# Drag Force and the Terminal Velocity of a Human

## 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.

*Note: In actual fact the Earth and the object are mutually attracted to each other and move towards each other. However the Earth is so massive that its movement is minuscule*

## Definitions of Quantities Used in Kinematics

**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 a mass to accelerate.**Acceleration.**The rate at which velocity changes.

## Does Velocity Keep Increasing?

If an object falls in a vacuum outside Earth's atmosphere, its velocity continues to increase because of the acceleration due to gravity. 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. *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.

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 F

_{g }= mgwhere m is the mass of an object

and g is the acceleration due to gravity.

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

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.*

## Terminal Velocity of an Object

Terminal velocity is the maximum velocity attainable by an object as it falls through a fluid

## 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 is given by the equation:

v = √(2gh)

where g is the acceleration due to gravity (9.81 m/s^{2})

and h is the distance fallen

In terms of time t 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 *p* is the density of the fluid

*u* is the velocity of the object relative to the fluid

*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. This can be visualized as the area of the shadow of the object cast on a surface if a light 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} = the weight F_{g}

We know F_{d} = ½ ρ u^{2} C_{d} A

and F_{g} = mg

At equilibrium, the velocity u = the terminal velocity V_{t}

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

Rearranging 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.

## 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

**© 2019 Eugene Brennan**

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