Free Fall

The Free Fall Research Page

The Influence of Weight on Terminal Velocity

Mathias Svensson, a student of Computer Science at Copenhagen University, has provided the following explanation:

On the Questions page on the Free Fall Research Page, there is a question about the influence of weight on the survival chance of a free fall . I can confirm that weight does indeed have an influence on your terminal velocity and thus your survival chance.

There are two forces that affect you in a free fall: gravity and drag. The gravitational force (Fg) of an object in earth's gravity can be described as:

Fg = 9.8*m

Here mass (m) is measured in kilograms and Fg is measured in newtons.

The force of drag (Fd) can be described by the equation:

Fd = K*v^2

Here velocity (v) is measured in meters per second and Fd is measured in newtons.

K is a variable whose calculation is a bit complex, as it depends on the temperature, air pressure, and the person's surface area among other things. A good approximation for K is 0.25 for a grown man trying to generate as much surface area as possible (by using only his own body). However you can make this quite a bit larger by grabbing stuff with a large surface area.

The terminal velocity is the point at which your velocity no longer increases. This is the point where the gravitational force minus the force of drag equals zero. Thus:

Fg - Fd = 0
(9.8*m) - (0.25*v^2) = 0
v = the square root of ((9.8*m)/0.25)

If you weigh 60 kilograms, your terminal velocity would be about 50 meters per second, while it would be about 70 meters per second if you weighed 120 kilograms. Of course the K-value of a 120 kilogram person would also be bigger than the K-value of a 60 kilogram person - but not very much so.

(Note: 60 kilograms is approximately 132 pounds. 120 kilograms is approximately 265 pounds. 50 meters per second is approximately 112 miles per hour. 70 meters per second is approximately 156 miles per hour.)

Here is how the two calculations look:

For a 60 kilogram human:
v = the square root of ((9.8*60)/0.25) = 48.5

For a 120 kilogram human:
v = the square root of ((9.8*120)/0.25) = 68.6

The difference between these two velocities, approximately 20 meters per second, makes quite a bit of difference, since the energy of your impact is proportional with the square of your velocity. Therefore the higher your velocity, the greater your energy of impact.

The formula for kinetic energy is:

Ekin = 1/2 * m * v^2

Here Ekin (energy) is measured in joules, m (mass) is measured in kilograms and v (velocity) is measured in meters per second.

So if we have a 60 kg person this would mean:

At 50 meters per second:
Ekin = 1/2 * 60 * 50^2 = 75,000

At 70 meters per second:
Ekin = 1/2 * 60 * 70^2 = 147,000

Thus the impact energy is almost doubled from 75 kilojoules (kJ) to 147 kJ.

The formula means that the kinetic energy is larger for a heavier person, but he would also have a larger body to absorb the impact, so that might make a difference, though someone with expertise in the physiology of the human body would be in a better position to address that point than I.

This also means that you should only grab stuff that makes you float better. For example, grabbing a large beach ball would increase your drag quite a bit while not increasing your weight significantly. However if the beach ball was filled with water it would not be worth it.

You have probably heard about how a feather and a hammer would fall at the same rate in a vacuum, no matter their shape or weight. This is also true, but when an atmosphere is introduced, things become much more complex. In fact many engineers spend large parts of their lives calculating how things (such as airplanes) interact with the atmosphere. However the simplified version is this: In an atmosphere you will fall faster, the heavier you are and fall slower the more surface area you have.