficiently thick cushion? If so, how thick? (The person must fall unobstructed from 10 km; answers like “9999 m” are not acceptable.)

2 Analysis

During the fall the person is subject to gravity, buoyancy and air resistance. Before reaching the ground the forces can reach equilibrium, at which point the velocity stops increasing.

3 Model

3.1 Variables, parameters and symbols

F_g: gravity

F_b: buoyancy

F_d: air drag

A: human cross‑sectional area

v_eq: equilibrium velocity

C_d: drag coefficient

ρ_air: air density

ρ_body: human density

g: gravitational acceleration

When forces are balanced:

F_g = F_b + F_d

The body posture influences the equilibrium speed. Below we calculate the minimum and maximum speeds for a falling person.

3.2 Minimum speed

When the body lies flat, drag is maximal and the equilibrium speed is lowest. For a 60 kg person with a relatively large cross‑sectional area, the drag coefficient of various materials is shown below.

These coefficients are measured under ideal conditions; in reality they depend on the Reynolds number, which is related to object size and speed.

Approximating the human body as a sphere (head) plus several cylinders (limbs), the ideal drag coefficient lies between 0.4 and 0.9. For a slender person in a prone position we take C_d ≈ 0.7.

Substituting into the equilibrium equation yields the minimum terminal speed (value omitted in the source).

3.3 Maximum speed

If the body falls straight down, the projected area is minimal, giving a lower drag coefficient, roughly C_d ≈ 0.5.

This leads to a maximum equilibrium speed (value omitted in the source). The ratio between the minimum and maximum speeds can be as high as 4.

Next we estimate the cushion thickness needed to keep the impact injury‑free.

Human tolerance to sustained acceleration is low, but short‑duration impacts can endure higher accelerations. In rescue scenarios, a maximum impact of about 20 g (≈12 kN for a 60 kg person) is considered safe.

Using this limit, the required effective cushioning height for the minimum‑speed impact is about 6 m of loosely packed cotton or feather‑down. To account for uneven distribution and to avoid the material scattering, a more realistic thickness would be at least 10 m. For the maximum‑speed impact (head‑first), a safety cushion of roughly 100 m would be necessary.

Note: The calculations are intended for mathematical modeling practice only and are not recommended for real‑world implementation.

Reference:

https://www.zhihu.com/question/539854942/answer/2548935062