What is the Terminal Velocity of a Human?

When a skydiver jumps from a plane, his velocity increases due to acceleration due to gravity. However, since he is falling through air he experiences drag force due to air resistance. As a result, his velocity is limited and reaches a constant value, known as terminal velocity. The following equation gives the terminal velocity of an object falling through the air.

\[v_{T} = \sqrt{\frac{2mg}{C\rho_{air} A}} \]

From this equation, one can see that the terminal velocity depends on:

Density of air, ρ_air
Mass of the object, m
Area A of the object that is projected vertically. In other words, the object’s orientation.
Drag coefficient C, which depends on the shape of the object

Thus, the speed achieved by a human during a free fall depends upon two factors – mass and orientation. Based on this, let us calculate the terminal velocity for two different positions.

1. Belly-down Position

For a human, the drag coefficient is 1 for a belly-down position, and the projected area is 0.7 m². The density of air is 1.29 kg/m³. Let us assume that an average skydiver weighs 75 kg or 165 lbs.

Given

m = 75 kg

ρ_air = 1.29 kg/m³

C = 1

A = 0.7 m²

g = 9.81 m/s²

Therefore, the terminal velocity is given by

\[ v_T = \sqrt{\frac{2 \cdot 75 \hspace{1mm} kg \cdot 9.81\hspace{1mm} m/s^2}{1 \cdot 1.29\hspace{1mm} kg/m^3 \cdot 0.7\hspace{1mm} m^2}}\\ \Rightarrow v_T = 40 \hspace{1mm} m/s \hspace{1mm} or \hspace{1mm} 89 \hspace{1mm} mph \]

2. Head Down Position

Experienced skydivers can increase the above value substantially by diving head first with their arms against the sides of their bodies, legs held tightly together, and toes pointed. This position offers a minimum projected area perpendicular to the direction of motion and reduces the aerodynamic drag.

For the head-down dive, the drag coefficient is 0.7 and the projection area is 0.18 m2. Therefore, the terminal velocity comes out to be

\[ v_T = \sqrt{\frac{2 \cdot 75 \hspace{1mm} kg \cdot 9.81\hspace{1mm} m/s^2}{0.7 \cdot 1.29\hspace{1mm} kg/m^3 \cdot 0.18\hspace{1mm} m^2}}\\ \Rightarrow v_T = 95 \hspace{1mm} m/s \hspace{1mm} or \hspace{1mm} 213 \hspace{1mm} mph \]

This is typically the fastest a human can skydive. However, the drag can be further minimized by streamlining the body such that a speed up to 310 mph can be reached.

How can the Terminal Velocity be Increased

From the above equation, we can see that the terminal velocity can be increased by:

being heavier
diving in thin, low-density air
reducing the projected area by diving head first
reducing the drag coefficient by diving head first
wearing clothing to improve streamlining and reduce drag

FAQs

Q.1. How long does it take for a human to reach terminal velocity?

Ans. It takes about 12 seconds to achieve terminal velocity.

References
- 1. Speed Skydiving – Fai.org
  2. How big a fall can a person survive? –Theguardian.com
  3. Speed of a Skydiver (Terminal Velocity) – Hypertextbook.com
  4. Terminal Velocity in Skydiving – Skydivecalifornia.com