Main concept

A parachute jump from an aircraft can be modeled using the basic principles of fluid dynamics in gravitational field. A skydiver with mass $m$, altitude $h$ and vertical velocity $v$ is subject to gravitational force $F_g$ and the drag force $F_d$.

$m \frac{{\ⅆ}^{}}{\ⅆ t} v\=F_g\+F_d$

The gravitational force can be evaluated as $F_g\=-m\cdot g$, where $g\=9.81 m\/s^2$ and remains relatively unchanged for typical skydiving altitudes. The drag on an object with a drag coefficient $C_d$ which presents orthographic projection $A$ on a plane perpendicular to the motion in a fluid of density $\mathrm{\ρ}$ can be modeled as

$F_d\=\frac{1}{2}\mathrm{\ρ}\cdot C_d\cdot A\cdot v^2$

This model is accurate for large values of Reynolds number $\mathrm{Re}\=\frac{\mathrm{\ρ}\cdot d\cdot v}{\mathrm{\μ}}\gg {10}^3$ (where $d$ is a characteristic length, and $\mathrm{\μ}$ is the fluid viscosity) which is typically true for both free fall and with parachute.

The air density in Earth's atmosphere $\mathrm{\ρ}$ drops exponentially with the altitude and can be calculated using reference values of $h_{\mathrm{ref}}\=7500 m$ and ${\mathrm{\ρ}}_0\=1.2 \mathrm{kg}\/m^3$ according to

$\mathrm{\ρ}\={\mathrm{\ρ}}_0\cdot e^{-\frac{h}{h_{\mathrm{ref}}}}$

In our model, the jump consists of three phases (depending on the initial conditions, one or two of them might be missing):

1. Skydiver is in free fall, both his drag coefficient $C_d$  and his cross-sectional area  $A$  is that of a cylinder or a flat strip, depending on the body position.

2. Skydiver pulls his ripcord at the opening altitude and his parachute starts to open. The drag coefficient $C_d\=1.33$ (inside of a hemisphere) and the cross-sectional area  $A$ increases linearly over the period of 10 seconds to the full size of $45 m^2$ (corresponding to a typical military-grade round parachute).

3. The parachute is fully inflated and descents to the ground.

Adjust the sliders to set the initial and opening altitudes, as well as the skydiver's mass and free fall position. Observe how the fall rate is affected in all three phases.

By adjusting the position and ground speed of the plane and speed of the wind, you can try to land inside the red target (50 $m$ across, a typical landing area).  Speed of the animation can be controlled on the fly.

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