# Terminal Velocity of a Human, Free Fall and Drag Force

*Eugene is a qualified control/instrumentation engineer Bsc (Eng) and has worked as a developer of electronics & software for SCADA systems.*

## What Happens to an Object Falling in a Vacuum?

When an object is released from a certain height we all know that it starts to fall. This of course is due to gravity, or more specifically the gravitational force of attraction between the object and the Earth. The force of gravity causes the object to accelerate and increase in velocity as it falls downwards towards the Earth. In actual fact both the Earth and the object are mutually attracted to each other and the Earth moves upwards at the same time. However since it's so massive in comparison to a small object and the force is so small, its movement is minuscule.

## Definitions of Quantities Used in Kinematics

Before we go any further, first let's define some of the terms used in *kinematics* which is an area of physics concerned with motion of objects.

**Mass.**The amount of matter in an object. The greater the mass of an object, the greater the amount of*inertia*it has and reluctance to move.**Speed.**Speed is the rate of change of position of an object (How fast something moves).**Velocity.**Speed in a given direction. Velocity is a*vector*quantity which means it has a magnitude called speed and also a direction. In physics, we generally talk about velocity rather than speed.**Force.**A push or pull. A force causes an object with mass to accelerate.**Acceleration.**The rate at which velocity changes.**Free Fall.**When an object falls under the influence of gravity alone without other forces acting on it.

See my beginners guide to mechanics for a more detailed understanding of the basics of forces and motion:

## Does Velocity Keep Increasing When Something Falls?

If an object falls in a vacuum outside Earth's atmosphere, its velocity continues to increase because of the acceleration due to gravity. This is called *free fall*. However if the object falls through air (or another fluid such as water), this limits the maximum velocity it can reach.

## Drag Force

When an object moves through a fluid, it experiences a force which opposes motion and tends to slow it down. This force is called *drag.* The fluid could be a liquid such as water or mixture of gasses such as air. * *If you put your hand out the window of a moving car, or try to wade through water, you can feel this force.

Drag increases on an object as it moves faster. In fact it increases exponentially, which means if velocity doubles, drag increases four times and if velocity triples, drag goes up nine times and so on.

When an object is dropped in a vacuum it free falls, acted on by gravity alone. However if it is dropped within Earth's atmosphere, it experiences drag which slows it down.

The force of gravity acts downwards and the drag force acts upwards.

## What is Weight?

Mass is the amount of matter in a body but in physics, mass and weight have very specific meanings. While the mass of an object is the same, irrespective of where it is located in the Universe, weight varies. Weight is the gravitational force between objects and equals mass multiplied by the acceleration due to gravity g.

So the force of gravity or weight is

F_{g }= mg

Where *F _{g}* is the force measured in Newtons (N)

*m* is the mass of an object in kilograms (kg)

and *g* is the acceleration due to gravity in metres per second squared (m/s^{2})

g is approximately 9.81 metres per second per second written as 9.81 m/s/s or m/s^{2}

## At Equilibrium, Drag Force Equals Weight of the Object

The net force acting on a free falling body is the difference between weight acting down and the drag force acting upwards. As long as this is positive, the body keeps accelerating downwards.

Since the drag force increases with velocity, eventually at some stage it equals the weight of the falling body (which isn't changing and staying constant at F_{g} = mg).

Once this equilibrium point is reached since the two forces are equal, there is no net force on the object. No net force means no more force to keep accelerating the body so its velocity reaches a maximum known as the *terminal velocity.*

## Velocity of a Falling Object With No Drag

As an aside, let's look at the equation for velocity of a falling object when there's no drag. If an object falls through a vacuum without being slowed down by a drag force, its velocity v in m/s is given by the equation:

v = √(2gh)

where g is the acceleration due to gravity.

and h is the distance fallen in metres (m)

In terms of time t in seconds since the object was dropped, another equation for velocity is:

v = gt

To put this into perspective after 10 seconds of free fall in a vacuum, an object would be traveling at:

v = gt= 9.81 x 10 = 98.1 m/s or 355 km/hr (219 miles per hour)

However as we shall see, drag puts an upper limit on velocity.

## The Drag Equation

The drag equation describes the force experienced by an object moving through a fluid:

If F_{d} is the drag force, then:

F_{d}= ½ ρ u^{2}C_{d}A

Where *F _{d}* is the force in newtons (N)

*p* is the density of the fluid in kilograms per cubic metre (kg/m^{3})

*u* is the velocity of the object relative to the fluid in metres per second (m/s)

*Cd* is the drag coefficient that depends on the shape of the object and the nature of its surface

and *A* is the area of the orthogonal projection of the object in m^{2}. This can be visualized as the area of the shadow of the object cast on a surface if a light with a parallel beam was shone on it and landed perpendicular to the surface.

Because of the u^{2} term in the equation, drag increases with the square of the velocity.

## Derivation of Terminal Velocity

At equilibrium, the drag force F_{d} acting upwards equals the weight F_{g }acting downwards

We know *F _{d} = ½ ρ u^{2} C_{d} A*

and *F _{g} = mg*

At equilibrium, the velocity becomes the terminal velocity. Let's call it V_{t}

Equate *F _{g}* to

*F*

_{d}and replace

*u*by

*V*

_{t}giving:

mg = ½ ρ u^{2}C_{d}A = ½ ρ V_{t}^{2}C_{d}A

So:

2mg = ρ V_{t}^{2}C_{d}A

Divide both sides by ρ C_{d} A giving:

2mg / ρ C_{d}A = V_{t}^{2}

Taking the square root of both sides gives us:

V_{t}= √((2mg) / (ρAC_{d})

## Terminal Velocity of a Human

From the equation for terminal velocity, we see it depends on several factors:

- Density of the air.
- Mass of the object
- Area of the object
- Acceleration due to gravity (this doesn't really change, so it can be assumed to be practically constant)
- The shape of the object

For a human, the drag coefficient C_{d} is about 1 in a belly down, horizontal orientation and 0.7 in head down position.

Typically in this position, terminal velocity is about 120 mph or 54 m/s.

## How Long Does it Take to Reach Terminal Velocity and How Far Does a Human Fall?

It takes about 12 seconds to reach 97% of terminal velocity. During that period, a human would fall about 455 metres.

## What Increases Terminal Velocity?

Speed skydivers compete by trying to reach the highest possible terminal velocity. From the equation, we can see that it can be increased by:

- being heavier
- diving in thinner, low density air
- reducing the projected area by diving head first
- reducing the drag coefficient by diving head first.
- wearing clothing that improves streamlining and reduces drag

## References

Hannah, J. and Hillerr, M. J., (1971) *Applied Mechanics* (First metric ed. 1971) Pitman Books Ltd., London, England.

*This article is accurate and true to the best of the author’s knowledge. Content is for informational or entertainment purposes only and does not substitute for personal counsel or professional advice in business, financial, legal, or technical matters.*

**© 2019 Eugene Brennan**